GROTHENDIECK-NEEMAN DUALITY AND THE WIRTHMÜLLER ISOMORPHISM

PAUL BALMER, IVO DELL’AMBROGIO, AND BEREN SANDERS

Abstract. We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk-Hu-May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: There exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin-Matlis duality à la Dwyer-Greenless-Iyengar in the theory of ring spectra, and of Brown-Comenetz duality à la Neeman in stable homotopy theory.

Contents

1. Introduction and statement of results 1 2. Brown representability and the three basic functors 7 3. Grothendieck-Neeman duality and ur-Wirthmüller 11 4. The Wirthmüller isomorphism 17 5. Grothendieck duality on subcategories 21 6. Categories over a base and relative compactness 29 7. Examples beyond Grothendieck duality 31 References 34

1. Introduction and statement of results

A tale of adjoint functors. Consider a tensor-exact functor f^{∗} : 𝒟 → 𝒞 between tensor-triangulated categories. As the notation f^{∗} suggests, one typically obtains such functors by pulling-back representations, sheaves, spectra, etc., along some suitable “underlying” map f : X → Y of groups, spaces, schemes, etc. (The actual underlying map f is not relevant for our discussion. Moreover, our choice of notation f^{∗} is geometric in spirit, i.e. dual to the ring-theoretic one; see Example 3.23.)

Date: December 12, 2015. 2010 Mathematics Subject Classification. 18E30; 14F05, 55U35. Key words and phrases. Grothendieck duality, ur-Wirthmüller, dualizing object, adjoints, com-

pactly generated triangulated category, Serre functor.

First-named author partially supported by NSF grant DMS-1303073. Second-named author partially supported by the Labex CEMPI (ANR-11-LABX-0007-01). Third-named author partially supported by the Danish National Research Foundation through

the Centre for Symmetry and Deformation (DNRF92).

We are interested in the existence of adjoints to f^{∗} and of further adjoints to these adjoints, and so on:

(1.1)  [figure: 𝒞; ···; f^{∗}; f_{∗}; ···; 𝒟]

Such questions arise in examples because certain geometric properties of the underlying f : X → Y can sometimes be translated into the existence, or into properties, of such adjoints. This is illustrated for instance in Neeman’s approach to Grothendieck duality [Nee96]. Our main motivation is to provide a systematic treatment of these adjoints in the context of compactly generated categories, while simultaneously clarifying the relationship between so-called Wirthmüller isomorphisms and Grothendieck duality. In that respect, our work is a continuation of Fausk-Hu-May [FHM03]. It turns out that the more adjoints exist, the more strongly related they must be to each other. Also remarkable is the existence of a tipping point after which there must exist infinitely many adjoints on both sides. This will happen for instance as soon as we have the six consecutive adjoints pictured in (1.1) above.

Let us be more precise. Here is our basic set-up:

1.2. Hypothesis. Throughout the paper, we assume that both tensor-triangulated categories 𝒞 and 𝒟 are rigidly-compactly generated. See Section 2 for details. In short, this means that 𝒞 admits arbitrary coproducts, its compact objects coincide with the rigid objects (a. k. a. the strongly dualizable objects) and 𝒞 is generated by a set of those rigid-compacts; and similarly for 𝒟. Such categories are the standard “big” tensor-triangulated categories in common use in algebra, geometry and homotopy theory. They are the unital algebraic stable homotopy categories of [HPS97] (with “algebraic” understood broadly since it includes, for example, the topological stable homotopy category SH). See Examples 2.9–2.13.

Moreover, we assume that f^{∗} : 𝒟 → 𝒞 is a tensor-exact functor (i.e. strong symmetric monoidal and triangulated) which preserves arbitrary coproducts. These hypotheses are quite natural and cover standard situations; see Examples 3.22–3.26 and 4.5–4.7. (Such f^{∗} are called geometric functors in [HPS97, Def. 3.4.1].)

By Neeman’s Brown Representability Theorem, these basic hypotheses already imply the existence of two layers of adjoints to the right of the given f^{∗} : 𝒟 → 𝒞.

1.3. Theorem (Cor. 2.14). Under Hypothesis 1.2, the functor f^{∗} : 𝒟 → 𝒞 admits a right adjoint f_{∗} : 𝒞 → 𝒟, which itself admits a right adjoint f^{(1)} : 𝒟 → 𝒞. Moreover, we have a projection formula d ⊗ f_{∗}(c) ≅ f_{∗}(f^{∗}(d) ⊗ c) and a couple of other relations detailed in Proposition 2.15.

In other words, we get f^{∗} ⊣ f_{∗} ⊣ f^{(1)} essentially “for free”. This includes the unconditional existence of a special object that we want to single out:

1.4. Definition. Writing 𝟙 for the ⊗-unit, the object ω_{f} := f^{(1)}(𝟙) in 𝒞 will be called the relative dualizing object (for f^{∗} : 𝒟 → 𝒞) in reference to the dualizing complexes of algebraic geometry; see [Lip09] and [Nee96, Nee10]. This object ω_{f} of 𝒞 is uniquely characterized by the existence of a natural isomorphism

(1.5)  Hom_{𝒟} (f_{∗}(−), 𝟙) ≅ Hom_{𝒞} (−, ω_{f}).

Equivalently, ω_{f} is characterized by the existence of a natural isomorphism

(1.6)  hom_{𝒟} (f_{∗}(−), 𝟙) ≅ f_{∗} hom_{𝒞} (−, ω_{f}),

where hom_{𝒞} and hom_{𝒟} are the internal hom functors on 𝒞 and 𝒟 respectively. In other words, ω_{f} allows us to describe the usual (untwisted) dual Δ := hom(−, 𝟙) of the direct image f_{∗} as the direct image of the ω_{f}-twisted dual Δ_{ω_{f}} := hom(−, ω_{f}).

Armed with this object ω_{f} ∈ 𝒞, we return to our three functors f^{∗} ⊣ f_{∗} ⊣ f^{(1)}. We prove that the existence of one more adjoint on either side forces adjoints on both sides f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)} and strong relations between these five functors. This is one of the main clarifications of the paper.

1.7. Theorem (Grothendieck-Neeman Duality, Theorem 3.3). Let f^{∗} : 𝒟 → 𝒞 be as in our basic Hypothesis 1.2 and consider the automatic adjoints f^{∗} ⊣ f_{∗} ⊣ f^{(1)} (Thm. 1.3). Then the following conditions are equivalent:

(GN1) Grothendieck duality: There is a natural isomorphism

ω_{f} ⊗ f^{∗}(−) ≅ f^{(1)}(−).

(GN2) Neeman’s criterion: The functor f_{∗} preserves compact objects, or equivalently its right adjoint f^{(1)} preserves coproducts, or equivalently by Brown Representability f^{(1)} admits a right adjoint f_{(−1)}.

(GN3) The original functor f^{∗} : 𝒟 → 𝒞 preserves products, or equivalently by

Brown Representability f^{∗} admits a left adjoint f_{(1)}.

Moreover, when these conditions hold, the five functors f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)}

[figure: 𝒞; f_{(1)}; f^{∗}; f_{∗}; f^{(1)}; f_{(−1)}; 𝒟]

are related by an armada of canonical isomorphisms, detailed in Theorem 3.3 and Example 3.17. Most notably, we have what we call the ur-Wirthmüller isomorphism

(1.8)  f_{(1)}(−) ≅ f_{∗}(ω_{f} ⊗ −)

and we have a canonical isomorphism 𝟙_{𝒞} ≅ hom_{𝒞} (ω_{f}, ω_{f}).

The equivalence between (GN 1) and (GN 2) was established by Neeman [Nee96]. We name the theorem after him since he has been the main architect of compactly generated categories and since several of our techniques have been pioneered by him, in the algebro-geometric context, like in [Nee10]. Our main input is to show that Grothendieck-Neeman duality can be detected on the original functor f^{∗}, namely by the property (GN 3) that f^{∗} preserves products. In other words, the existence of Neeman’s right adjoint f_{(−1)} on the far-right is equivalent to the existence of a left adjoint f_{(1)} four steps to the left. Our Lemma 2.6 is the tool which allows us to move from left to right via the duality on the subcategory of compact objects. This lemma is the key to the proof of the new implication (GN 3)⇒(GN 2) above and appears again in the proof of Theorem 1.9 below.

The ur-Wirthmüller formula (1.8) plays a fundamental role in our approach and connects with similar formulas in [FHM03], as discussed in Remark 1.12 below. In algebraic geometry, an isomorphism as in (1.8) is mentioned in [Nee10, Rem. 4.3].

Our Grothendieck-Neeman Duality Theorem 1.7 leaves one question open, made very tempting by the isomorphism hom(ω_{f}, ω_{f}) ≅ 𝟙: When is the relative dualizing object ω_{f} ⊗-invertible? Amusingly, this is related to another layer of adjoints, on either side of f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)}. We reach here the tipping point from which infinitely many adjoints must exist on both sides.

1.9. Theorem (Wirthmüller Isomorphism; see Section 4). Suppose that we have the five adjoints f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)} of Grothendieck-Neeman duality (Thm. 1.7). Then the following conditions are equivalent:

(W1) The left-most functor f_{(1)} admits itself a left adjoint, or equivalently by

Brown Representability it preserves arbitrary products. (W2) The right-most functor f_{(−1)} admits itself a right adjoint, or equivalently

by Brown Representability it preserves arbitrary coproducts, or equivalently its left adjoint f^{(1)} preserves compact objects.

(W3) The relative dualizing object ω_{f} (Def. 1.4) is a compact object of 𝒞.

(W4) The relative dualizing object ω_{f} is ⊗-invertible in 𝒞.

(W5) There exists a (strong) Wirthmüller isomorphism between f_{∗} and f_{(1)}; that is, there exists a ⊗-invertible object ω ∈ 𝒞 such that f_{(1)} ≅ f_{∗}(ω ⊗ −), or equivalently such that f_{∗} ≅ f_{(1)}(ω^{−1} ⊗ −).

(W6) There exists an infinite tower of adjoints on both sides:

[figure: 𝒞; ···; f^{(−n)}; f_{(n)}; ···; f^{(−1)}; f_{(1)}; f^{∗}; f_{∗}; f^{(1)}; f_{(−1)}; f^{(2)}; ···; f^{(n)}; f_{(−n)}; ···; 𝒟]

which necessarily preserve all coproducts, products and compact objects.

Moreover, when these conditions hold, the tower of adjoints appearing in (W6) is necessarily given for all n ∈ ℤ by the formulas

(1.10)  f^{(n)} = ω_{f}^{⊗n} ⊗ f^{∗}  and  f_{(n)} = f_{∗}(ω_{f}^{⊗n} ⊗ −) .

Finally, (W1)-(W6) hold true as soon as the functor f_{∗} : 𝒞 → 𝒟 satisfies, in addition to Grothendieck-Neeman duality, any one of the following three properties:

(1) The functor f_{∗} is faithful (i.e. f^{∗} is surjective up to direct summands).

(2) The functor f_{∗} detects compact objects: any x ∈ 𝒞 is compact if f_{∗}(x) is.

(3) Any x ∈ 𝒞 is compact if f_{∗}(x ⊗ y) is compact for every compact y ∈ 𝒞. (These conditions are ordered in increasing generality, for (1)⇒(2)⇒(3).)

1.11. Remark. We opted for the notation f^{(n)} ⊣ f_{(−n)} ⊣ f^{(n+1)} after trying everything else. As is well-known, notations of the form f^{!}, f_{!}, f^{×}, f_{#}, etc., have flourished in various settings, sometimes with contradictory meanings. Instead of risking collision, we propose a systematic notation which allows for an infinite tower of adjoints, following the tradition that f^{(n)} is numbered with n going up ⋯ f^{(n)}, f^{(n+1)} ⋯ and f_{(n)} with n going down ⋯ f_{(n)}, f_{(n−1)} · · ·. Our notation also recalls that f^{(n)} and f_{(n)} are n-fold twists of f^{(0)} = f^{∗} and f_{(0)} = f_{∗} by ω_{f}; see (1.10).

1.12. Remark. In the literature, Property (W5) is usually simply called a Wirthmüller isomorphism, referring to the original [Wir74]. Such a strong relation between the left and right adjoints to f^{∗} is very useful, for then f_{∗} and f_{(1)} will share all properties which are stable under pre-tensoring with an invertible object (e. g., being full, faithful, etc.). Similarly, most formulas valid for one of them will easily transpose into a formula for the other one. Here, we sometimes add the adjective “strong” to avoid collision with Fausk-Hu-May’s slightly different notion of “Wirthmüller context” [FHM03]; see more in Remark 4.3. Let us point out that the existence of any Wirthmüller isomorphism (W5) is not independent of Grothendieck duality but actually requires it. This is because our new condition (GN 3) tells us that the mere existence of the left adjoint f_{(1)} forces Grothendieck duality. Furthermore, the Wirthmüller isomorphism itself and the twisting object ω_{f} are borrowed from the earlier ur-Wirthmüller isomorphism (1.8), since ur-Wirthmüller (1.8) clearly implies Wirthmüller (W5) when ω_{f} is invertible.

In conclusion, we have the following picture:

f^{∗}⊣f_{∗}⊣f^{(1)}

in general_{(Thm. 1.3)}

f_{(1)}⊣f^{∗}⊣f_{∗}⊣f^{(1)}⊣f_{(−1)}

Grothendieck-Neeman duality

(Thm. 1.7)

[figure]

··· ⊣f_{(1)}⊣f^{∗}⊣f_{∗}⊣f^{(1)}⊣f_{(−1)}⊣ ···

Wirthmüller isomorphism

(Thm. 1.9)

[figure]

1.13. Corollary (Trichotomy of adjoints). If f^{∗} is a coproduct-preserving tensor triangulated functor between rigidly-compactly generated tensor triangulated categories, then exactly one of the following three possibilities must hold:

(1) There are two adjunctions as follows and no more: f^{∗} ⊣ f_{∗} ⊣ f^{(1)}.

(2) There are four adjunctions as follows and no more: f_{(1)} ⊣f^{∗} ⊣f_{∗} ⊣f^{(1)} ⊣f_{(−1)}.

(3) There is an infinite tower of adjunctions in both directions:

⋯ f^{(−1)} ⊣ f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)} ⋯ f^{(n)} ⊣ f_{(−n)} ⊣ f^{(n+1)} ⋯

1.14. Remark. The dualizing object ω_{f} could be invertible even in case (1) above, i.e. without Grothendieck-Neeman duality. See Example 3.25. Of course, there is no Wirthmüller isomorphism in such cases, since f_{(1)} does not even exist, by (GN 3).

1.15. Remark. In case (3), the invertible object ω_{f} can be trivial: ω_{f} ≃ 𝟙. This happens precisely when f^{∗} is a Frobenius functor [Mor65], i.e. admits a simultaneous left-and-right adjoint f^{∗} ⊣ f_{∗} ⊣ f^{∗}. This is also called an ambidextrous adjunction.

* * *

Abstract Grothendieck duality. In the literature, the phrase “Grothendieck duality” can refer to several different things. In its crudest form, it is the isomorphism ω_{f} ⊗ f^{∗} ≅ f^{(1)} of (GN 1) – hence the name twisted inverse image for f^{(1)}. “Grothendieck duality” can also refer to the compatibility Δ ∘ f_{∗} = f_{∗} ∘ Δ_{ω_{f}} given in (1.6), between direct image f_{∗} and the two dualities Δ = hom(−, 𝟙) and Δ_{ω_{f}} = hom(−, ω_{f}). However, this is usually formulated for certain proper subcategories 𝒞_{0} ⊂ 𝒞 and 𝒟_{0} ⊂ 𝒟 on which these duality functors earn their name by inducing equivalences 𝒞0^{op} →^{∼} 𝒞_{0} and 𝒟0^{op} →^{∼} 𝒟_{0}. Then “Grothendieck duality” refers to the situation where the functor f_{∗} : 𝒞 → 𝒟 maps 𝒞_{0} to 𝒟_{0} and intertwines the two dualities. The initial example was 𝒞_{0} = D^{b}(coh X) and 𝒟_{0} = D^{b}(coh Y) for a suitable morphism of schemes f : X → Y; here X and Y are assumed noetherian and D^{b}(coh X) is the bounded derived category of coherent O_{X}-modules. Since in general hom_{𝒞} (−, 𝟙) might not preserve 𝒞_{0} (as in the geometric example just mentioned), one should also try to replace the naive duality Δ = hom(−, 𝟙) of (1.6) by a more friendly one, say Δ_{κ} := hom(−, κ) for some object κ ∈ 𝒞_{0} having the property that hom_{𝒞} (−, κ) : 𝒞_{0}^{op} →^{∼} 𝒞_{0} is an equivalence. In algebraic geometry, for 𝒞_{0} = D^{b}(coh X), such κ are called dualizing complexes.

In Sections 5 and 6, we follow this approach to Grothendieck duality in our abstract setting, with an emphasis on the trichotomy of Corollary 1.13. Let f^{∗} be a functor as in our basic Hypothesis 1.2. As before, we write Δ_{κ} = hom(−, κ) for the κ-twisted duality functor, for any object κ. We prove:

(1) In the general situation, f_{∗} always intertwines dualities: we have

Δ_{κ} ∘ f_{∗} ≅ f_{∗} ∘ Δ_{κ′}

where κ ∈ 𝒟 is any object and κ′ := f^{(1)}(κ) ∈ 𝒞; see Theorem 5.15.

