WITT GROUPS OF GRASSMANN VARIETIES

PAUL BALMER AND BAPTISTE CALMÈS

Abstract. We compute the Witt groups of split Grassmann varieties, over any regular base X. The answer is that the total Witt group of the Grassmannian is a free module over the total Witt ring of X. We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams.

Contents

Introduction 1 1. Combinatorics of Grassmann and flag varieties 4 2. Even Young diagrams 8 3. Total bases and lax-similitude 11 4. Construction of the total basis 15 5. Cellular decomposition 18 6. Main result 22 7. Corollaries and examples 27 References 30

Introduction

At first glance, it might be surprising for the non-specialist that more than thirty years after the definition of the Witt group of a scheme, by Knebusch [14], the Witt group of such a well-known variety as a Grassmannian has not been computed yet. This is especially striking since analogous results for ordinary cohomologies, for K-theory and for Chow groups, have been settled for even longer. The goal of this article is to solve this problem and explain what made it so hard in the first place.

Main Theorem (See Thm. 6.1). Let X be a regular noetherian and separated scheme over ℤ[\frac{1}{2}], of finite Krull dimension. Let 0 < d < n be integers and let Gr_{X}(d, n) be the Grassmannian of d-dimensional subbundles of the trivial ndimensional vector bundle V = O_{X}^{n} over X. (More generally, we treat any vector bundle V admitting a complete flag of subbundles.)

Then the total Witt group of Gr_{X}(d, n) is a free graded module over the total Witt group of X with an explicit basis indexed by so-called “even” Young diagrams. The basis element corresponding to an even Young diagram is essentially the pushforward of the unit along the inclusion of the corresponding Schubert variety. The

cardinal of this basis equals 2 · \frac{(d′ + e′)!}{d′! · e′!} where d′ = [\frac{d}{2}] and e′ = [\frac{n − d}{2}].

Date: April 29, 2011.

1991 Mathematics Subject Classification. 19G99, 11E81, 18E30.

Key words and phrases. Witt group, Grassmann variety, triangulated category, cellular.

The first author is supported by NSF grant DMS-0969644.

Before explaining the statement in more detail, recall that the Grothendieck group, or the Chow group, of Gr_{X}(d, n) would also be free over that of X but with a basis indexed by all Young diagrams. We shall explain below why only some Young diagrams “make it to the Witt group”.

The total Witt group refers to the sum of the Witt groups W^{i}(X, L)

W^{tot}(X) = ⨁_{{i ∈ ℤ/4 \atop [L] ∈ Pic(X)/2}} W^{i}(X, L)

for all possible shifts i ∈ ℤ/4 and all possible twists [L] ∈ Pic(X)/2 in the duality. Details about this total Witt group, including the dependency on choosing L in its class [L] ∈ Pic(X)/2, can be found in Section 3. For this introduction, let us keep things simple : The total Witt group of X wraps up all Witt groups of X, for all possible shifts i and all twists L.

For X = Spec(F), the spectrum of a field, the total Witt group boils down to the classical Witt group W(F) but even in that case the above Theorem is new and the total Witt group of Gr_{F} (d, n) involves non-trivial shifted and twisted Witt groups. The result has a very round form when stated for total Witt groups but Knebusch’s classical unshifted Witt groups W^{0}(Gr_{X}(d, n), L) can be isolated, as well as the unshifted and untwisted Witt group W(Gr_{X}(d, n)) = W^{0}(Gr_{X}(d, n), O). Indeed, the announced basis consists of homogeneous elements and we describe below how to read their explicit shifts i and twists L directly on the corresponding Young diagram. For instance, it is worth noting that there are no new interesting antisymmetric forms in the Witt groups of Gr_{X}(d, n), that is, except for those extended from X, see Corollary 7.2.

To describe our basis explicitly, we need to introduce even Young diagrams. We first consider ordinary Young diagrams sitting in the upper left corner of a rectangle with d rows and e columns, which we call the frame of the diagram. See Figure 1.

\begin{matrix}d \\ e \\ Λ \end{matrix}

Figure 1. Young diagram Λ in (d×e)-frame

We say that such a framed Young diagram Λ is even if all the segments of the boundary of Λ which are strictly inside the frame have even length. That is, we allow Λ to have odd-length segments on its boundary only where it touches the outside frame. See Figure 2 for examples. (In Figures 14, 15 and 16 we further give all even diagram in (d×e)-frame for (d, e) = (4, 4), (4, 5) and (5, 5), respectively.)

We shall see that basis elements of W^{tot} ( Gr_{X}(d, n)) are in bijective correspondence with even Young diagrams in (d×e)-frame, for e := n − d. Moreover, as explained in Section 4, the Witt class ϕ_{d,e}(Λ) corresponding to such a diagram Λ lives in the Witt group W^{i} ( Gr_{X}(d, n), L) for the shift i = |Λ| ∈ ℤ/4 equal to the area of the diagram and for the twist [L] ∈ Pic(Gr_{X}(d, n))/2 equal to the class of t(Λ) · Δ where Δ is the determinant of the tautological bundle and where the integer t(Λ) is half the perimeter of the diagram Λ (see Figure 9 in Section 4). More 4 2 2 2 2 2 2 4 2 2 4

Figure 2. Five examples of even Young diagrams

generally, when V is not free but admits a complete flag of subbundles, the twist of ϕ_{d,e}(Λ) also involves a multiple of the determinant of V, in the direct summand of Pic(Gr_{X}(d, n))/2 coming from Pic(X)/2.

Let us put our result in perspective. In its modern form, see for instance Laksov [15], the computation of the cohomology (or K-theory, or Chow groups) of a cellular variety uses essentially only localization long exact sequences and homotopy invariance, applied to the classical cellular decomposition of the Grassmannian. It took some time for Witt groups to reach the necessary cohomological maturity. Indeed, the localization long exact sequence could only be established by defining first “higher” or “shifted” Witt groups, as was done in [3] and [4] by the first author, using the framework of triangulated categories. Then, homotopy invariance was proved by Gille [11]. To be more precise, one actually also needs some form of dévissage, that allows us to compare the theory with supports on a closed subset Z with the theory of Z itself. This piece of theory was a hard nut to crack because dualities come in the way but is now well understood. (We return to these dualities below.)

This construction of the cohomological machines accounts for most of the delay that Witt groups accumulated in comparison to sister theories. However, this is not the end of the story. What makes the computation of the Witt groups of Gr_{X}(d, n) much harder than that of classical theories is the following very interesting phenomenon. The classical computation proceeds by induction, using the closed embedding of a smaller Grassmannian Gr_{X}(d, n − 1) inside Gr_{X}(d, n), whose open complement U = Gr_{X}(d, n)∖Gr_{X}(d, n−1) is an affine bundle over another smaller Grassmannian, Gr_{X}(d−1, n). (See more in Section 5.) Now the true miracle in the classical proof is that the restriction homomorphism, from the big Grassmannian Gr_{X}(d, n) to the open U, is split surjective. This holds more precisely for any oriented cohomology theory in the sense of Levine-Morel [16] or Panin [18]. In other words, there are no real localization long exact sequences involved in the classical proof, no real connecting homomorphisms, just split short exact sequences.

The interesting point is that this miracle ceases to happen for Witt groups : For a general (i, [L]) ∈ ℤ/4 × Pic(Gr_{X}(d, n))/2, the restriction homomorphism

W^{i}(Gr_{X}(d, n), L) ⟶ W^{i}(U, L)

does not admit a section. Worse, it is not even surjective ! That is, the connecting homomorphism in the localization long exact sequence is not zero in general. This makes the Witt group computation not a mere technical adaptation of the classical methods but a completely different story : One needs to compute a connecting homomorphism in geometric enough terms, in order to follow what happens on the given varieties. This very last pocket of Witt resistance has been cleared in the companion article [6] and Grassmannians are the first known examples where this phenomenon can be described explicitly.