(2) Assume that f^{∗} : 𝒟 → 𝒞 satisfies Grothendieck-Neeman duality (Theorem 1.7). Let 𝒟_{0} ⊂ 𝒟 be a subcategory admitting a dualizing object κ ∈ 𝒟_{0}, in which case κ induces an equivalence Δ_{κ} : 𝒟0^{op} →^{∼} 𝒟_{0}. Provided 𝒟_{0} is a 𝒟^{c}-submodule (meaning 𝒟^{c} ⊗ 𝒟_{0} ⊆ 𝒟_{0}), the object κ′ = f^{(1)}(κ) ≅ ω_{f} ⊗ f^{∗}(κ) is dualizing for the following subcategory of 𝒞:

𝒞_{0} := {x ∈ 𝒞 | f_{∗}(c ⊗ x) ∈ 𝒟_{0} for all c ∈ 𝒞^{c}},

that is, κ′ ∈ 𝒞_{0} and Δ_{κ′} : 𝒞0^{op} →^{∼} 𝒞_{0}. Thus, by the formula in (1), f_{∗} : 𝒞_{0} → 𝒟_{0} is a morphism of categories with duality. See Theorem 5.25 for details, and see Theorem 6.4 for a more general relative version.

(3) Assume moreover that we have the Wirthmüller isomorphism of Theorem 1.9. Because of the monoidal adjunction f^{∗} : 𝒟 ⇄ 𝒞 : f_{∗}, we may consider 𝒞 as an enriched category over 𝒟, i.e. we may equip 𝒞 with Hom objects \underline{𝒞}(x, y) := f_{∗} hom_{𝒞} (x, y) in 𝒟. Then the equivalence (−) ⊗ ω_{f} : 𝒞^{c} →^{∼} 𝒞^{c} behaves like a Serre functor relative to 𝒟, meaning that there is a natural isomorphism

Δ\underline{𝒞}(x, y) ≅ \underline{𝒞}(y, x ⊗ ω_{f})

for all x, y ∈ 𝒞^{c}, where Δ = hom_{𝒟} (−, 𝟙) is the plain duality of 𝒟. If 𝒟 = D(𝕜) is the derived category of a field 𝕜, this reduces to an ordinary Serre functor on the 𝕜-linear category 𝒞^{c}. See Theorem 6.9 and Corollary 6.12. In algebraic geometry, we prove that if X is a projective scheme over a regular

noetherian base then the category 𝒞_{0} of (2) specializes to D^{b}(coh X); see Theorem 5.21. Thus in this case the results in (1) and (2) specialize to the classical algebro-geometric Grothendieck duality. Similarly, (3) specializes to the classical Serre duality for smooth projective varieties (cf. Example 6.14). But of course now these results apply more generally, for instance in representation theory, equivariant stable homotopy, and so on, ad libitum.

* * *

Further examples. Let us illustrate the broad reach of our setup with two additional examples, now taken from algebra and topology.

Still consider a tensor-exact functor f^{∗} : 𝒟 → 𝒞 satisfying Hypothesis 1.2 and the adjoints f^{∗} ⊣ f_{∗} ⊣ f^{(1)}. Instead of starting with a subcategory 𝒟_{0} ⊂ 𝒟 as we did above, we may reverse direction and consider a subcategory 𝒞_{0} ⊂ 𝒞 with dualizing object κ′ and ask under what circumstances may we “push” the subcategory with duality (𝒞_{0}, κ′) along f_{∗} : 𝒞 → 𝒟 to obtain a subcategory with duality in 𝒟.

We prove that if κ′ admits a “Matlis lift”, that is, an object κ ∈ 𝒟 such that f^{(1)}(κ) ≅ κ′, then κ is dualizing for the thick subcategory of 𝒟 generated by f_{∗}(𝒞_{0}); see Theorem 7.1. This result specializes to classical Matlis duality for commutative noetherian local rings; see Example 7.2. (For this we now allow dualizing objects to be external, i.e. we only assume that κ′ ∈ 𝒞 induces an equivalence Δ_{κ′} : 𝒞_{0} →^{∼} 𝒞_{0} while possibly κ′ ∉ 𝒞_{0}. Indeed, even when κ′ ∈ 𝒞_{0}, the lift κ ∈ 𝒟 need not be in 𝒟_{0}; see Example 7.3.) Dwyer, Greenlees and Iyengar [DGI06] have developed a rich framework which captures several dualities in the style of Pontryagin-Matlis duality, and we show how this connects with our theory in Example 7.4.

Finally, we conclude our article by showing that Neeman’s improved version [Nee92] of Brown-Comenetz duality [BC76] can also be expressed in our framework: It is given by the ω_{f}-twisted duality Δ_{ω_{f}} for a certain tensor-exact functor f^{∗} satisfying our basic hypothesis; see Theorem 7.10. Interestingly, it is possible to show that this functor f^{∗} is not induced by any underlying map f; see Remark 7.11.

2. Brown representability and the three basic functors

We begin by recollecting some well-known definitions and results.

Perhaps the most basic fact about adjoints of exact functors on triangulated categories is that they are automatically exact; see [Nee01, Lemma 5.3.6].

A triangulated category 𝒯 is said to be compactly generated if it admits arbitrary coproducts, and if there exists a set of compact objects 𝒢 ⊂ 𝒯 such that 𝒯(𝒢, t) = 0 implies t = 0 for any t ∈ 𝒯. An object t ∈ 𝒯 is compact (a.k.a. finite) if the functor 𝒯(t, −) : 𝒯 → Ab sends coproducts in 𝒯 to coproducts of abelian groups. We denote by 𝒯^{c} the thick subcategory of compact objects of 𝒯. A (contravariant) functor 𝒯 → 𝒜 to an abelian category is called (co)homological if it sends exact triangles to exact sequences. The notion of a compactly generated category is extremely useful, thanks to the following result of Neeman:

2.1. Theorem (Brown representability; see [Nee96, Kra02]). Let 𝒯 be a compactly generated triangulated category. Then:

(a) A cohomological functor 𝒯^{op} → Ab is representable — i.e., is isomorphic to one of the form 𝒯(−, t) for some t ∈ 𝒯 — if and only if it sends coproducts in 𝒯 to products of abelian groups.

(b) A homological functor 𝒯 → Ab is corepresentable — i.e., is isomorphic to one of the form 𝒯(t, −) for some t ∈ 𝒯 — if and only if it sends products in 𝒯 to products of abelian groups.

2.2. Remark. Theorem 2.1 (a) already implies that 𝒯 admits products (apply it to the functor ∏_{i} 𝒯(−, t_{i})). In turn, this allows for “dual” statements, such as (b).

2.3. Corollary. Let F : 𝒯 → 𝒮 be an exact functor between triangulated categories, and assume that 𝒯 is compactly generated. Then:

(a) F admits a right adjoint if and only if it preserves coproducts.

(b) F admits a left adjoint if and only if it preserves products.

Proof. As F is exact, the functors 𝒮(F (−), s) : 𝒯^{op} → Ab and 𝒮(s, F (−)) : 𝒯 → Ab are (co)homological for each s ∈ 𝒮, so we can feed them to Theorem 2.1. □

2.4. Remark. For 𝒯 compactly generated, in order to show that a natural transformation α : F → F′ between two coproduct-preserving exact functors F, F′ : 𝒯 → 𝒮 is an isomorphism, it suffices to prove so for the components α_{x} at x ∈ 𝒯^{c} compact. In some cases, this involves giving an alternative definition of α_{x}, valid for x compact, and showing by direct computation that the two definitions coincide. Such computations can become rather involved. We shall leave the easiest of these verifications to the reader but sketch the most difficult ones, hopefully to the benefit of the careful reader.

We will also make frequent use of the following two general facts about adjoints on compactly generated categories.

2.5. Proposition ([Nee96, Thm. 5.1]). Let F : 𝒮 ⇄ 𝒯 : G be an adjoint pair of exact functors between triangulated categories 𝒮 and 𝒯, and assume 𝒮 compactly generated. Then F preserves compact objects iff G preserves coproducts. □

The second general fact will play a crucial role in this paper:

2.6. Lemma. Let F : 𝒮 ⇄ 𝒯 : G be an adjoint pair of exact functors between triangulated categories. Assume 𝒮 compactly generated and that F preserves compacts.

(a) If the restriction F |_{𝒮^{c}} : 𝒮^{c} → 𝒯^{c} admits a right adjoint G_{0}, then G preserves compacts, and its restriction to compacts is isomorphic to G_{0}.

(b) If the restriction F |_{𝒮^{c}} : 𝒮^{c} → 𝒯^{c} admits a left adjoint E_{0} and if 𝒯 is compactly generated, then F preserves products.

Proof. For every compact t ∈ 𝒯^{c} and every compact s ∈ 𝒮^{c}, we have a natural bijection 𝒮^{c}(s, G_{0}(t)) ≅ 𝒯^{c}(F |_{𝒮^{c}} (s), t) = 𝒯(F (s), t) ≅ 𝒮(s, G(t)). By plugging s := G_{0}(t), the identity map of G_{0}(t) corresponds to a certain morphism γ_{t} : G_{0}(t) → G(t). Varying t ∈ 𝒯^{c}, we obtain a natural morphism γ : G_{0} → G|_{𝒯^{c}} by the naturality in t of the bijection. By its naturality in s, it actually follows that the bijection is obtained by composing maps f ∈ 𝒮(s, G_{0}(t)) with γ_{t}. In particular, for any fixed t ∈ 𝒯^{c} the induced map 𝒮(−, γ_{t}) : 𝒮(−, G_{0}(t)) → 𝒮(−, G(t)) is invertible on all s ∈ 𝒮^{c} by construction, and since 𝒮 is compactly generated, it is therefore invertible on all s ∈ 𝒮 (cf. Remark 2.4). It follows by Yoneda that γ_{t} is an isomorphism. Hence G(t) ≃ G_{0}(t) ∈ 𝒮^{c} for every t ∈ 𝒯^{c}, which gives (a).

For (b), let η : Id_{𝒯^{c}} → F ∘ E_{0} be the unit of the adjunction E_{0} ⊣ F |_{𝒮^{c}}. For every x ∈ 𝒯^{c} compact and s ∈ 𝒮 arbitrary, we can consider the morphism

α_{x,s} : 𝒮(E_{0}(x), s) ⟶^{F} 𝒯(F E_{0}(x), F (s)) ⟶^{η^{∗}} 𝒯(x, F (s)).

It is an isomorphism when s ∈ 𝒮^{c}, by the adjunction. Both functors 𝒮(E_{0}(x), −) and 𝒯(x, F (−)) are homological 𝒮 → Ab and preserve coproducts because F does (it has a right adjoint) and because x and E_{0}(x) are compact. By Remark 2.4, α_{x,s} is an isomorphism for every x ∈ 𝒯^{c} and every s ∈ 𝒮. This kind of “partial adjoint” suffices to prove that F preserves products, as usual : Let {s_{i}}_{i∈I} be a set of objects of 𝒮 and x ∈ 𝒯^{c} be compact and consider the isomorphism

𝒯(x, F ( ∏_{i∈I} s_{i})) ≅_{α} 𝒮(E_{0}(x), ∏_{i} s_{i}) ≅ ∏_{i} 𝒮(E_{0}(x), s_{i}) ≅_{α} ∏_{i} 𝒯(x, F s_{i}) ≅ 𝒯(x, ∏_{i∈I} F s_{i}) .

One verifies that this is the morphism induced by the canonical map F (∏_{i∈I} s_{i}) → ∏_{i∈I} F (s_{i}) and since 𝒯 is compactly generated, this map is an isomorphism. □

* * *

We now let the tensor ⊗ enter the game.

2.7. Definition. A tensor-triangulated category 𝒞 (i.e. a triangulated category with a compatible closed symmetric monoidal structure, see [HPS97, App. A.2]) is called rigidly-compactly generated if it is compactly generated and if compact objects and rigid objects coincide; in particular, the tensor unit object 𝟙 is compact. We denote the tensor by ⊗ : 𝒞 × 𝒞 ⟶ 𝒞 and its right adjoint by hom : 𝒞^{op} × 𝒞 ⟶ 𝒞 (internal Hom). An object x is rigid if the natural map hom(x, 𝟙) ⊗ y → hom(x, y) is an isomorphism for all y. Rigid objects are often called “(strongly) dualizable” in the literature but we avoid this terminology to prevent any possible confusion with our “dualizing objects”.

2.8. Remark. When 𝒞 is rigidly-compactly generated, its subcategory of compact objects 𝒞^{c} ⊂ 𝒞 is a thick subcategory, closed under ⊗. It admits the canonical duality Δ = hom(−, 𝟙) : (𝒞^{c})^{op} → 𝒞^{c} satisfying Δ^{2} ≅ Id. See details in [HPS97, App. A] for instance, where our rigidly-compactly generated tensor-triangulated categories are called “unital algebraic stable homotopy categories”.

Let us mention at this point a few important examples of rigidly-compactly generated categories 𝒞 arising in various fields of mathematics.

2.9. Example. Let X be a quasi-compact and quasi-separated scheme. Let 𝒞 := D_{Qcoh}(X) be the derived category of complexes of O_{X}-modules having quasicoherent homology (see [Lip09]). It is rigidly-compactly generated, and its compact objects are precisely the perfect complexes: (D_{Qcoh}(X))^{c} = D^{perf}(X) (see [BvdB03]). The latter are easily seen to be rigid for the derived tensor product ⊗ = ⊗_{O_{X}}^{L} . If moreover X is separated, there is an equivalence D_{Qcoh}(X) ≃ D(Qcoh X) with the derived category of complexes of quasi-coherent O_{X}-modules (see [BN93]). If X = Spec(A) is affine, then D(Qcoh X) ≃ D(A - Mod) with compacts D(A - Mod)^{c} ≃ K^{b}(A - proj), the homotopy category of bounded complexes of finitely generated projectives.

2.10. Example. Let G be a compact Lie group. Then 𝒞 := SH(G), the homotopy category of “genuine” G-spectra indexed on a complete G-universe (see [HPS97, §9.4]), is rigidly-compactly generated. The suspension G-spectra Σ_{+}^{∞}G/H, with H running through all closed subgroups of G, form a set of rigid-compact generators which includes the tensor unit 𝟙 = Σ_{+}^{∞}G/G.

2.11. Example. Let G be a finite group and let 𝕜 be a field. Then 𝒞 := Stab(𝕜G), the stable category of 𝕜G-modules modulo projectives, is rigidly-compactly generated. (Note that the derived category D(𝕜G), though compactly generated, is not rigidlycompactly generated because its unit 𝟙 = 𝕜 is not compact). More generally, G could be a finite group scheme over 𝕜 (see e.g. [HPS97, Theorem 9.6.3]).

2.12. Example. Let 𝕜 be a field and let 𝒞 := SH^{𝔸^{1}}(𝕜) denote the stable 𝔸^{1}-homotopy category. Twists of smooth projective 𝕜-varieties are rigid-compact in 𝒞. They generate the whole category under resolution of singularities (see [Rio05]). Hence if 𝕜 has characteristic zero, SH^{𝔸^{1}}(𝕜) is rigidly-compactly generated.

2.13. Example. Let A be a “Brave New” commutative ring, that is, a structured commutative ring spectrum. To fix ideas, we can understand A to be a commutative 𝕊-algebra in the sense of [EKMM97]. Then its derived category D(A), i.e. the homotopy category of A-modules, is a rigidly-compactly generated category, which is generated by its tensor unit A (see e.g. [HPS97, Example 1.2.3(f)] and [SS03, Example 2.3(ii)]). For example, every commutative dg-ring has an associated commutative 𝕊-algebra (its Eilenberg-MacLane spectrum) whose derived category is equivalent, as a tensor triangulated category, to the derived category of dg-modules (see [Shi07] and [SS03, Theorem 5.1.6]). Thus, derived categories of commutative dg-rings are also rigidly-compactly generated.

2.14. Corollary. Let 𝒞 and 𝒟 be rigidly-compactly generated categories, and let f^{∗} : 𝒟 → 𝒞 be as in our basic Hypothesis 1.2. Then f^{∗} preserves compacts and admits a right adjoint f_{∗} : 𝒞 → 𝒟, which itself admits a right adjoint f^{(1)} : 𝒟 → 𝒞.

Proof. Since f^{∗} preserves coproducts by assumption, f_{∗} exists by Brown Representability, Cor. 2.3 (a). Since f^{∗} is symmetric monoidal by assumption, it must send rigid objects of 𝒟 to rigid objects of 𝒞 (see e.g. [LMSM86, §III.1]). Hence it must preserve compacts (= rigids). By Proposition 2.5, f_{∗} preserves coproducts and we can apply another layer of Brown Representability to f_{∗} in order to get f^{(1)}. □

Our three functors f^{∗} ⊣ f_{∗} ⊣ f^{(1)} automatically satisfy some basic formulas.

2.15. Proposition. Let f^{∗} ⊣ f_{∗} ⊣ f^{(1)} be as in Corollary 2.14. Then there is a canonical natural isomorphism

(2.16)  π : x ⊗ f_{∗}(y) ⟶^{∼} f_{∗}(f^{∗}(x) ⊗ y)

for all x ∈ 𝒟 and y ∈ 𝒞, obtained from f^{∗}(x⊗f_{∗}(y)) ≅ f^{∗}(x)⊗f^{∗}f_{∗}(y) → f^{∗}(x)⊗y by adjunction. We also have three further canonical isomorphisms as follows:

(2.17)  hom_{𝒟} (x, f_{∗}y) ≅ f_{∗} hom_{𝒞} (f^{∗}x, y)
(2.18)  hom_{𝒟} (f_{∗}x, y) ≅ f_{∗} hom_{𝒞} (x, f^{(1)}y)
(2.19)  f^{(1)}hom_{𝒟} (x, y) ≅ hom_{𝒞} (f^{∗}x, f^{(1)}y) .

2.20. Terminology. We call (2.16) the (right) projection formula. Equations (2.17) and (2.18) are internal realizations of the two adjunctions f^{∗} ⊣ f_{∗} ⊣ f^{(1)}, from which the adjunctions can be recovered by applying hom_{𝒟} (𝟙_{𝒟} , −). Note that (2.18) specializes to (1.6) by inserting y = 𝟙_{𝒟} .

Proof. The map π is clearly well-defined for all x and y and is automatically invertible whenever x is rigid (cf. [FHM03, Prop. 3.2]). Fixing an arbitrary y ∈ 𝒞, note that both sides of (2.16) are exact and commute with coproducts in the variable x. As 𝒟 is generated by its compact (= rigid) objects, π is an isomorphism for all x ∈ 𝒟 (Rem. 2.4). This proves the first isomorphism, i.e. the projection formula.

Now we can derive from it two of the other equations by taking adjoints. (Recall that if F_{i} ⊣ G_{i} for i = 1, 2 then F_{1}F_{2} ⊣ G_{2}G_{1}. Note the order-reversal.) First, by fixing x we see two composite adjunctions

x ⊗ f_{∗} = (x ⊗ −) ∘ f_{∗} ⊣ f^{(1)} ∘ hom_{𝒟} (x, −)
f_{∗}(f^{∗}(x) ⊗ −) = f_{∗} ∘ (f^{∗}(x) ⊗ −) ⊣ hom_{𝒞} (f^{∗}x, −) ∘ f^{(1)} = hom_{𝒞} (f^{∗}x, f^{(1)}(−)) .

Since π is an isomorphism of the left adjoints, by the uniqueness of right adjoints it induces an isomorphism between the right ones, i.e. we get (2.19). (The naturality in x is guaranteed by the fact that the two adjunctions above are actually natural families of adjunctions parametrized by x.) If we fix y instead, we get adjunctions

(−) ⊗ f_{∗}(y) ⊣ hom_{𝒟} (f_{∗}y, −)
f_{∗}(f^{∗}(−) ⊗ y) = f_{∗} ∘ (− ⊗ y) ∘ f^{∗} ⊣ f_{∗} ∘ hom_{𝒞} (y, −) ∘ f^{(1)} = f_{∗} hom_{𝒞} (y, f^{(1)}(−))

from which we derive the natural isomorphism (2.18). By fixing x in the isomorphism f^{∗}(x) ⊗ f^{∗}(y) ≅ f^{∗}(x ⊗ y) given by the monoidal structure of f^{∗}, we obtain

f^{∗}(x) ⊗ f^{∗} = (f^{∗}(x) ⊗ −) ∘ f^{∗} ⊣ f_{∗} hom_{𝒞} (f^{∗}x, −)
f^{∗}(x ⊗ −) = f^{∗} ∘ (x ⊗ −) ⊣ hom_{𝒟} (x, −) ∘ f_{∗} = hom_{𝒟} (x, f_{∗}(−))

from which we derive the remaining relation (2.17). □

2.21. Remark. The reasoning of the previous proof will be used several times, so it is worth spending a little thought on it. Let’s say we have some formula, by which we mean a natural isomorphism F_{1}∘…∘F_{n} ≅ F′_{1}∘…∘F′_{m} between composite functors, in which every factor is part of an adjunction F_{i} ⊣ G_{i} and F′_{j} ⊣ G′_{j}. By taking right adjoints on both sides we derive a formula G_{n}G_{n−1} ⋯ G_{1} ≅ G′_{m}G′_{m−1} ⋯ G′_{1}. Actually the two formulas are equivalent, since we may recover the first one by taking left adjoints in the second one. Following [FHM03], we can say that the two formulas are conjugate, or adjunct. Note however that if the original formula admits two different factor-decompositions as above, we would obtain a different conjugate formula from each choice of decomposition. This is illustrated by the previous proposition, in which (2.18) and (2.19) are obtained from two different decompositions of (2.16). In this case, (2.16) is a formula between functors of two variables x and y, and the two decompositions have been obtained by first fixing either x or y. Note that the tensor formula f^{∗}(x ⊗ y) ≅ f^{∗}(x) ⊗ f^{∗}(y) is symmetric in x and y, hence the two resulting decompositions yield the same conjugate formula (2.17). All our conjugate formulas will come in such couplets or triplets and will be obtained in this way from a starting formula in either one or two variables. The systematic exploitation of this principle will greatly simplify the search for new relations. When repeating this reasoning below we will mostly leave the straightforward details to the reader.

3. Grothendieck-Neeman duality and ur-Wirthmüller

We want to prove Theorem 1.7, for which we need some preparation. Recall the basic set-up as in Hypothesis 1.2 and the three functors f^{∗} ⊣ f_{∗} ⊣ f^{(1)} (Cor. 2.14). We first focus on the new, slightly surprising facts. The following lemma should be compared to the well-known property presented in Proposition 2.5.

3.1. Lemma. If f^{∗} : 𝒟 → 𝒞 has a left adjoint f_{(1)} ⊣ f^{∗}, i.e. if f^{∗} preserves products, then its right adjoint f_{∗} preserves compact objects: f_{∗}(𝒞^{c}) ⊆ 𝒟^{c}.

Proof. Recall that f^{∗} preserves coproducts by our standing hypothesis, hence f_{(1)} preserves compacts (Prop. 2.5). Therefore f_{(1)} ⊣ f^{∗} restricts to an adjunction f_{(1)} : 𝒞^{c} ⇄ 𝒟^{c} : f^{∗} on compact objects. Since compacts are rigid, duality provides equivalences of (tensor) categories Δ := hom_{𝒞} (−, 𝟙) : (𝒞^{c})^{op} → 𝒞^{c} and Δ := hom_{𝒟} (−, 𝟙) : (𝒟^{c})^{op} → 𝒟^{c} which are quasi-inverse to themselves (i.e. Δ^{−1} = Δ^{op}). Moreover, the symmetric monoidal functor f^{∗} preserves rigid objects c and their tensor duals Δ(c) (cf. [LMSM86, §III.1]), so that the following square commutes (up to isomorphism of functors):

\xymatrix{(𝒟^{c})^{op} \ar[d]_{(f^{∗})^{op}} \ar[r]^{Δ}_{∼} & 𝒟^{c} \ar[d]^{f^{∗}} \\ (𝒞^{c})^{op} \ar[r]^{Δ}_{∼} & 𝒞^{c}.}

This self-duality implies that the composite functor f_{∗}^{c} := Δ ∘ (f_{(1)})^{op} ∘ Δ^{−1} = Δf_{(1)}Δ: 𝒞^{c} → 𝒟^{c} is right adjoint to f^{∗}: 𝒟^{c} → 𝒞^{c}. By Lemma 2.6 (a) (applied to F := f^{∗}), the right adjoint f_{∗} to f^{∗} must preserve compact objects. □

3.2. Proposition. Suppose that f_{∗} : 𝒞 → 𝒟 preserves compacts. Then there is a canonical natural isomorphism f^{(1)}(x) ⊗ f^{∗}(y) ⟶^{∼} f^{(1)}(x ⊗ y) for all x, y ∈ 𝒟.

Proof. The natural comparison map f^{(1)}(x) ⊗ f^{∗}(y) → f^{(1)}(x ⊗ y) can always be constructed out of the counit ϵ : f_{∗}f^{(1)} → Id_{𝒞} of the adjunction f_{∗} ⊣ f^{(1)} as follows:

ϵ_{x} ⊗ id_{y} ∈ 𝒟(f_{∗}f^{(1)}(x) ⊗ y, x ⊗ y)
≅ 𝒟(f_{∗}(f^{(1)}(x) ⊗ f^{∗}(y)), x ⊗ y) projection formula Prop. 2.15
≅ 𝒞(f^{(1)}(x) ⊗ f^{∗}(y), f^{(1)}(x ⊗ y)) adjunction f_{∗} ⊣ f^{(1)} .

If y ∈ 𝒟^{c} is rigid, we have for all z ∈ 𝒞 a natural isomorphism:

𝒞(z, f^{(1)}(x) ⊗ f^{∗}(y)) ≅ 𝒞(z ⊗ Δf^{∗}(y), f^{(1)}(x)) f^{∗}(y) is rigid
≅ 𝒞(z ⊗ f^{∗}Δ(y), f^{(1)}(x)) f^{∗}Δ ≅ Δf^{∗} on rigids
≅ 𝒟(f_{∗}(z ⊗ f^{∗}Δ(y)), x) adjunction f_{∗} ⊣ f^{(1)}
≅ 𝒟(f_{∗}(z) ⊗ Δ(y), x)  projection formula Prop. 2.15
≅ 𝒟(f_{∗}(z), x ⊗ y)  y is rigid
≅ 𝒞(z, f^{(1)}(x ⊗ y))  adjunction f_{∗} ⊣ f^{(1)}.

A tedious but straightforward diagram chase verifies that this isomorphism is merely post-composition by the general comparison map f^{(1)}(x) ⊗ f^{∗}(y) → f^{(1)}(x ⊗ y) previously defined. Hence, by Yoneda, we conclude that the general comparison map is an isomorphism whenever y is rigid. By Proposition 2.5, the hypothesis on f_{∗} is equivalent to f^{(1)} preserving coproducts. Hence both sides of the comparison map f^{(1)}(x) ⊗ f^{∗}(y) → f^{(1)}(x ⊗ y) are coproduct-preserving exact functors in both variables. Hence this comparison map is invertible for all x, y ∈ 𝒟 (Remark 2.4). □

We are now ready to prove our generalized Grothendieck-Neeman duality theorem. Recall from Definition 1.4 that ω_{f} := f^{(1)}(𝟙) ∈ 𝒞 is the relative dualizing object associated with the given functor f^{∗} : 𝒟 → 𝒞.

3.3. Theorem. Let f^{∗} : 𝒟 → 𝒞 be as in our basic Hypothesis 1.2 and consider the adjoints f^{∗} ⊣ f_{∗} ⊣ f^{(1)} (Cor. 2.14). Then the following conditions are equivalent:

(a) The functor f^{∗} admits a left adjoint f_{(1)}.

(b) The functor f^{∗} preserves products.

(c) The functor f^{(1)} admits a right adjoint f_{(−1)}.

(d) The functor f^{(1)} preserves coproducts.

(e) The functor f_{∗} preserves compact objects. Furthermore, if (a)-(e) hold true then f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)} satisfy the following additional relations given by canonical natural isomorphisms:

(3.4)  f^{(1)} ≅ ω_{f} ⊗ f^{∗}(−)  (Grothendieck duality)
(3.5)  f_{(−1)} ≅ f_{∗} hom_{𝒞} (ω_{f}, −)
(3.6)  f^{(1)}(x ⊗ y) ≅ f^{(1)}(x) ⊗ f^{∗}(y)
(3.7)  hom_{𝒟} (x, f_{(−1)}y) ≅ f_{∗} hom_{𝒞} (f^{(1)}x, y)
(3.8)  hom_{𝒟} (x, f_{(−1)}y) ≅ f_{(−1)} hom_{𝒞} (f^{∗}x, y)

(3.9)  f^{∗}(−) ≅ hom_{𝒞} (ω_{f}, f^{(1)}(−))
(3.10)  f_{(1)}(−) ≅ f_{∗}(ω_{f} ⊗ −)  (ur-Wirthmüller)

(3.11)  x ⊗ f_{(1)}(y) ≅ f_{(1)}(f^{∗}(x) ⊗ y)  (left projection formula)
(3.12)  f^{∗}hom_{𝒟} (x, y) ≅ hom_{𝒞} (f^{∗}x, f^{∗}y)
(3.13)  hom_{𝒟} (f_{(1)}x, y) ≅ f_{∗} hom_{𝒞} (x, f^{∗}y).

3.14. Remark. The existence of any natural isomorphism as in Grothendieck duality (3.4) implies that f^{(1)} preserves coproducts (i.e. property (d) holds). Hence (3.4) is not only a consequence of, but is equivalent to, conditions (a)-(e) of the theorem. Similarly, the more general (3.6) is also equivalent to (a)-(e). Finally, if there exists any isomorphism as in (3.9), then f^{∗} must preserve products, since so do the left adjoints hom_{𝒞} (ω_{f}, −) and f^{(1)}. Hence (3.9) is also an equivalent condition for Theorem 3.3 to hold. We note this for completeness but it is unlikely that such isomorphisms can be established in practice before any of (a)-(e) is known.

3.15. Remark. We will see in the proof that each group of equations in (3.4)-(3.13), as displayed above, forms a conjugate set of formulas in the sense of Remark 2.21.

3.16. Remark. All of the adjunctions f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)} now have an internal realization in 𝒟 by (3.13), (2.17), (2.18), and (3.7), respectively.

3.17. Remark. We can further combine the fundamental formulas of Theorem 3.3, for instance by composing Grothendieck duality (3.4) and the ur-Wirthmüller (3.10) isomorphism and then variating by conjugation:

f_{(1)}(f^{∗}(x) ⊗ y) ≅ f_{∗}(f^{(1)}(x) ⊗ y)
hom_{𝒞} (f^{∗}x, f^{∗}y) ≅ hom_{𝒞} (f^{(1)}x, f^{(1)}y)
f_{∗} hom_{𝒞} (x, f^{∗}y) ≅ f_{(−1)}hom_{𝒞} (x, f^{(1)}y) .

Or we may plug Grothendieck duality into ur-Wirthmüller’s adjunct (3.9) to obtain

f^{∗} ≅ hom_{𝒞} (ω_{f}, ω_{f} ⊗ f^{∗}(−))

which, when applied to the tensor unit, specializes to the important relation

(3.18)  𝟙_{𝒞} ≅ hom_{𝒞} (ω_{f}, ω_{f}) .

Further combinations of the original formulas are left to the interested reader.

Proof of Theorem 3.3. We already know that (a)⇔(b) and (c)⇔(d)⇔(e) by Brown representability (Cor. 2.3) and by Proposition 2.5. We also isolated the non-obvious parts of the equivalences in Lemma 2.6 (b) and Lemma 3.1, which give (e)⇒(b) and (a)⇒(e) respectively. So we can assume that (a)-(e) hold true and we now turn to proving Formulas (3.4)-(3.13).

Proposition 3.2 already gives (3.6), which then specializes to (3.4) by setting x := 𝟙_{𝒟} . We now construct the canonical ur-Wirthmüller isomorphism (3.10). For every x ∈ 𝒞, consider the composite (3.19)

f_{∗}(x ⊗ ω_{f}) ⟶^{f_{∗}(η⊗1)} f_{∗}(f^{∗}f_{(1)}x ⊗ ω_{f}) ≅ f_{(1)}x ⊗ f_{∗}ω_{f} = f_{(1)}x ⊗ f_{∗}f^{(1)}𝟙 ⟶^{1⊗ϵ} f_{(1)}x

where the middle map is the right projection formula (2.16) and η : Id → f^{∗}f_{(1)} and ϵ : f_{∗}f^{(1)} → Id are the unit and counit of these adjunctions. By Remark 2.4, it suffices to show that this map is an isomorphism for x ∈ 𝒞^{c} compact, because both ends of (3.19) preserve coproducts in 𝒞 (being composed of left adjoints). Now, for every d ∈ 𝒟 and for x ∈ 𝒞^{c} rigid we compute:

𝒟(d, f_{∗}(x ⊗ ω_{f})) ≅ 𝒞(f^{∗}(d), x ⊗ ω_{f})  f^{∗} ⊣ f_{∗}
≅ 𝒞(Δ(x) ⊗ f^{∗}(d), ω_{f}) x ∈ 𝒞^{c} is rigid
≅ 𝒟(f_{∗}(Δx ⊗ f^{∗}d), 𝟙) ω_{f} = f^{(1)}(𝟙) and f_{∗} ⊣ f^{(1)}
≅ 𝒟(f_{∗}(Δx) ⊗ d, 𝟙)  projection formula (2.16)
≅ 𝒟(d, Δf_{∗}Δ(x))  f_{∗}Δ(x) ∈ 𝒟^{c} by Lemma 3.1
≅ 𝒟(d, f_{(1)}(x)).

The last isomorphism holds because, thanks to (e), the adjunction f_{(1)} ⊣ f^{∗} ⊣ f_{∗} restricts to the categories of compact objects, so that the dual Δ intertwines the two restricted functors: Δf_{∗}Δ ≅ f_{(1)} on 𝒞^{c}. By Yoneda, we obtain from the above an isomorphism f_{∗}(x ⊗ ω_{f}) ≅ f_{(1)}(x) for x ∈ 𝒞^{c} and we “only” need to show that it coincides with the canonical map (3.19). This is an adventurous diagram chase that we outline in more detail this time, since it might be harder to guess.

Following through the chain of isomorphisms, we can reduce the problem to checking the commutativity of the following diagram:

(3.20)  \xymatrix{f_{∗}(x ⊗ ω_{f}) \ar[r]^{coev⊗1} \ar[d]_{f_{∗}(η⊗1)} & Δf_{∗}Δx ⊗ f_{∗}Δx ⊗ f_{∗}(x ⊗ ω_{f}) \ar[d]^{1⊗lax} \\ f_{∗}(f^{∗}f_{(1)}x ⊗ ω_{f}) \ar[d]_{π} & Δf_{∗}Δx ⊗ f_{∗}(Δx ⊗ x ⊗ ω_{f}) \ar[d]^{1⊗f_{∗}(ev⊗1)} \\ f_{(1)}x ⊗ f_{∗}ω_{f} \ar[r]^{≅} & Δf_{∗}Δx ⊗ f_{∗}ω_{f}}

where π is the projection formula isomorphism (2.16). Using an explicit description of the bottom isomorphism f_{(1)}x ≅ Δf_{∗}Δx in terms of the unit and counit of f_{(1)} ⊣ f^{∗} and the duality maps, one can check that the composite along the left and bottom edges is equal to

\xymatrix{f_{∗}(x ⊗ ω_{f}) ≅ f_{∗}(f^{∗}𝟙 ⊗ x ⊗ ω_{f}) \ar[r]^{f_{∗}(f^{∗}coev⊗1⊗1)} & f_{∗}(f^{∗}(Δf_{∗}Δx ⊗ f_{∗}Δx) ⊗ x ⊗ ω_{f}) \ar[d]^{f_{∗}(f^{∗}π⊗1)} \\  & f_{∗}(f^{∗}f_{∗}(f^{∗}Δf_{∗}Δx ⊗ Δx) ⊗ x ⊗ ω_{f}) \ar[d]^{f_{∗}(ϵ⊗1)} \\  & f_{∗}(f^{∗}Δf_{∗}Δx ⊗ Δx ⊗ x ⊗ ω_{f}) \ar[d]^{f_{∗}(1⊗ev⊗1)} \\  & f_{∗}(f^{∗}Δf_{∗}Δx ⊗ ω_{f}) \ar[d]^{π} \\  & Δf_{∗}Δx ⊗ f_{∗}ω_{f}.}

This composite can then be checked to agree with

\xymatrix{f_{∗}(x ⊗ ω_{f}) = 𝟙 ⊗ f_{∗}(x ⊗ ω_{f}) \ar[r]^{coev⊗1} & Δf_{∗}Δx ⊗ f_{∗}Δx ⊗ f_{∗}(x ⊗ ω_{f}) \ar[d]^{π⊗1} \\  & f_{∗}(f^{∗}Δf_{∗}Δx ⊗ Δx) ⊗ f_{∗}(x ⊗ ω_{f}) \ar[d]^{lax} \\  & f_{∗}(f^{∗}Δf_{∗}Δx ⊗ Δx ⊗ x ⊗ ω_{f}) \ar[d]^{f_{∗}(1⊗ev⊗1)} \\  & f_{∗}(f^{∗}Δf_{∗}Δx ⊗ ω_{f}) \ar[d]^{π} \\  & Δf_{∗}Δx ⊗ f_{∗}ω_{f}.}

Using this last description, the commutativity of diagram (3.20) can be established. In carrying out these verifications, the commutativity of the following diagrams

(3.21) [figure: f_{∗}a ⊗ f_{∗}b; π; lax; f_{∗}(f^{∗}f_{∗}a ⊗ b); f_{∗}(ϵ⊗1); f_{∗}(a ⊗ b); and; a ⊗ f_{∗}b ⊗ f_{∗}c; π⊗1; 1⊗lax; f_{∗}(f^{∗}a ⊗ b) ⊗ f_{∗}c; lax; a ⊗ f_{∗}(b ⊗ c); π; f_{∗}(f^{∗}a ⊗ b ⊗ c)]

will prove to be useful. The remaining details are now left to the careful reader.

We have now established (3.4), (3.6) and (3.10), from which we derive the other ones by the general method of Remark 2.21. Taking right adjoints of the functors in (3.10) yields (3.9). Taking right adjoints of the functors in (3.6) for each fixed x yields (3.7), and taking right adjoints for each fixed y yields (3.8). (The left-hand-sides of the two latter formulas coincide, because f^{(1)}(x ⊗ y) is symmetric in x and y.) On the other hand, taking right adjoints in (3.4) yields (3.5). Also, (3.18) is (3.9) evaluated at 𝟙_{𝒟} . The left projection formula (3.11) follows by conjugating the right projection formula (2.16) by the ur-Wirthmüller isomorphism (3.10):

\xymatrix{x ⊗ f_{(1)}(y) \ar[r]^{≅} \ar@{.>}[d]_{≅} & x ⊗ f_{∗}(ω_{f} ⊗ y) \ar[d]^{≅} \\ f_{(1)}(f^{∗}(x) ⊗ y) \ar[r]^{≅} & f_{∗}(f^{∗}(x) ⊗ ω_{f} ⊗ y).}

This new two-variable equation (3.11) has two conjugate formulas (3.12) and (3.13), obtained by taking right adjoints while fixing x or y, respectively. □

3.22. Example (Algebraic geometry). Let f : X → Y be a morphism of quasicompact and quasi-separated schemes, as in Example 2.9, and consider the (derived) inverse image functor f^{∗} : 𝒟 = D_{Qcoh}(Y) → D_{Qcoh}(X) = 𝒞. It is easy to see that f^{∗} satisfies our basic Hypothesis 1.2; its right adjoint is the derived pushforward f_{∗} = Rf_{∗}, whose right adjoint f^{(1)} is the twisted inverse image functor, usually written f^{×} or f^{!} (see [Lip09]). Then the functor f^{∗} satisfies Grothendieck-Neeman duality precisely when the morphism f is quasi-perfect [LN07, Def. 1.1]. Indeed, the latter means by definition that Rf_{∗} preserves perfect complexes, i.e. compact objects. In this context, our Theorem 3.3 recovers the original results of Neeman that have inspired us; see [LN07, Prop. 2.1] for a geometric statement in the same generality as we obtain here by specializing our abstract methods. Yet, even when specialized to algebraic geometry, our theorem is somewhat stronger, because it includes the extra information about the left adjoint f_{(1)} of f^{∗}, whose existence is equivalent to the quasi-perfection of f and which is necessarily given by the ur-Wirthmüller formula f_{(1)} ≅ ω_{f} ⊗ f_{∗}.

The article [LN07] contains a thorough geometric study of quasi-perfection. Among other things, it is shown that f is quasi-perfect iff it is proper and of finite tor-dimension. In particular if f : X → Y is finite then it is quasi-perfect iff f_{∗}(𝟙_{𝒞}) = Rf_{∗}(O_{X}) is a perfect complex. See Examples 2.2 of loc. cit. for more.

3.23. Example (Affine case). Let ϕ : B → A be a morphism of commutative rings. We may specialize Example 3.22 to f := Spec(ϕ) : X := Spec(A) → Spec(B) =: Y. (As pointed out in the introduction, we do not use the notation ϕ^{∗} for the restriction of rings f_{∗}, to avoid confusion with extension of scalars f^{∗}.) Since a proper affine morphism is necessarily finite, we see that f is quasi-perfect if and only if A admits a finite resolution by finitely generated projective B-modules. Since Rf_{∗} = f_{∗} ≅ BA ⊗_{A} −, the right adjoint f^{(1)} has the following reassuring description as a right derived functor: f^{(1)} = R Hom_{B}(_{B}A, −).

3.24. Example. Of course, not all scheme maps f : X → Y are quasi-perfect. For instance, the affine morphism f := Spec(ϕ), with ϕ : ℤ[t]/(t^{2}) → ℤ sending the variable t to zero, is not; see [Nee96, Ex. 6.5]. Note that f is finite, but indeed Rf_{∗}(O_{X}) is not compact, as ℤ has infinite projective dimension over ℤ[t]/(t^{2}).

3.25. Example. Let R be a Gorenstein local ring of Krull dimension d and k its residue field. Consider as above the morphism f : Spec(k) → Spec(R) and the induced functor f^{∗} : D(R) → D(k). It does not satisfy Grothendieck-Neeman duality in general, that is, unless f_{∗}(k) is perfect, i.e. R is regular. However, ω_{f} = f^{(1)}(𝟙) = RHom_{R}(k, R) ≃ Σ^{−d}k because R is Gorenstein, see [Mat86, § 18]. In this case, ω_{f} is invertible despite failure of Grothendieck-Neeman duality.

3.26. Example. The affine situation of Example 3.23 can be generalized in the Brave New direction (or in the Differential Graded direction), as in Example 2.13. That is, we may consider ϕ : B → A to be a morphism of commutative 𝕊-algebras (or commutative dg-rings). One can check that ϕ induces a functor f^{∗} := A ⊗_{B} −: D(B) → D(A) satisfying our basic Hypotheses. As before, its right adjoint f_{∗} is obtained simply by considering A-modules as B-modules through ϕ and the next right adjoint f^{(1)} is given by the formula f^{(1)} = Hom_{B}(A, −). (All functors considered here are derived from appropriate Quillen adjunctions.) Since D(A)^{c} is the thick subcategory generated by A, we see by Neeman’s criterion that, as before, f^{∗} satisfies Grothendieck-Neeman duality iff f_{∗}(A) is compact.

3.27. Example. Let k be a field and consider the inclusion ϕ : k[x^{n}] ↪ k[x] of graded k-algebras. Since k[x] is a free k[x^{n}]-module with homogeneous basis {1, x, …, x^{n−1}} we see that f_{∗}(𝟙) is compact, and hence f^{∗} satisfies Grothendieck-Neeman duality. Moreover, one easily checks that ω_{f} = Hom_{k[x^{n}]}(k[x], k[x^{n}]) ≅ Σ^{n−1}k[x] = Σ^{n−1}𝟙. From a more abstract point of view, f_{∗}(𝟙) ≅ ⨁_{i=0}^{n−1} Σ^{−i}𝟙, and since D(k[x]) is generated by the unit, the relative dualizing object ω_{f} is characterized (cf. (1.6)) by f_{∗}(ω_{f}) ≅ [f_{∗}(𝟙), 𝟙] = [⨁_{i=0}^{n−1} Σ^{−i}𝟙, 𝟙] = ⨁_{i=0}^{n−1} Σ^{i}𝟙. Hence ω_{f} ≅ Σ^{n−1}𝟙. Note that if we regard k[x^{n}] → k[x] as a map of ungraded commutative rings and consider the extension-of-scalars functor f^{∗} between ordinary derived categories, then ω_{f} ≅ 𝟙.

4. The Wirthmüller isomorphism

When we are in the Grothendieck-Neeman situation, i.e. when we have five adjoints f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)}, the relative dualizing object ω_{f} is remarkably “close” to being ⊗-invertible, a fact which perhaps deserves separate statement.

4.1. Proposition. Assume Grothendieck-Neeman duality (Thm. 3.3). Then f_{∗}(ω_{f}) is compact in 𝒟. Moreover, ω_{f} is compact in 𝒞 if and only if it is ⊗-invertible.

Proof. We have f_{∗}(ω_{f}) ≅ f_{(1)}(𝟙) by the ur-Wirthmüller isomorphism (3.10). Moreover, 𝟙 is assumed to be compact and f_{(1)} preserves compact objects by Proposition 2.5, because its right adjoint f^{∗} preserves coproducts by hypothesis.

Invertible objects are always rigid, hence compact under our assumptions. Conversely, if ω_{f} is rigid there is an isomorphism Δ(ω_{f}) ⊗ ω_{f} →^{∼} hom_{𝒞} (ω_{f}, ω_{f}) and therefore by (3.18) an isomorphism Δ(ω_{f}) ⊗ ω_{f} ≃ 𝟙, hence ω_{f} is invertible. □

We are now ready to prove Theorem 1.9, which abundantly characterizes the situations when ω_{f} does become invertible.

Proof of the Wirthmüller isomorphism Theorem 1.9. The equivalent formulations of (W1) hold by Corollary 2.3 and similarly for (W2), together with Proposition 2.5. The equivalence (W3)⇔(W4) holds by Proposition 4.1. If f^{(1)} preserves compacts then obviously ω_{f} = f^{(1)}(𝟙) is compact, hence we have (W2)⇒(W3). Conversely, we can see from Grothendieck duality f^{(1)} ≅ ω_{f} ⊗ f^{∗} that (W3)⇒(W2), as our f^{∗} always preserves compacts.

Let us show (W1)⇒(W2). Thus we now have six adjoints f^{(−1)} ⊣ f_{(1)} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{(1)} ⊣ f_{(−1)} and we want to show that f^{(1)} preserves compacts. Since f^{(−1)}, f_{(1)}, f^{∗} and f_{∗} have two-fold right adjoints they must preserve compacts by Proposition 2.5 (their right adjoints preserve coproducts). By restricting to compacts, we have four consecutive adjoints f^{(−1)}|_{𝒟^{c}} ⊣ f_{(1)}|_{𝒞^{c}} ⊣ f^{∗}|_{𝒟^{c}} ⊣ f_{∗}|_{𝒞^{c}} . Now recall from Remark 2.8 that Δ = hom(−, 𝟙) defines a duality on compact objects, hence conjugating with it turns left adjoints into right adjoints. Furthermore, since f^{∗}Δ = Δf^{∗}, the original functor f^{∗}|_{𝒟^{c}} is fixed by conjugation by Δ. The above four adjoints therefore yield (several isomorphisms, like Δ(f_{(1)}|_{𝒞^{c}} )Δ ≅ f_{∗}|_{𝒞^{c}} , and) the following five consecutive adjoints between 𝒟^{c} and 𝒞^{c}

f^{(−1)}|_{𝒟^{c}} ⊣ f_{(1)}|_{𝒞^{c}} ⊣ f^{∗}|_{𝒟^{c}} ⊣ f_{∗}|_{𝒞^{c}} ⊣ Δ(f^{(−1)}|_{𝒟^{c}} )Δ .

The right-most functor is the unpredicted one. We can now apply Lemma 2.6 (a) for F := f_{∗} to show that its right adjoint f^{(1)} preserves compacts, as desired.

Clearly (W4)⇒(W5) because of the ur-Wirthmüller isomorphism (1.8). Now assume (W5) instead, i.e. that there exists a ⊗-invertible ω ∈ 𝒞 such that f_{∗}(ω ⊗−) is left adjoint to f^{∗}. Then the formulas (1.10) make sense with ω instead of ω_{f}, yielding well-defined functors f^{(n)} : 𝒟 → 𝒞 and f_{(n)} : 𝒞 → 𝒟 for all n ∈ ℤ:

f^{(n)} := ω^{⊗n} ⊗ f^{∗} f_{(n)} := f_{∗}(ω^{⊗n} ⊗ −) with f^{(0)} := f^{∗} and f_{(0)} := f_{∗} .

Moreover, for all n ∈ ℤ we obtain the required adjunctions f^{(n)} ⊣ f_{(−n)} ⊣ f^{(n+1)} by variously composing the adjunction f^{∗} ⊣ f_{∗} with the appropriate power of (ω_{f} ⊗ −) ⊣ (ω_{f}^{−1} ⊗ −) or (ω_{f}^{−1} ⊗ −) ⊣ (ω_{f} ⊗ −). Thus (W5)⇒(W6).

If (W6) holds then we have (W1)-(W2) because then every functor in the tower must preserve products, coproducts and compacts. The uniqueness of adjoints implies that whenever f^{∗} ⊣ f_{∗} sprouts a doubly infinite tower of adjoints this is necessarily given by the formulas (1.10), because in that case ω_{f} is invertible.

As the alert (or record-keeping) reader must have noticed, we have proved that the conditions (W1)-(W6) are all equivalent, and that they imply (1.10). It remains to verify the claimed sufficient conditions, i.e. the “finally” part.

For completeness, let us first recall (see Lemma 4.2 below) that f_{∗} is faithful if and only if f^{∗} is surjective on objects, up to direct summands. Moreover, if this is the case then the counit ϵ_{x} : f^{∗}f_{∗}(x) → x is a split epi for all x ∈ 𝒞. Therefore if f_{∗}(x) is compact then x must be as well, because f^{∗} preserves compacts. This shows the implication (1)⇒(2). Clearly (2) implies (3), as they have the same conclusion but the hypothesis in (2) is weaker. To conclude the proof of the theorem, it suffices to show that (3) implies (W3). As we are in the context of Grothendieck-Neeman duality, we can use the ur-Wirthmüller equation f_{∗}(ω_{f} ⊗ x) ≅ f_{(1)}(x). Since f_{(1)} preserves compacts, it implies that f_{∗}(ω_{f} ⊗ x) is compact whenever x is a compact object of 𝒞, so by (3) we can conclude that ω_{f} is compact, that is (W3). □

4.2. Lemma. Let F : 𝒮 ⇄ 𝒯 : G be adjoint exact functors between triangulated categories. Then G is faithful if and only if F is surjective up to direct summands, that is, if and only if every object x ∈ 𝒯 is a retract of F (y) for some y ∈ 𝒮. Moreover, this is equivalent to the counit of adjunction admitting a (possibly unnatural) section at each object.

Proof. (Cf. [Bal11, Prop. 2.10].) For any x ∈ 𝒯, let F G(x) →^{ϵ_{x}} x →^{α} y → ΣF G(x) be an exact triangle containing ϵ_{x} : F G(x) → x, the counit of adjunction at x. Then G(ϵ_{x}) is a split epi by one of the unit-counit relations. Also, αϵ_{x} = 0 hence G(α)G(ϵ_{x}) = 0. Together these facts imply G(α) = 0. Now if G is faithful we have α = 0 and therefore, by the exact triangle, ϵ_{x} is a split epi (cf. [Nee01, Cor. 1.2.7]). In particular, x is a retract of F G(x). Hence G faithful implies that F is surjective up to direct summands.

Conversely, assume F is surjective up to direct summands: for every x ∈ 𝒯 we can find an x′ ∈ 𝒯, a y ∈ 𝒮, and an isomorphism x ⊕ x′ ≅ F (y). By the other unit-counit relation, the morphism ϵ_{F (y)} : F GF (y) → F (y) is a split epi. By the naturality of ϵ and the additivity of the functors, the morphisms ϵ_{F (y)} and ϵ_{x} ⊕ ϵ_{x′} are isomorphic, hence ϵ_{x} must be an epi. As x ∈ 𝒯 was arbitrary, this proves G faithful by [ML98, Thm. IV.3.1]. □

4.3. Remark. The important article Fausk-Hu-May [FHM03] also deals with Grothendieck duality and Wirthmüller isomorphisms, without assuming the categories to be triangulated until their final section. In some sense, we take over where they leave things and the picture becomes much simpler, as we now explain.

Fausk-Hu-May assume given two pairs of adjoints: the original f^{∗} ⊣ f_{∗} and another one f_{!} ⊣ f^{!}. This is motivated by “Verdier-Grothendieck duality” in algebraic geometry (which we do not consider here). They mainly study two special cases:

(1) The case f_{!} = f_{∗} or “Grothendieck context”, which reads f^{∗} ⊣ f_{∗} ⊣ f^{!}.

(2) The case f^{!} = f^{∗} or “Wirthmüller context”, which reads f_{!} ⊣ f^{∗} ⊣ f_{∗}. Although they explicitly say that both cases can happen simultaneously, their paper rather stresses the separation between the two contexts (first in the notation, since f_{!} ⊣ f^{∗} ⊣ f_{∗} ⊣ f^{!} would collide with f_{!} ⊣ f^{!}, but more systematically in the presentation: the two cases are deemed “deceptively similar, but genuinely different”). However, for triangulated categories, our new condition (GN 3) says that the mere existence of a left adjoint to f^{∗} already forces Grothendieck duality, even before asking whether this left adjoint is a twisted form of f_{∗}. In other words, (2) is not genuinely different from (1), it actually implies (1)!

Also, the Wirthmüller context (2) assumes the existence of some object C with f_{∗}(𝟙) ≃ f_{(1)}(C). Then [FHM03, Thm. 8.1] establishes a Wirthmüller isomorphism f_{∗} ≃ f_{(1)}(C ⊗ −) comparable to our (W5). However, in each example, such a “Wirthmüller object” C needs to be constructed by hand. For instance, in equivariant stable homotopy such a construction is done in a separate article [May03], sequel to [FHM03]. Moreover, the relation f_{∗}(𝟙) ≃ f_{(1)}(C) does not characterize C uniquely, a priori. Our approach avoids the mysterious object C altogether: The relative dualizing object ω_{f} and the ur-Wirthmüller isomorphism (1.8) exist in the more general setting of Grothendieck duality, and when a Wirthmüller isomorphism exists, we can simply take C to be the inverse of the relative dualizing object ω_{f}.

We did not find any example with non-rigid Wirthmüller object C. Hence, the following result essentially subsumes the Wirthmüller context of [FHM03] into ours (again, for rigidly-compactly generated categories).

4.4. Proposition. Suppose that the basic adjunction f^{∗} : 𝒟 ⇄ 𝒞 : f_{∗} as in Hypothesis 1.2 fits in a “Wirthmüller context” in the sense of [FHM03], i.e. suppose that f^{∗} has a left adjoint f_{(1)} (denoted f_{!} in [FHM03]) and that there exists an object C ∈ 𝒞 such that f_{(1)}(C) ≃ f_{∗}(𝟙). Then its dual hom_{𝒞} (C, 𝟙) is isomorphic to ω_{f}.

If moreover C is compact (i.e. rigid), as is commonly the case in examples, then ω_{f} and C are invertible and C ≅ ω_{f}^{−1}. In other words, a Wirthüller context with compact Wirthmüller object C only happens in the case of the infinite tower of adjoints (Theorem 1.9) and then C must be the inverse of the canonical object ω_{f}.

Proof. By [FHM03, Thm. 8.1], the Wirthmüller context yields an isomorphism f_{(1)}(C ⊗ −) ≃ f_{∗}. Taking right adjoints (which exist by Theorem 1.3), we get

hom_{𝒞} (C, f^{∗}(−)) ≃ f^{(1)}.

Evaluating at 𝟙 ∈ 𝒟, we obtain the desired hom_{𝒞} (C, 𝟙) ≃ f^{(1)}(𝟙) = ω_{f}. If moreover C is rigid, then so is its dual ω_{f}. So, by Theorem 1.9, ω_{f} must be invertible. □

4.5. Example (Equivariant homotopy theory). Let H be a closed subgroup of a compact Lie group G and let f^{∗} : SH(G) → SH(H) denote the restriction functor from the equivariant stable homotopy category of (genuine) G-spectra to that of H-spectra, as in Example 2.10. Then f^{∗} provides an example of Theorem 1.9. The relative dualizing object ω_{f} is the H-sphere S^{L} where L denotes the tangent H-representation of the smooth G-manifold G/H at the identity coset eH (see [LMSM86, Chapter III]). The ur-Wirthmüller isomorphism reads G_{+} ∧_{H} X ≃ F_{H}(G_{+}, X ∧ S^{L}) and provides the well-known Wirthmüller isomorphism between induction and coinduction, up to a twist by S^{L}. If H has finite index in G (e. g. if G is a finite group) then L = 0 and ω_{f} ≅ 𝟙.

4.6. Example (Finite group schemes). Let H be a closed subgroup of a finite group scheme G and consider their stable representation categories, as in Example 2.11. As discussed in [Jan87, Chapter 8], the restriction functor f^{∗} : Stab(𝕜G) → Stab(𝕜H) provides another example of Theorem 1.9. If δ_{G} denotes the unimodular character of the finite group scheme G then the relative dualizing object ω_{f} is δ_{G}|_{H} · δ_{H}^{−1}. A finite group scheme is said to be “unimodular” if its unimodular character is trivial, which is equivalent to the group algebra being a symmetric algebra. This is the case for instance for (discrete) finite groups.

4.7. Example (Motivic homotopy theory). Let 𝕜 be a field of characteristic zero, and let SH^{𝔸^{1}}(𝕜) denote the stable 𝔸^{1}-homotopy category over 𝕜, as in Example 2.12. For any finite extension i : 𝕜 ↪ 𝕜′, the base change functor i^{∗} : SH^{𝔸^{1}}(𝕜) → SH^{𝔸^{1}}(𝕜′) provides another example of Theorem 1.9. In this example, the relative dualizing object ω_{f} is the unit object 𝟙. See [Hu01] for further details.

4.8. Example (Cohomology rings of classifying spaces). Let H ↪ G be an inclusion of connected compact Lie groups, and consider the restriction map H^{∗}(BG; ℚ) → H^{∗}(BH; ℚ) on rational cohomology rings – regarded as commutative dg-algebras over ℚ with zero differentials. We are then in the situation of Example 3.26, with f^{∗} : D(H^{∗}(BG; ℚ)) → D(H^{∗}(BH; ℚ)) the derived extension-of-scalars. Since the cohomology rings are polynomial algebras, it follows from [DW98, §11] and Venkov’s theorem that H^{∗}(BH; ℚ) is a finitely generated free H^{∗}(BG; ℚ)-module, and thus a compact object of the derived category D(H^{∗}(BG; ℚ)). Hence we have GN-duality. In fact, ω_{f} = Σ^{−d}𝟙 where d = dim(G/H). Indeed, the Eilenberg-Moore spectral sequence associated to the fibration G/H → BH → BG collapses and provides an isomorphism H^{∗}(BH; ℚ) ≃ H^{∗}(BG; ℚ) ⊗_{ℚ} H^{∗}(G/H; ℚ) (cf. [McC01, Thm. 8.1]). Moreover, since our groups are connected, the manifold G/H is orientable. Hence, by Poincaré duality its Betti numbers are symmetric and one sees that the graded dual of H^{∗}(G/H; ℚ) is Σ^{−d}H^{∗}(G/H; ℚ). Extending scalars along ℚ → H^{∗}(BG; ℚ), it follows that Σ^{−d}H^{∗}(BH; ℚ) is the dual of H^{∗}(BH; ℚ) in the derived category D(H^{∗}(BG; ℚ)) and the identification ω_{f} ≅ Σ^{−d}𝟙 follows (cf. (1.6)).

As a concrete example, take G = U(n) and let T ≤ G be the maximal torus consisting of the diagonal unitary matrices. The Weyl group W = N_{G}(T)/T is the symmetric group S_{n} and acts on T by permuting the (diagonal) entries. The cohomology ring H^{∗}(BT; ℚ) is the polynomial algebra ℚ[x_{1}, …, x_{n}] with all generators in degree 2 and with the Weyl group acting by permuting the generators. Similarly, H^{∗}(BU(n); ℚ) is a polynomial algebra ℚ[c_{1}, …, c_{n}] with generators the universal Chern classes c_{i} (of degree 2i). The map H^{∗}(BU(n); ℚ) → H^{∗}(BT; ℚ) is a monomorphism and provides an isomorphism ℚ[c_{1}, …, c_{n}] ⟶^{∼} ℚ[x_{1}, …, x_{n}]^{W} onto the subalgebra of W -invariant polynomials (a. k. a. symmetric polynomials), sending c_{i} to the ith elementary symmetric polynomial. Here d = dim(G/T) = n^{2} − n. In this example, one can see directly that ω_{f} = Σ^{−n(n−1)}𝟙 by the same method as in Example 3.27 using the fact that ℚ[x_{1}, …, x_{n}] is a free ℚ[c_{1}, …, c_{n}]-module with basis given by the collection of monomials {x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}} | 0 ≤ a_{i} ≤ n − i}. Just note that the degrees of the monomial generators range from 0 to n(n − 1) (recall that |x_{i}| = 2) and that the number of monomial generators of degree i equals the number of generators of degree n(n − 1) − i.

4.9. Example (Highly-structured cochains). Fix a field 𝕜 and let H𝕜 denote the associated Eilenberg-MacLane spectrum. For any space X, the function spectrum C^{∗}(X) := F (Σ^{∞}X_{+}, H𝕜) is a highly structured commutative H𝕜-algebra and we can consider its derived category D(C^{∗}(X)) as in Example 2.13. Note that π_{−n}(C^{∗}(X)) ≅ H^{n}(X; 𝕜) and we think of C^{∗}(X) as the spectrum of “cochains” on X. In particular, for a compact Lie group G we can consider the derived category D(C^{∗}(BG)). For any closed subgroup H ≤ G, the map BH → BG induces a map of commutative H𝕜-algebras C^{∗}(BG) → C^{∗}(BH) and we are in the situation of Example 3.26. It follows from the fact that G/H is a finite CW-complex, that C^{∗}(BH) is compact when regarded as a C^{∗}(BG)-module. Hence we have GNduality. Moreover, as discussed in [BG14, §6], the relative dualizing object is invertible and we have an isomorphism ω_{f} ≅ F (BH^{−L}, H𝕜) where L denotes the tangent H-representation of G/H at the identity coset (already seen in Example 4.5) and BH^{−L} is the associated Thom spectrum. Moreover, if our groups are connected then this simplifies to ω_{f} ≅ Σ^{−d}(C^{∗}(BH)) = Σ^{−d}𝟙 where d = dim(G/H).

In fact, when 𝕜 = ℚ this example can be united with the previous Example 4.8. Indeed, for any compact connected Lie group G, the results of [Shi07], together with the intrinsic formality of the polynomial algebra H^{∗}(BG; ℚ), provides an equivalence D(C^{∗}(BG)) ≅ D(H^{∗}(BG; ℚ)) under which the (derived) extensions of scalars along C^{∗}(BG) → C^{∗}(BH) and along H^{∗}(BG; ℚ) → H^{∗}(BH; ℚ) coincide.

4.10. Remark. In fact, the last two examples can be connected with Example 4.5 by using Greenlees and Shipley’s work on algebraic models for free rational G-spectra. See [GS14, Gre14] for details.

Finally, we provide an example of a functor f^{∗} : 𝒟 → 𝒞 satisfying Grothendieck-Neeman duality for which ω_{f} is not invertible.

4.11. Example. Let ϕ : B → A be a morphism of commutative rings, as in Example 3.23, and assume that the induced pull-back functor f^{∗} = Spec(ϕ)^{∗} = A ⊗_{B}^{L} − : D(B) → D(A) satisfies Grothendieck-Neeman duality, i.e. that A is (finitely generated and) perfect over B (see Ex. 3.23). For simplicity assume that B = 𝕜 is a field, so that A is a finite-dimensional commutative 𝕜-algebra. In this case ω_{f} is the A-module R Hom_{B}(A, B) = Hom_{𝕜}(A, 𝕜) =: A^{⋆}, the 𝕜-linear dual of A, seen as an object of D(A). As A has Krull dimension zero, the Picard group of A is trivial, so ω_{f} is invertible (i.e. perfect!) only if A^{⋆} ≅ A as A-modules. For an explicit example where this is not the case, we can take the three-dimensional 𝕜-algebra A := 𝕜[t, s]/(s^{2}, t^{2}, st), which has the basis {1, s, t}; then A^{⋆} has the dual basis {1^{⋆}, s^{⋆}, t^{⋆}} and the A-action determined by s · s^{⋆} = 1^{⋆}, t · t^{⋆} = 1^{⋆}, s · t^{⋆} = s · 1^{⋆} = t · s^{⋆} = t · 1^{⋆} = 0. Since (s · A^{⋆}) ∩ (t · A^{⋆}) ∋ 1^{⋆} ≠ 0, this intersection is non-zero. On the other hand, (s · A) ∩ (t · A) = 0, hence A^{⋆} ≄ A as an A-module. (From a traditional point of view: if A is a finite-dimensional algebra, A^{⋆} has finite A-injective dimension and in fact any of its finite injective resolutions is a dualizing complex for A, in the classical sense; see [Bou07, Prop. X.9.1(b) and Ex. X.9.10(b)].)

5. Grothendieck duality on subcategories

In this section we consider subcategories 𝒞_{0} ⊂ 𝒞 admitting a dualizing object κ and study the behavior of such structures under our functors f^{∗} and f_{∗}.

5.1. Definition. Let 𝒞_{0} ⊂ 𝒞 be a 𝒞^{c}-submodule, i.e. a thick triangulated subcategory of our big category 𝒞 such that c ⊗ x ∈ 𝒞_{0} for all x ∈ 𝒞_{0} and all compact c ∈ 𝒞^{c}. An object κ ∈ 𝒞_{0} is called a dualizing object for 𝒞_{0} if the κ-twisted duality Δ_{κ} := hom_{𝒞} (−, κ) defines an anti-equivalence on 𝒞_{0}:

(5.2)  Δ_{κ} = hom_{𝒞} (−, κ) : 𝒞_{0}^{op} ⟶^{∼} 𝒞_{0}.

In Section 7, we will consider the more general situation of an “external” dualizing object κ ∈ 𝒞 by dropping the assumption that κ belongs to the subcategory 𝒞_{0} itself. (Note that if 𝟙 belongs to 𝒞_{0} then necessarily κ ≅ Δ_{κ}(𝟙) ∈ 𝒞_{0}.)

5.3. Remark. Because 𝒞(x, Δ_{κ}(y)) ≅ 𝒞(x ⊗ y, κ) ≅ 𝒞(y ⊗ x, κ) ≅ 𝒞(y, Δ_{κ}(x)) ≅ 𝒞^{op}(Δ_{κ}(x), y), we see that Δ_{κ} is adjoint to itself and we have a canonical natural morphism

(5.4)  ϖ_{κ} : x ⟶ Δ_{κ}Δ_{κ}(x)

for all x ∈ 𝒞, which is both the unit and the counit of this self-adjunction. It satisfies Δ_{κ}(ϖ) ∘ ϖ_{Δ_{κ}} = id_{Δ_{κ}} : Δ_{κ} → Δ_{κ}Δ_{κ}Δ_{κ} → Δ_{κ} by the unit-counit relation. We say that x ∈ 𝒞 is κ-reflexive if this morphism ϖ_{κ} is an isomorphism (at x).

5.5. Lemma. For a 𝒞^{c}-submodule 𝒞_{0} ⊂ 𝒞, an object κ ∈ 𝒞_{0} is a dualizing object if and only if the following two conditions are satisfied:

(i) For every x ∈ 𝒞_{0}, the κ-twisted dual Δ_{κ}(x) = hom_{𝒞} (x, κ) belongs to 𝒞_{0}. (ii) Every x ∈ 𝒞_{0} is κ-reflexive, i.e. ϖ_{κ} : x →^{∼} Δ_{κ}Δ_{κ}(x).

Proof. In the adjunction Δ_{κ} : 𝒞_{0}^{op} ⇄ 𝒞_{0} : Δ_{κ} the unit and counit are isomorphisms if and only if Δ_{κ} is an equivalence. □

5.6. Example. For the subcategory 𝒞_{0} = 𝒞^{c} of all rigids, an object κ ∈ 𝒞_{0} is dualizing if and only if it is invertible (cf. [FHM03, Cor. 5.9]). In particular 𝒞_{0} = 𝒞^{c} always admits κ = 𝟙 as dualizing object.

5.7. Lemma. There is a canonical natural isomorphism Δ_{κ}(x) ⊗ Δ(c) ≅ Δ_{κ}(x ⊗ c) for all x, κ ∈ 𝒞 and all rigid c ∈ 𝒞^{c}, where Δ = Δ_{𝟙} is the usual dual.

Proof. This is standard; see [HPS97, Thm. A.2.5.(d)]. The map hom(x, κ) ⊗ hom(c, 𝟙) → hom(x ⊗ c, κ) is adjoint to the map hom(x, κ) ⊗ hom(c, 𝟙) ⊗ x ⊗ c ≅

hom(x, κ) ⊗ x ⊗ hom(c, 𝟙) ⊗ c ⟶^{ev ⊗ ev} κ ⊗ 𝟙 ≅ κ. □

5.8. Remark. It follows that if κ is a dualizing object for 𝒞_{0} then so is κ ⊗ u for every ⊗-invertible u. In algebraic geometry, dualizing complexes are unique up to tensoring by an invertible object; see [Nee10, Lem. 3.9]. For a general 𝒞_{0} this seems to be over-optimistic, although we can prove the following variant, replacing an equivalence of the form u ⊗ − by one of the form hom_{𝒞} (u, −).

5.9. Proposition. Let 𝒞_{0} ⊆ 𝒞 be a 𝒞^{c}-submodule containing 𝒞^{c} (that is, 𝟙 ∈ 𝒞_{0}). Let κ and κ′ be two dualizing objects for 𝒞_{0}. Let u := Δ_{κ′}(κ) = hom_{𝒞} (κ, κ′) and v := Δ_{κ}(κ′) = hom_{𝒞} (κ′, κ), both in 𝒞_{0}. Then v ≅ Δu and u ≅ Δv and the restrictions to 𝒞_{0} of the functors F_{u} := hom_{𝒞} (u, −) and F_{v} := hom_{𝒞} (v, −) yield mutually inverse equivalences F_{u} : 𝒞_{0} ⟷^{∼} 𝒞_{0} : F_{v} such that κ ≅ F_{u}(κ′). Moreover, we have Δ_{κ′} ≅ F_{v} ∘ Δ_{κ} ≅ Δ_{κ} ∘ F_{u}.

Proof. Let us write [−, −] for hom_{𝒞} (−, −) to save space. From the equivalence [−, κ] : 𝒞_{0}^{op} →^{∼} 𝒞_{0}, one deduces a natural isomorphism

(5.10)  β : [x, y] →^{∼} [[y, κ], [x, κ]]

for x, y ∈ 𝒞_{0}. Indeed, the morphism β is (double) adjoint to the (double) evaluation [x, y] ⊗ [y, κ] ⊗ x ⟶ κ. When tested on Hom_{𝒞} (c, −) for c ∈ 𝒞^{c}, we obtain

[figure: Hom_{𝒞} (c, [x, y]); ≅; Hom_{𝒞} (c,β); Hom_{𝒞} (c ⊗ x, y); Hom_{𝒞} ([y, κ], [c ⊗ x, κ]); Hom_{𝒞} (c, [[y, κ], [x, κ]]); ≅; Hom_{𝒞} (c ⊗ [y, κ], [x, κ]); ≅; Hom_{𝒞} ([y, κ], Δc ⊗ [x, κ]).; ≅]

The dotted arrow thus obtained is nothing but the map on morphisms sets induced by the contravariant functor [−, κ] : 𝒞_{0}^{op} → 𝒞_{0}. Since the latter is an equivalence, this dotted map is a bijection hence so is the left vertical map. Since this holds for every c ∈ 𝒞^{c} and since 𝒞 is compactly generated, the morphism β is an isomorphism. As the same is true for [−, κ′] we have a natural isomorphism

(5.11)  [[y, κ], [x, κ]] ≅ [x, y] ≅ [[y, κ′], [x, κ′]]

for all x, y ∈ 𝒞_{0}. Since 𝟙 ∈ 𝒞_{0}, we have [κ, κ] ≅ Δ_{κ}^{2}(𝟙) ≅ 𝟙 and similarly [κ′, κ′] ≅ 𝟙. Thus, plugging x = κ and y = κ′ (resp. x = κ′ and y = κ) in (5.11) we get the announced isomorphisms u ≅ Δv and v ≅ Δu. Plugging instead x = 𝟙 and y = κ (resp. y = κ′) in (5.11), we obtain

(5.12)  κ ≅ [u, κ′]  and  κ′ ≅ [v, κ].

Now compute the composite equivalence \xymatrix{𝒞_{0} \ar[r]_{≃}^{[−,κ]} & 𝒞_{0}^{op} \ar[r]_{≃}^{[−,κ′]} & 𝒞_{0}.} For all x ∈ 𝒞_{0},

[[x, κ], κ′] ≅^{(5.12)} [[x, κ], [v, κ]] ≅^{(5.10)} [v, x].

This proves that F_{v} := [v, −] ≅ Δ_{κ′}Δ_{κ} and F_{u} := [u, −] ≅ Δ_{κ}Δ_{κ′} define equivalences on 𝒞_{0}, which satisfy the desired relations since Δ_{κ}^{2} ≅ Id ≅ Δ_{κ′}^{2}. □

5.13. Remark. One can deduce from Proposition 5.9 that u = hom_{𝒞} (κ, κ′) is ⊗- invertible, and that κ ⊗ u ≅ κ′, if 𝒞_{0} satisfies any of the following properties:

(i) u ⊗ 𝒞_{0} ⊂ 𝒞_{0} (for instance if u belongs to 𝒞^{c}); (ii) 𝒞_{0} cogenerates 𝒞, i.e. for t ∈ 𝒞 if Hom_{𝒞} (t, x) = 0 for all x ∈ 𝒞_{0} then t = 0; (iii) if an object a ∈ 𝒞, not necessarily in 𝒞_{0}, admits a natural isomorphism

hom_{𝒞} (a, x) ≅ x for all x ∈ 𝒞_{0}, then a ≅ 𝟙. This is left as an easy exercise for the interested reader. Note in particular that condition (i) holds if 𝒞_{0} satisfies 𝒞_{0} ⊗ 𝒞_{0} ⊆ 𝒞_{0}. This assumption appears, for example, in [BD13]; however, 𝒞_{0} ⊗ 𝒞_{0} ⊆ 𝒞_{0} is not true in algebraic geometry for 𝒞_{0} = D^{b}(coh X). Nevertheless, this important example can also be derived from Proposition 5.9:

5.14. Corollary ([Nee10, Lem. 3.9]). If X is a noetherian scheme admitting two dualizing complexes κ and κ′, then there exists a ⊗-invertible ℓ ∈ D^{perf}(X) (a shift of a line bundle on each connected component of X) such that κ′ ≅ κ ⊗ ℓ.

Proof. According to the definition used in [Nee10], which is slightly more general than the classical one, a dualizing complex for X is an (internal) dualizing object for 𝒞_{0} = D^{b}(coh X). It suffices to prove that ℓ := u = hom(κ, κ′) is ⊗-invertible, since Proposition 5.9 then implies that κ ≅ hom(u, κ′) ≅ Δu ⊗ κ′ = u^{−1} ⊗ κ′. To this end, we appeal to the following criterion: Over a noetherian scheme X, a complex x ∈ D^{b}(coh X) is ⊗-invertible iff both

(a) x is (𝟙-)reflexive, x →^{∼} Δ^{2}(x), and

(b) 𝟙 is x-reflexive, 𝟙 →^{∼} Δ_{x}^{2}(𝟙) = hom(x, x). For the affine case, see [AIL10, Cor. 5.7]. The criterion globalizes because the canonical morphism ϖ commutes with localization to an open subscheme [AIL11, §1.3 and Rem. 1.5.5]; cf. [CH09, Thm. 4.1.2].

By Proposition 5.9, we have isomorphisms u ≅ Δ(v) ≅ Δ^{2}(u) and 𝟙 ≅ hom(u, u). In order to conclude that u is ⊗-invertible by the above criterion, we must prove that the latter isomorphisms are instances of the canonical maps ϖ. This verification is easy for the second map, but for the first one it appears to be rather involved. Fortunately, we can avoid it altogether: By the local nature of ϖ, we may reduce to the affine case where, by [AIL10, Prop. 2.3], the existence of any isomorphism x ≅ Δ_{y}^{2}(x) implies that x is y-reflexive (for x, y ∈ D^{b}(coh X)). □

* * *

Let us now “move” the above subcategories with duality under f^{∗} and f_{∗}. In order to do this, and for later use in Theorem 7.1, we need to clarify the following:

5.15. Theorem. Let f^{∗} : 𝒟 → 𝒞 be as in our basic Hypothesis 1.2 and κ ∈ 𝒟. Recall the two adjunctions f^{∗} ⊣ f_{∗} ⊣ f^{(1)} of Corollary 2.14, as well as their internal realizations (2.17) and (2.18). The latter yields a canonical natural isomorphism

ζ : Δ_{κ} ∘ f_{∗} →^{∼} f_{∗} ∘ Δ_{f^{(1)}(κ)}. (5.16)

This isomorphism is compatible with the canonical maps ϖ of Δ_{κ} and Δ_{κ′} for κ′ = f^{(1)}(κ). This means that the following diagram commutes, for all x ∈ 𝒞:

\xymatrix{f_{∗}(x) \ar[d]_{ϖ_{f∗(x)}} \ar[r]^{f_{∗}(ϖ_{x})} & f_{∗}Δ_{κ′}Δ_{κ′}(x) \ar[d]_{≅}^{ζ_{Δ_{κ′} (x)}} \\ Δ_{κ}Δ_{κ}f_{∗}(x) \ar[r]^{≅}_{Δ_{κ}(ζ)} & Δ_{κ}f_{∗}Δ_{κ′}(x).}

(5.17)

In other words, f_{∗} : 𝒞 → 𝒟 is a duality-preserving functor in the sense of [CH09].

Proof. It is far from obvious to verify that (5.17) commutes, but fortunately this has already been proved, in even greater generality, in [CH09, Thm. 4.2.9]. Indeed, we are proving that the functor f_{∗} : 𝒞 → 𝒟, or rather the pair (f_{∗}, ζ), is a “dualitypreserving functor” in the sense of [CH09, Def. 2.2.1] between the “categories with (weak) duality” (𝒞, Δ_{κ′}) and (𝒟, Δ_{κ}). The hypotheses (A_{f}), (B_{f}) and (C_{f}) of the cited theorem are all satisfied in our situation by virtue of Corollary 2.14 (note that our f^{(1)} is denoted f^{!} in loc. cit.). To see that the conclusion of the cited theorem applies here, we must still verify that the natural maps we denote by π and ζ coincide with the homonymous maps of [CH09]. For π, it suffices to inspect the definitions in Proposition 2.15 and [CH09, Prop. 4.2.5] and note that they agree. For ζ, we must compare our definition of (2.18) as a conjugate of π (which then specializes to (5.16)) with the definition of ζ given in [CH09, Thm. 4.2.9]. In more detail, we must show that our map (2.18) coincides with the composite

f_{∗}hom(x, f^{(1)}y) → hom(f_{∗}x, f_{∗}f^{(1)}y) ⟶^{hom(1,ϵ)} hom(f_{∗}x, y)

where the first map is the canonical map f_{∗}hom(a, b) → hom(f_{∗}a, f_{∗}b) induced by the (lax) monoidal structure on f_{∗}. This is readily checked from the definition of (2.18) in terms of π, together with the first diagram in (3.21). □

5.18. Definition. If 𝒟_{0} is a 𝒟^{c}-submodule of 𝒟, we define its compact pull-back along f_{∗} as the following full subcategory of 𝒞:

f^{#}(𝒟_{0}) := {x ∈ 𝒞 | f_{∗}(c ⊗ x) ∈ 𝒟_{0} for all c ∈ 𝒞^{c}} .

One sees immediately that f^{#}(𝒟_{0}) is a 𝒞^{c}-submodule of 𝒞, because 𝒞^{c} is one.

We can use this compact pull-back f^{#} to rephrase Grothendieck-Neeman duality:

5.19. Proposition. Let f^{∗} : 𝒟 → 𝒞 be as in Hypothesis 1.2.

(a) The functor f^{∗} satisfies Grothendieck-Neeman duality (Theorem 3.3) if and only if the following inclusion holds: 𝒞^{c} ⊂ f^{#}(𝒟^{c}).

(b) If 𝒞^{c} = f^{#}(𝒟^{c}) then we have the Wirthmüller isomorphism (Theorem 1.9).

Proof. Hypothesis (GN 2) of Theorem 1.7 reads: f_{∗}(c) ∈ 𝒟^{c} for all c ∈ 𝒞^{c}. Since 𝟙 ∈ 𝒞^{c} and since 𝒞^{c} is stable under the tensor product, this condition is equivalent to: f_{∗}(c ⊗ x) ∈ 𝒟^{c} for all x, c ∈ 𝒞^{c}. The latter exactly means 𝒞^{c} ⊂ f^{#}(𝒟^{c}) by Definition 5.18. Hence (a). For (b), it suffices to note that 𝒞^{c} ⊃ f^{#}(𝒟^{c}) is precisely the sufficient condition (3) in Theorem 1.9. □

Furthermore, compact pullback is compatible with composition of functors:

5.20. Proposition. Consider two composable functors ℰ ⟶^{g^{∗}} 𝒟 ⟶^{f^{∗}} 𝒞, both satisfying Hypothesis 1.2 and with composite f^{∗}g^{∗} =: (gf)^{∗}, and let ℰ_{0} be any ℰ^{c}- submodule of ℰ. Then (gf)^{#}(ℰ_{0}) = f^{#}(g^{#}(ℰ_{0})).

Proof. Notice that the composite (gf)^{∗} also satisfies Hypothesis 1.2, and that its right adjoint must be (gf)_{∗} ≅ g_{∗}f_{∗} by the uniqueness of adjoints. Thus x ∈ 𝒞 belongs to (gf)^{#}(ℰ_{0}) iff g_{∗}f_{∗}(c ⊗ x) ∈ ℰ_{0} for all c ∈ 𝒞^{c}.

On the other hand: x ∈ f^{#}(g^{#}(ℰ_{0})) iff f_{∗}(c ⊗ x) ∈ g^{#}(ℰ_{0}) for all c ∈ 𝒞^{c}, iff g_{∗}(d ⊗ f_{∗}(c ⊗ x)) ∈ ℰ_{0} for all c ∈ 𝒞^{c} and d ∈ 𝒟^{c}. By the projection formula (2.16), we see that g_{∗}(d ⊗ f_{∗}(c ⊗ x)) ≅ g_{∗}f_{∗}(f^{∗}(d) ⊗ c ⊗ x).

Since each f^{∗}(d) ⊗ c as above is compact in 𝒞, and since every compact of 𝒞 has this form (choose d = 𝟙_{𝒟} ), the two conditions on x are equivalent. □

Let us give an example of this compact pullback f^{#} in algebraic geometry.

5.21. Theorem. Let f : X → Y be a morphism of noetherian schemes and let f^{∗} : 𝒟 = D_{Qcoh}(Y) ⟶ D_{Qcoh}(X) = 𝒞 be the induced functor (see Ex. 3.22).

(a) Suppose that f : X → Y is proper. Then f_{∗} : 𝒞 → 𝒟 maps D^{b}(coh X) into D^{b}(coh Y). Moreover, for every object x ∈ D^{b}(coh X) and every perfect c ∈ D_{Qcoh}(X)^{c} we have f_{∗}(c ⊗ x) ∈ D^{b}(coh Y).

(b) Suppose that f : X → Y is projective. Then the following converse to (a) holds : If x ∈ D_{Qcoh}(X) is such that f_{∗}(c ⊗ x) ∈ D^{b}(coh Y) for every perfect c ∈ D_{Qcoh}(X)^{c} then x ∈ D^{b}(coh X).

In the notation of Definition 5.18, we have for f : X → Y projective that

f^{#}( D^{b}(coh Y)) = D^{b}(coh X).

Proof. For (a), the question being local in the base Y, we can assume that Y = Spec(A) is affine. For any coherent sheaf F ∈ coh(X), the A-modules R^{i}f_{∗}F are finitely generated and vanish for i >> 0, by [Gro63, III.3.2.3]. It follows that f_{∗}F = Rf_{∗}F is bounded coherent. Hence so is f_{∗}(x) for any x ∈ D_{Qcoh}(X) contained in the thick subcategory generated by coherent sheaves, which is precisely D^{b}(coh X). The “moreover part” follows immediately since D^{b}(coh X) is a D^{perf}(X)-submodule of D_{Qcoh}(X), where D^{perf}(X) = D_{Qcoh}(X)^{c}.

For (b), in view of Proposition 5.20, and since we can decompose f into a closed immersion followed by the structure morphism ℙ_{Y}^{n} → Y, we treat the two cases separately. For f : X → Y a closed immersion, the result is straightforward since f_{∗} itself detects boundedness and coherence. (One can reduce to Y affine – in any case, f_{∗} commutes with taking homology and detects coherence, so f_{∗}(x) ∈ D^{b}(coh Y) forces x to be bounded with coherent homology, i.e. to be in D^{b}(coh X).) For the projection f : ℙ_{Y}^{n} → Y, we can use the resolution of the diagonal O_{Δ} in ℙ^{n}×_{Y} ℙ^{n} by objects of the form p_{1}^{∗}d_{i} ⊗ p_{2}^{∗}c_{i} where p_{1}, p_{2} : ℙ^{n} × ℙ^{n} → ℙ^{n} are the two projections and d_{i} = Ω^{i}(i) and c_{i} = O(−i) are vector bundles over ℙ^{n} and in particular compact objects; see [Beĭ84]. Hence, since every x ∈ D_{Qcoh}(ℙ_{Y}^{n}) is isomorphic to (p_{2})_{∗}(O_{Δ} ⊗ p_{1}^{∗}(x)), we see that x belongs to the thick subcategory generated by the (p_{2})_{∗}(p_{1}^{∗}d_{i} ⊗ p_{2}^{∗}c_{i} ⊗ p_{1}^{∗}(x)). Computing the latter using the projection formula and flat base change (in the cartesian square for ℙ^{n} ×_{Y} ℙ^{n} over Y), we get

(p_{2})_{∗}(p_{1}^{∗}d_{i} ⊗ p_{2}^{∗}c_{i} ⊗ p_{1}^{∗}(x)) ≅ (p_{2})_{∗}(p_{1}^{∗}(x ⊗ d_{i})) ⊗ c_{i} ≅ f^{∗}(f_{∗}(x ⊗ d_{i})) ⊗ c_{i} .

In particular, as soon as f_{∗}(x ⊗ d_{i}) ∈ D^{b}(coh Y) then all the objects above belong to D^{b}(coh ℙ_{Y}^{n}), hence so does x itself. □

5.22. Corollary. Let f : X → S be a projective morphism of noetherian schemes, with S regular (for instance S = Spec(𝕜) for 𝕜 a field). Then D^{b}(coh X) is equal to { x ∈ D_{Qcoh}(X) | f_{∗}(c ⊗ x) ∈ D^{perf}(S) for all c ∈ D^{perf}(X) }.

Proof. In this case, D^{perf}(S) = D^{b}(coh S) and we can apply Theorem 5.21. □

* * *

We now turn to the interaction between the two notions discussed above, namely that of dualizing object and that of compact pullback of subcategories.

5.23. Theorem. Let f^{∗}: 𝒟 → 𝒞 be as in Hypothesis 1.2. Let 𝒟_{0} ⊂ 𝒟 be a 𝒟^{c}- submodule equipped with a dualizing object κ ∈ 𝒟_{0} (Def. 5.1) and consider the following two possible properties of an object x ∈ 𝒞:

(i) x ∈ f^{#}(𝒟_{0}).

(ii) x is f^{(1)}(κ)-reflexive: x →^{∼} Δ_{f^{(1)}(κ)}Δ_{f^{(1)}(κ)}(x).

Then we have :

(a) If 𝟙 ∈ 𝒟_{0} then (i) implies (ii).

(b) If 𝒟_{0} consists precisely of the κ-reflexive objects of 𝒟, then (ii) implies (i).

Proof. Let us prove (i)⇒(ii) when 𝟙 ∈ 𝒟_{0}, and write κ′ := f^{(1)}(κ) for short. Consider x, y, c ∈ 𝒞 with c compact, y arbitrary, and x ∈ f^{#}(𝒟_{0}), which implies in particular that f_{∗}(x ⊗ Δc) ∈ 𝒟^{c}. We obtain the following isomorphism:

𝒞(c, x) ≅ 𝒞(𝟙, x ⊗ Δc)  c ∈ 𝒞^{c} is rigid
≅ 𝒟(𝟙, f_{∗}(x ⊗ Δc))  𝟙 ≅ f^{∗}𝟙 and f^{∗} ⊣ f_{∗}
≅ 𝒟(Δ_{κ}f_{∗}(x ⊗ Δc), κ) 𝟙 and f_{∗}(x ⊗ Δc) ∈ 𝒟_{0} and Δ_{κ} : 𝒟_{0}^{op} →^{∼} 𝒟_{0}
≅ 𝒟(f_{∗}Δ_{κ′}(x ⊗ Δc), κ) by (5.16), special case of (2.18)
≅ 𝒞(Δ_{κ′}(x ⊗ Δc), κ′) f_{∗} ⊣ f^{(1)} and f^{(1)}(κ) = κ′
≅ 𝒞(Δ_{κ′}(x) ⊗ c, κ′)  c ∈ 𝒞^{c} is rigid and Lemma 5.7
≅ 𝒞(c, Δ_{κ′}Δ_{κ′}(x))  ⊗ ⊣ hom_{𝒞} .

By following through this composite isomorphism, one can check that it is induced by the canonical map (5.4). Indeed, choosing an arbitrary morphism φ : c → x and writing [−, −] for hom(−, −), the relevant diagram can be checked using the naturality of the isomorphism f_{∗}[a, f^{(1)}b] ≅ [f_{∗}a, b] in (2.18) with respect to the morphism 𝟙 →^{η} c ⊗ Δc ⟶^{φ⊗1} x ⊗ Δc, together with the following three diagrams:

[figure: [c, κ′] ⊗ c; ev; ∼; 5.7; [c ⊗ Δc, κ′]; [η,1]; [𝟙, κ′]; ∼; f_{∗}⊣f^{(1)}; f^{(1)}f_{∗}[𝟙, κ′]; ∼; (2.18); f^{(1)}[f_{∗}𝟙, κ]; f^{∗}⊣f_{∗}; κ′; ∼; [𝟙, κ′]; κ′; f^{(1)}κ; ∼; f^{(1)}[𝟙, κ]]

\xymatrix{c \ar[r]^{φ} \ar[d]^{coev} & x \ar[r]^{coev} & [[x, κ′], [x, κ′] ⊗ x] \ar[d]^{[1,ev]} \\ [[x, κ′], [x, κ′] ⊗ c] \ar[r]^{[1,[φ,1]⊗1]} & [[x, κ′], [c, κ′] ⊗ c] \ar[r]^{[1,ev]} & [[x, κ′], κ′]].}

The first diagram can be checked using the definition of [c, κ′] ⊗ c ≅ [c ⊗ Δc, κ′] (cf. Lemma 5.7) and dinaturality of coevaluation with respect to η : 𝟙 → c ⊗ Δc. Similarly, the second diagram can be checked using the definition of f_{∗}[𝟙, κ′] ≅ [f_{∗}𝟙, κ] together with dinaturality of coevaluation with respect to 𝟙 → f_{∗}𝟙. Finally, the last diagram follows from dinaturality of (co)evaluation and naturality of evaluation with respect to φ : c → x. As 𝒞 is compactly generated, the canonical map (5.4) is invertible for any x, as claimed.

For (b), in order to prove the conditional implication (ii)⇒(i), assume that x ∈ 𝒞 is κ′-reflexive and let c ∈ 𝒞^{c} be a compact object. We must show that f_{∗}(x ⊗ c) belongs to 𝒟_{0}. By the extra hypothesis, it suffices to show that f_{∗}(x ⊗ c) ∈ 𝒟 is κreflexive. We have the following composite isomorphism starting with κ′-reflexivity of x and 𝟙-reflexivity of c:

f_{∗}(x ⊗ c) ≅ f_{∗}(Δ_{κ′}Δ_{κ′}(x) ⊗ ΔΔ(c))
≅ f_{∗}Δ_{κ′}Δ_{κ′}(x ⊗ c)  Lemma 5.7 twice
≅ Δ_{κ}Δ_{κ}f_{∗}(x ⊗ c)  (5.16) twice.

To check that this isomorphism f_{∗}(x ⊗ c) →^{∼} Δ_{κ}Δ_{κ}f_{∗}(x ⊗ c) coincides with the canonical map (5.4), it suffices to check that it is adjoint to the evaluation map [f_{∗}(x ⊗ c), κ] ⊗ f_{∗}(x ⊗ c) ⟶^{ev} κ. This can be accomplished from the definitions by using dinaturality of evaluation with respect to f_{∗}Δ_{κ′}(x ⊗ c) ≅ Δ_{κ}(f_{∗}(x ⊗ c)) together with the following two commutative diagrams:

\xymatrix{f_{∗}[a, f^{(1)}(b)] ⊗ f_{∗}(a) \ar[r]^{lax} \ar[d]^{≅} & f_{∗}([a, f^{(1)}(b)] ⊗ a] \ar[r]^{f_{∗} ev} & f_{∗}f^{(1)}(b) \ar[d]^{ϵ} \\ [f_{∗}(a), b] ⊗ f_{∗}(a) \ar[rr]^{ev} &  & b}

and

\xymatrix{Δ_{κ}(x ⊗ c) ⊗ x ⊗ c \ar[r]^{1⊗ϖ_{κ}⊗ϖ} \ar[dd]_{≅} & Δ_{κ}(x ⊗ c) ⊗ Δ_{κ}^{2}x ⊗ Δ^{2}c \ar[d]^{switch} \\  & Δ_{κ}^{2}x ⊗ Δ^{2}c ⊗ Δ_{κ}(x ⊗ c) \ar[d]_{≅} \\ κ & Δ_{κ}^{2}(x ⊗ c) ⊗ Δ_{κ}(x ⊗ c). \ar[l]_{ev}}

The first diagram can be checked in a straightforward manner by using the definition of f_{∗}[a, f^{!}b] ≅ [f_{∗}a, b] in (2.18). The second diagram can be checked using the definition of the maps ϖ_{κ} : x → Δ_{κ}^{2}x and ϖ : c → Δ^{2}c together with the fact that

\xymatrix{[a, b] ⊗ [a′, b′] ⊗ a ⊗ a′ \ar[r]^{switch} \ar[d]_{≅} & [a, b] ⊗ a ⊗ [a′, b′] ⊗ a′ \ar[d]^{ev ⊗ ev} \\ [a ⊗ a′, b ⊗ b′] ⊗ a ⊗ a′ \ar[r]^{ev} & b ⊗ b′}

commutes (which can be checked from the definition).

We conclude that indeed f_{∗}(x⊗c) is κ-reflexive and therefore belongs to 𝒟_{0}. □

Let us give an example illustrating part (b) of the theorem.

5.24. Corollary. Let f : X → Spec(k) be a projective scheme over a field, and let ω_{f} ∈ D_{Qcoh}(X) denote the relative dualizing object for f^{∗}. Then x ∈ D_{Qcoh}(X) is ω_{f}-reflexive iff dim_{k} H^{i}f_{∗}(c ⊗ x) < ∞ for all c ∈ D^{perf}(X) and i ∈ ℤ.

Proof. An object of D(k) is k-reflexive iff its homology groups are all finite dimensional. Now apply Theorem 5.23 (a) & (b) to f^{∗} : D(k) → D_{Qcoh}(X). □

The next theorem is the main result of this section.

5.25. Theorem (Grothendieck duality). Let f^{∗}: 𝒟 → 𝒞 be as in our basic Hypothesis 1.2 and let κ ∈ 𝒟. Recall f^{∗} ⊣ f_{∗} ⊣ f^{(1)} from Corollary 2.14. Suppose that f^{∗} satisfies Grothendieck-Neeman duality (Theorem 1.7) and that 𝒟_{0} ⊂ 𝒟 is a 𝒟^{c}-submodule which admits κ ∈ 𝒟_{0} as dualizing object (Definition 5.1). Then

κ′ := f^{(1)}(κ) ≅ ω_{f} ⊗ f^{∗}(κ)

is a dualizing object for the 𝒞^{c}-submodule f^{#}(𝒟_{0}) ⊂ 𝒞 (Definition 5.18). In particular, f_{∗} : f^{#}(𝒟_{0}) ⟶ 𝒟_{0} is a duality-preserving exact functor between categories with duality, where f^{#}(𝒟_{0}) is equipped with the duality Δ_{κ′} and 𝒟_{0} with Δ_{κ}.

Proof. We have κ′ ≅ ω_{f} ⊗ f^{∗}(κ) by formula (3.4). Moreover, by Theorem 5.23, every object of f^{#}(𝒟_{0}) is κ′-reflexive, hence by Lemma 5.5 it remains to prove that κ′ = ω_{f} ⊗ f^{∗}κ belongs to f^{#}(𝒟_{0}) and that Δ_{κ′} preserves f^{#}(𝒟_{0}). Indeed for every compact c ∈ 𝒞^{c}, we can use the projection formula (2.16) and the ur-Wirthmüller formula (1.8) to compute

f_{∗}(ω_{f} ⊗ f^{∗}κ ⊗ c) ≅ f_{∗}(ω_{f} ⊗ c) ⊗ κ ≅ f_{(1)}(c) ⊗ κ;

this object belongs to 𝒟_{0} since κ does, since f_{(1)} preserves compacts and since 𝒟_{0} is a 𝒞^{c}-submodule. Hence κ′ = ω_{f} ⊗ f^{∗}κ belongs to f^{#}(𝒟_{0}). Finally, let us show that Δ_{κ′} preserves f^{#}(𝒟_{0}). For x ∈ f^{#}(𝒟_{0}) and c ∈ 𝒞^{c} we compute, using first Lemma 5.7 and the rigidity of c ∈ 𝒞^{c} and then (5.16):

f_{∗}(c ⊗ Δ_{κ′}(x)) ≅ f_{∗}(Δ_{κ′}(x ⊗ Δc)) ≅ Δ_{κ}(f_{∗}(x ⊗ Δc)) .

The latter belongs to 𝒟_{0} since f_{∗}(x ⊗ Δc) does by definition of x ∈ f^{#}(𝒟_{0}) and since Δ_{κ} preserves 𝒟_{0} by hypothesis. This shows Δ_{κ′}(x) ∈ f^{#}(𝒟_{0}) as wanted. □

6. Categories over a base and relative compactness

We now want to analyse a relative setting.

6.1. Definition. Let ℬ be a rigidly-compactly generated tensor triangulated category that we call the “base”. We say that 𝒞 is a ℬ-category if it comes equipped with a structure functor p^{∗} : ℬ → 𝒞 satisfying Hypothesis 1.2.

6.2. Definition. A morphism of ℬ-categories f^{∗} : (𝒟, q^{∗}) → (𝒞, p^{∗}) is a functor f^{∗} : 𝒟 → 𝒞 satisfying Hypothesis 1.2 together with an isomorphism f^{∗}q^{∗} ≅ p^{∗}. By the uniqueness property of adjoint functors, this canonically spawns isomorphisms q_{∗}f_{∗} ≅ p_{∗} and f^{(1)}q^{(1)} ≅ p^{(1)}. In particular, we have an isomorphism in 𝒞:

(6.3)  f^{(1)}(ω_{q}) = f^{(1)}q^{(1)}(𝟙_{ℬ} ) ≅ p^{(1)}(𝟙_{ℬ} ) = ω_{p} .

We can then prove the following generalization of Theorem 5.25, in which we do not assume that f^{∗} satisfies Grothendieck-Neeman duality, but only that its source and target do, with respect to their base.

6.4. Theorem. Let f^{∗} : 𝒟 → 𝒞 be a morphism of ℬ-categories (Def. 6.2). Assume that the structure morphisms p^{∗} : ℬ → 𝒞 and q^{∗} : ℬ → 𝒟 satisfy Grothendieck-Neeman duality (Thm. 1.7). Let ℬ_{0} ⊂ ℬ be a ℬ^{c}-subcategory with dualizing object κ ∈ ℬ_{0} (Def. 5.1). Let 𝒞_{0} = p^{#}ℬ_{0} and 𝒟_{0} = q^{#}ℬ_{0} be its compact pullbacks in 𝒞 and 𝒟 respectively (Def. 5.18), which admit the dualizing objects

γ := ω_{p} ⊗ p^{∗}(κ) ∈ 𝒞_{0}  and  δ := ω_{q} ⊗ q^{∗}(κ) ∈ 𝒟_{0}

respectively, by Theorem 5.25. Then f^{#}(𝒟_{0}) = 𝒞_{0} and f_{∗} restricts to a well-defined exact functor f_{∗} : 𝒞_{0} → 𝒟_{0} which is duality-preserving with respect to Δ_{γ} and Δ_{δ}.

Proof. Note that γ = ω_{p} ⊗ p^{∗}(κ) ≅ p^{(1)}(κ) ≅ f^{(1)}q^{(1)}(κ) ≅ f^{(1)}(δ). So we already know from Theorem 5.15 that f_{∗} is compatible with the dualities Δ_{γ} and Δ_{δ}. By Proposition 5.20, we know that 𝒞_{0} = p^{#}ℬ_{0} = f^{#}q^{#}ℬ_{0} = f^{#}𝒟_{0}. It follows from this and the definition of f^{#}𝒟_{0} that f_{∗}(𝒞_{0}) ⊆ 𝒟_{0} yielding the desired f_{∗} : 𝒞_{0} → 𝒟_{0}. □

An example of the above relative discussion over a base category ℬ is the situation where ℬ_{0} = ℬ^{c} and κ = 𝟙. In other words, we can assume that the base is sufficiently simple that the duality question over ℬ is solved in the “trivial” way, as in Example 5.6. This is interesting in algebraic geometry when ℬ = D_{Qcoh}(S) for S regular, as we saw in Corollary 5.22, for instance when S = Spec(𝕜) for 𝕜 a field. In that case, it is not a restriction to consider the trivial duality on ℬ, with ℬ_{0} = ℬ^{c} and κ = 𝟙. We can then pull it back to obtain a more interesting subcategory with duality 𝒞_{0} = p^{#}(ℬ_{0}) in 𝒞.

6.5. Definition. Let 𝒞 be a ℬ-category as in Definition 6.1. We define the full subcategory of 𝒞 of ℬ-relatively compact objects to be

𝒞^{c/p} := p^{#}(ℬ^{c}) = { x ∈ 𝒞 | p_{∗}(c ⊗ x) ∈ ℬ^{c} for all compact c ∈ 𝒞^{c} } .

6.6. Example. Let p : X → S be a projective morphism with S regular and let p^{∗} : ℬ = D_{Qcoh}(S) → D_{Qcoh}(X) = 𝒞. Then 𝒞^{c/p} = D^{b}(coh X) by Corollary 5.22.

6.7. Corollary. Let f^{∗} : 𝒟 → 𝒞 be a morphism of ℬ-categories (Def. 6.2) with structure morphisms p^{∗} : ℬ → 𝒞 and q^{∗} : ℬ → 𝒟.

(a) Compact pullback (Def. 5.18) preserves the subcategories of ℬ-relatively compact objects: f^{#}(𝒟^{c/q}) = 𝒞^{c/p}.

(b) Suppose that p^{∗} : ℬ → 𝒞 satisfies Grothendieck-Neeman duality. Then ω_{p} = p^{(1)}(𝟙) is a dualizing object for the subcategory of relatively compact objects 𝒞^{c/p} (Def. 6.5).

(c) Suppose that p^{∗} and q^{∗} satisfy Grothendieck-Neeman duality. Then the functor f_{∗} restricts to an exact functor f_{∗} : 𝒞^{c/p} → 𝒟^{c/q} which is dualitypreserving with respect to the dualities Δ_{ω_{p}} and Δ_{ω_{q}}.

Proof. Proposition 5.20 gives (a). Theorem 5.25 applied to p^{∗} gives (b). Theorem 6.4 gives (c). □

* * *

Historically, Grothendieck duality arose in order to generalize Serre duality to the relative situation (i.e. to morphisms of schemes). However, another generalization of Serre duality is given by the notion of a Serre functor [BK89, BO01], and we next explain how a relative version of Serre functors naturally arises in our theory.

6.8. Remark. If f^{∗} : 𝒟 ⇄ 𝒞 : f_{∗} is an adjunction between closed tensor categories with f^{∗} a tensor functor, then 𝒞 inherits an enrichment over 𝒟 (see [Kel05]): the Hom-objects are given by \underline{𝒞}(x, y) := f_{∗} hom_{𝒞} (x, y) ∈ 𝒟, and the unit and composition morphisms 𝟙_{𝒟} → \underline{𝒞}(x, y) and \underline{𝒞}(y, z) ⊗_{𝒟} \underline{𝒞}(x, y) → \underline{𝒞}(x, z) in 𝒟 are obtained by adjunction in the evident way from the 𝒞-internal unit and composition maps.

6.9. Theorem (Relative Serre duality). Let f^{∗} : 𝒟 → 𝒞 be a functor as in Hypothesis 1.2 and let \underline{𝒞} denote the resulting 𝒟-enriched category as in Remark 6.8. Then there is a canonical natural isomorphism in 𝒟

σ_{x,y} : Δ\underline{𝒞}(x, y) →^{∼} \underline{𝒞}(y, x ⊗ ω_{f}) (6.10)

for all x ∈ 𝒞^{c} and y ∈ 𝒞, where we recall Δ := hom_{𝒟} (−, 𝟙). In particular, if we have the Wirthmüller isomorphism (Theorem 1.9), the pair (𝕊 := (−) ⊗ ω_{f}, σ) defines a Serre functor on 𝒞^{c} relative to 𝒟^{c}, by which we mean that 𝕊 is an equivalence 𝕊 : 𝒞^{c} →^{∼} 𝒞^{c} and that σ is a natural isomorphism Δ\underline{𝒞}(x, y) ≅ \underline{𝒞}(y, 𝕊x) in the tensor-category 𝒟^{c} for all x, y ∈ 𝒞^{c}.

Proof. Under our basic hypothesis, we have the adjunction f_{∗} ⊣ f^{(1)} and its internal version. If x is compact, and hence rigid, we obtain an isomorphism

Δ \underline{𝒞}(x, y) = hom_{𝒟} (f_{∗} hom_{𝒞} (x, y), 𝟙)  by definition
≅ f_{∗} hom_{𝒞} (hom_{𝒞} (x, y), ω_{f})  (2.18)
≅ f_{∗} hom_{𝒞} (y, x ⊗ ω_{f})  x ∈ 𝒞^{c}
= \underline{𝒞}(y, x ⊗ ω_{f})

(the second isomorphism uses [HPS97, Thm. A.2.5] again). This is the claimed natural isomorphism σ. When the object ω_{f} is invertible (Thm. 1.9), the functor 𝕊 = (−) ⊗ ω_{f} restricts to a self-equivalence on compacts. □

6.11. Remark. Usually, what one means by a “Serre functor” is a self-equivalence 𝕊 on a 𝕜-linear (triangulated) category 𝒞 together with an isomorphism as in (6.10), where 𝕜 is a field and Δ should be replaced by the 𝕜-linear dual. We can easily deduce such a structure from our result when the target category is 𝒟 = D(𝕜).

6.12. Corollary (Serre duality). Let f^{∗} : 𝒟 → 𝒞 satisfy the Wirthmüller isomorphism (Theorem 1.9), and assume moreover that 𝒟 = D(𝕜) is the derived category of a field 𝕜. Then 𝒞^{c} is 𝕜-linear and endowed with a Serre functor

𝕊 = (−) ⊗ ω_{f} : 𝒞^{c} →^{∼} 𝒞^{c} σ : 𝒞(x, y)^{⋆} →^{∼} 𝒞(y, 𝕊x)

in the sense of [BK89, BO01], where (−)^{⋆} = Hom_{𝕜}(−, 𝕜) denotes the 𝕜-linear dual.

Proof. Apply H^{0} = D(𝕜)(𝟙, −) to (6.10) and note that H^{0} ∘ Δ ≅ (−)^{⋆} ∘ H^{0}. □

6.13. Remark. If it exists, a Serre functor (𝕊, σ) on 𝒞^{c} relative to 𝒟^{c} as in Theorem 6.9 is unique. More precisely, if (𝕊, σ) and (𝕊′, σ′) are two of them then by Yoneda there is a unique isomorphism 𝕊 →^{∼} 𝕊′ of functors 𝒞^{c} → 𝒞^{c} inducing (σ′σ^{−1})_{∗} on the Hom sets. A similar remark holds for the usual Serre functors.

6.14. Example (Projective varieties). Let p : X → Spec(𝕜) be a projective variety over a field 𝕜, and let p^{∗} : ℬ = D(𝕜) → 𝒞 = D(Qcoh X) be the pull-back functor. Then 𝒞^{c} = D^{perf}(X), and moreover 𝒞^{c/p} = D^{b}(coh X) by Theorem 5.21. Thus we have the inclusion 𝒞^{c} = D^{perf}(X) ⊂ D^{b}(coh X) = 𝒞^{c/p} =^{def.}p^{#}(ℬ^{c}) and therefore p^{∗} must satisfy Grothendieck-Neeman duality by Proposition 5.19. We conclude that p is quasi-perfect. Moreover, by Theorem 5.23 we know that the subcategory D^{b}(coh X) consists of ω_{p}-reflexive objects in D(Qcoh X). Hence by our Grothendieck duality Theorem 5.25, the object ω_{p} is dualizing for D^{b}(coh X), i.e. in more classical language, it is a dualizing complex for X (as defined in [Nee10]). Can we describe ω_{p} more explicitly?

If X is Gorenstein (e.g. regular, or a complete intersection), then by [Har66, p. 299] the structure sheaf O_{X} is also a dualizing complex for X. But then, by the uniqueness of dualizing complexes (see Corollary 5.14), there exists a tensor invertible ℓ ∈ D(Qcoh X) and an isomorphism ω_{p} ≅ O_{X} ⊗ ℓ = ℓ, so in this case ω_{p} is invertible and therefore p^{∗} satisfies the Wirthmüller isomorphism (Thm. 1.9). Indeed, it can be shown in general that Gorenstein varieties are characterized by having an invertible dualizing complex (see [AIL10, §8.3]). Still, this does not yet determine ω_{p} up to isomorphism.

Assume further that X is regular, so that we have the equality 𝒞^{c} = D^{perf}(X) = D^{b}(coh X) = 𝒞^{c/p}. In this case, by condition (3) of Theorem 1.9, ω_{p} must be invertible. Moreover, Theorem 6.12 applies so that − ⊗ ω_{p} yields a Serre functor on 𝒞^{c}. But it is a basic classical result that 𝒞^{c} also admits a Serre functor −⊗Σ^{n}ω_{X}, where ω_{X} = Λ^{n}Ω_{X/𝕜} is the canonical sheaf on X (see e.g. [Rou10, Lemma 4.18]); here we assume X is of pure dimension n, for simplicity. Therefore ω_{p} ≅ Σ^{n}ω_{X} by Remark 6.13.

Suppose now that f: X → Y is a 𝕜-morphism of projective varieties. By Corollary 6.7 (c), we have a well-defined f_{∗} : D^{b}(coh X) → D^{b}(coh Y) compatible with the dualities Δ_{ω_{p}} and Δ_{ω_{q}} as discussed above (for q : Y → Spec(𝕜)).

This example illustrates how our abstract notions and results specialize to ones that are familiar to algebraic geometers, depending on the various additional assumptions that are available.

7. Examples beyond Grothendieck duality

In this final section we show that Matlis duality as well as Neeman’s version of Brown-Comenetz duality also fall under the scope of our theory.

7.1. Theorem. Let f^{∗} : 𝒟 → 𝒞 be a functor satisfying our basic Hypothesis 1.2. Let 𝒞_{0} be a subcategory of 𝒞 admitting a dualizing object κ′ ∈ 𝒞 (external or not, see Def. 5.1). Assume moreover that κ′ admits a Matlis lift κ, that is, an object κ ∈ 𝒟 such that f^{(1)}(κ) ≅ κ′. Then κ is a possibly external dualizing object for the subcategory 𝒟_{0} := thick(f_{∗}(𝒞_{0})), the thick subcategory generated by the image of 𝒞_{0} under push-forward.

Proof. Since Id_{𝒟} and Δ_{κ′}^{2} are triangulated functors, it suffices to show that the natural transformation ϖ_{f_{∗}(x)} : f_{∗}(x) → Δ_{κ′}^{2}(f_{∗}x) of (5.4) is invertible whenever x belongs to 𝒞_{0}. For this, recall that ϖ_{f_{∗}(x)} appears in the commutative square (5.17). Since the top horizontal map in (5.17) is invertible for x ∈ 𝒞_{0}, the commutativity of (5.17) implies that ϖ_{f_{∗}(x)} is also invertible, as desired. □

7.2. Example (Matlis duality). Let R be a commutative noetherian local ring, let R → k be the quotient map to the residue field and let f^{∗} : D(R) → D(k) be the induced functor as in Example 3.23. Then E(k), the injective hull of the R-module k, is a Matlis lift of k: f^{(1)}(E(k)) = RHom_{R}(k, E(k)) ≅ k in D(k). By Theorem 7.1, the functor Δ_{E(k)} = RHom_{R}(−, E(k)) induces a duality on the thick subcategory of D(R) generated by f_{∗}(k), i.e. on complexes whose homology is bounded and consists of finite length modules. As E(k) is injective, we may restrict this duality to the category of finite length modules.

7.3. Example (Pontryagin duality). The dualizing object E(k) of Example 7.2 is typically external, i.e. it often lies outside the subcategory it dualizes: E(k) ∉ thick(f_{∗}(k)). This already happens in the archetypical example of (discrete p-local) Pontryagin duality, where R → k is the quotient map ℤ_{(p)} → ℤ/p and E(k) is the Prüfer group ℤ[\frac{1}{p}]/ℤ ≅ ℚ/ℤ_{(p)}, which has infinite length.

7.4. Example (Generalized Matlis duality). Let R → k be a morphism of commutative 𝕊-algebras and consider the three induced functors

(7.5)  [figure: 𝒞 := D(k); k ⊗_{R} (−) = f^{∗}; f_{∗}; f^{(1)} = Hom_{R}(k, −); 𝒟 := D(R)]

as in Example 3.26. We write R → k rather then B → A in order to be consistent with the notation of Dwyer-Greenlees-Iyengar [DGI06]. In loc. cit., a Matlis lift of k is defined to be a (structured) R-module I such that D(k)(x, k) ≅ D(R)(f_{∗}x, I) naturally in x ∈ D(k), i.e. by Yoneda, such that f^{(1)}(I) ≅ k (see [DGI06, Def. 6.2 and Rem. 6.3]). Moreover, I is required to be “effectively constructible from k”, a property somewhat stronger than I belonging to the localizing subcategory generated by f_{∗}(k). In particular, a Matlis lift of k in the sense of Dwyer-Greenlees-Iyengar is also a Matlis lift, in the more modest sense of Theorem 7.1, of the dualizing object κ′ := k for the subcategory 𝒞_{0} := D(k)^{c} of D(k). Hence by Theorem 7.1 we immediately obtain the following generalization of Matlis duality.

7.6. Corollary. Let R → k be a morphism of commutative 𝕊-algebras, and assume that the R-module I is a Matlis lift of k in the sense of [DGI06]. Then I is a (possibly external) dualizing object for the thick subcategory of D(R) generated by f_{∗}(k). □

Similar ideas and results can also be found in [DGI11] [Yek10] [PSY14].

* * *

7.7. Example (Brown-Comenetz duality). Let SH denote the stable homotopy category of spectra and recall that the Brown-Comenetz dual I_{ℚ/ℤ}E of a spectrum E ∈ SH is defined, using Brown Representability, by the equation

(7.8)  Hom_{ℤ}(π_{0}(X ∧ E), ℚ/ℤ) ≅ Hom_{SH}(X, I_{ℚ/ℤ}E)  (X ∈ SH).

It follows immediately from the tensor-Hom adjunction that I_{ℚ/ℤ}E is given by the function spectrum Δ_{I_{ℚ/ℤ}}(E) = hom_{SH}(E, I_{ℚ/ℤ}) where I_{ℚ/ℤ} := I_{ℚ/ℤ}𝕊 denotes the Brown-Comenetz dual of the sphere. Classical Brown-Comenetz duality asserts that the functor Δ_{I_{ℚ/ℤ}} = hom_{SH}(−, I_{ℚ/ℤ}) : SH^{op} → SH restricts to a duality on the subcategory of “homotopy finite” spectra, i.e. spectra whose homotopy groups are all finite. Moreover, defining I_{ℤ} to be the homotopy fiber of the canonical map Hℚ → I_{ℚ/ℤ}, we obtain a functor Δ_{I_{ℤ}} = hom_{SH}(−, I_{ℤ}) which restricts to a duality on the larger subcategory consisting of those spectra whose homotopy groups are all finitely generated (cf. [HS14, §2]). Restricted to the subcategory of homotopy finite spectra, this “Anderson duality” agrees with Brown-Comenetz duality up to a shift: Δ_{I_{ℚ/ℤ}} ≅ ΣΔ_{I_{ℤ}}.

On the other hand, the main result of [Nee92] establishes the existence of a (unique) triangulated functor Π : SH → D(ℤ[\frac{1}{2}]) which lifts stable homotopy with 2 inverted: H^{0} ∘ Π = π_{0}(−)[\frac{1}{2}]. Neeman defines a new “Brown-Comenetz type” dual ĨE by the equation

(7.9)  Hom_{D(ℤ[ \frac{1}{2} ])}(Π(X ∧ E), ℤ[\frac{1}{2}]) ≅ Hom_{SH}(X, ĨE)  (X ∈ SH).

As before, we immediately see that ĨE = Δ_{Ĩ𝕊}(E). The next theorem demonstrates that Neeman’s Δ_{Ĩ𝕊} delivers a version of Brown-Comenetz duality with the same scope as Anderson duality – and with a completely conceptual origin – provided we also invert the prime 3. To this end, regard Π as a functor SH[\frac{1}{6}] → D(ℤ[\frac{1}{6}]) where SH[\frac{1}{6}] denotes the stable homotopy category localized away from 6; being a finite localization of the ordinary stable homotopy category SH, it is again a rigidly-compactly generated tensor-triangulated category generated by its unit.

Neeman proves that, even before 3 is inverted, the Moore spectrum construction extends to an exact functor f^{∗} (denoted by F in loc. cit.) which is left adjoint to Π and admits a natural isomorphism μ : f^{∗}(x) ∧ f^{∗}(y) ≅ f^{∗}(x ⊗ y) ([Nee92, Prop. 3.6 and 5.5]). He then shows that μ satisfies the hexagon axiom of a monoidal functor if and only if 3 is also inverted ([Nee92, Prop. 5.6 and Ex. 5.1]), and derives from this the second main result of his article: SH[\frac{1}{6}] admits an enrichment over D(ℤ[\frac{1}{6}]). Although he never states so, he really shows that the functor f^{∗} : D(ℤ[\frac{1}{6}]) → SH[\frac{1}{6}] is symmetric monoidal, with structure map f^{∗}(𝟙) → 𝟙 given by the identity map (in fact he deduces the enrichment from the monoidal adjunction f^{∗} ⊣ f_{∗} as in Remark 6.8). Since f^{∗} has a right adjoint it preserves coproducts, so it satisfies Hypothesis 1.2 and we deduce the existence of f^{(1)} by Corollary 2.14, as usual:

[figure: SH[\frac{1}{6}]; f_{∗} = Π; D(ℤ[\frac{1}{6}]); f^{(1)}; f^{∗}]

7.10. Theorem. The above triple f^{∗} ⊣ f_{∗} ⊣ f^{(1)} has the following properties:

(a) The relative dualizing object associated with f^{∗} is Neeman’s Brown-Comenetz dual of the sphere: ω_{f} = Ĩ𝕊. Hence we have ĨE = Δ_{ω_{f}} (E) for all E ∈ SH[\frac{1}{6}].

(b) The object Ĩ𝕊 is dualizing for the subcategory 𝒞_{0} ⊂ SH[\frac{1}{6}] of spectra whose homotopy groups are finitely generated ℤ[\frac{1}{6}]-modules. In other words, Neeman’s Brown-Comenetz duality restricts to an equivalence Ĩ(−) : (𝒞_{0})^{op} →^{∼} 𝒞_{0}.

(c) There is a canonical natural isomorphism

Π(ĨE) = f_{∗}Δ_{ω_{f}} (E) ≅ Δ_{𝟙}(f_{∗}E) = hom_{D(ℤ[ \frac{1}{6} ])}(ΠE, ℤ[\frac{1}{6}])

for all E ∈ SH[\frac{1}{6}], analogous to the isomorphism

π_{∗}(I_{ℚ/ℤ}E) ≅ Hom_{ℤ}(π_{∗}E, ℚ/ℤ)

of ordinary Brown-Comenetz duality.

(d) f^{∗} does not satisfy Grothendieck-Neeman duality: f^{(1)} ≅ ˜I̸𝕊 ∧ f^{∗}.

Proof. Part (a) holds simply because ω_{f} and Ĩ𝕊 are defined by the same natural isomorphism: (7.9) with E = 𝕊. Part (c) is (2.18). Part (d) holds because f_{∗} = Π does not preserve compact objects. Indeed, π_{∗}(𝕊)_{(p)} is well-known to be unbounded for every prime p (see e.g. [MN84]). Finally, to prove (b), consider the full subcategory 𝒟_{0} ⊂ 𝒟 = D(ℤ[\frac{1}{6}]) of complexes whose homology groups are finitely generated ℤ[\frac{1}{6}]-modules. As the ring ℤ[\frac{1}{6}] is principal (hence hereditary), every object of its derived category has (non-canonically) the form M = ∐_{i} Σ^{i}M_{i} for modules M_{i}, and every morphism between such objects has nonzero components only of degree 0 and −1. An object M = ∐_{i} Σ^{i}M_{i} belongs to the subcategory 𝒟_{0} if and only if every M_{i} is finitely generated. Using that the natural map ∐_{i} Σ^{i}M_{i} →^{∼} ∏_{i} Σ^{i}M_{i} is an isomorphism here, it is easy to see that hom(−, 𝟙) is a duality on 𝒟_{0}. By Theorem 5.23 (a) for κ = 𝟙, using that our 𝒞_{0} is nothing but f^{#}(𝒟_{0}), we get the result. □

7.11. Remark. As explained in [Nee92, Rem. 4.2], there are K-theoretic obstructions for the functor f^{∗} : D(ℤ[\frac{1}{2}]) → SH[\frac{1}{2}] to derive from a functor of the underlying (Waldhausen) model categories. This shows that there can be interesting functors f^{∗} with no underlying map f.

Acknowledgements: We thank Greg Stevenson for helpful discussions, Henning Krause for the reference to [Jan87], and an anonymous referee for a careful reading and several useful comments.

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URL: http://math.univ-lille1.fr/∼dellambr

Beren Sanders, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

E-mail address: sanders@math.ku.dk
URL: http://beren.blogs.ku.dk