Let us now comment on the organization of the paper. Sections 1 and 2 contain preparatory material on Grassmann varieties, desingularizations of Schubert varieties and even Young diagrams.

Section 3 formalizes the use of total Witt groups, since the group W^{∗}(X, L) does not truly depend only on the class [L] ∈ Pic(X)/2 but really on the representative L in this class. This is an old problem, rooting back to Knebusch [14], to which we proposed a solution in [7]. We recall this formalism in Section 3 with emphasis on the case of Grassmannians. Until then, this Introduction should therefore be read cum grano salis.

Our generators of W^{tot} ( Gr_{X}(d, n)) are defined in Section 4 as push-forwards of the unit forms of certain desingularized Schubert varieties. The reader should observe that pushing the unit form is not always possible, due to the presence of line bundles in the definition of the push-forward. Indeed, for a proper morphism f : Ȳ → Y of constant relative dimension dim(f), between regular noetherian schemes Ȳ and Y (think dim f = dim Ȳ − dim Y), the push-forward along f is defined between the following Witt groups :

W^{i+dim f}(Ȳ , ω_{f} ⊗ f^{∗}L) ⟶^{f_{∗}} W^{i}(Y, L) ,  (1)

where the special line bundle ω_{f} on the left is the relative line bundle. So, if unfortunately OȲ is not (up to squares) of the form ω_{f} ⊗ f^{∗}L for any line bundle L over Y, then one simply cannot push-forward the unit form of Ȳ, which lives in W^{0}(Ȳ , OȲ). This is why we start Section 4 by discussing the “parity” of the relevant canonical bundles ω_{f}. Although somewhat heavy, these computations are elementary and are all based on a repeated application of the computation of the relative canonical bundle of a Grassmann bundle (Prop. 1.5). The condition for a Young diagram to be even implies the existence of such a push-forward for the unit of the desingularized Schubert cell into the Grassmannian. Actually, we could pushforward the unit form for more Schubert cells but these additional generators would be redundant. The even Young diagrams are chosen so that the corresponding forms are also linearly independent.

Then, in Section 5, we recall the classical relative cellular structure of the Grassmann varieties. In Section 6, we compute how our candidate-generators behave under the morphisms in the long exact sequence, especially under the connecting homomorphism, which is most of the time not zero (Cor. 7.3). The proof of the main theorem (Thm. 6.1) then follows by induction on the rank of the vector bundle V. The last section contains corollaries and examples.

This article is written in the language of “functors of points”, which means that we describe schemes in terms of their points (which are here flags) and morphisms of schemes as how they act on those points. This method is completely rigorous in this case. The original source is [1] and we also refer the reader to [9, § I.1] and [13, Part 2] for general considerations on this subject. This language is customary when dealing with flag varieties, see for instance [15] in which it is used for the computation of Chow groups of Grassmann varieties.

1. Combinatorics of Grassmann and flag varieties

We recall elementary facts about Grassmann varieties and desingularizations of Schubert cells. We also provide the necessary material about canonical bundles to treat the push-forward homomorphisms for Witt groups in Section 4.

1.1. Definition. A subbundle P ⊲ V of a vector bundle V over a scheme X is an O_{X}-submodule which is locally a direct summand, i.e. P and V/P are vector bundles.

1.2. Definition. Let V be a vector bundle of rank n > 0 over a scheme X and let d be an integer 0 ⩽ d ⩽ n. We denote by Gr_{X}(d, V) the Grassmann bundle over X parameterizing the subbundles of rank d of V. In the language of functors of points, it means that for any morphism f : Spec(R) → X, the set Gr_{X}(d, V)(R) consists of the R-submodules P ⊲ V(R) = f^{∗}(V) which are direct summands of rank d.

The scheme Gr_{X}(d, V) comes equipped with a smooth structural morphism π : Gr_{X}(d, V) → X and a tautological bundle T_{d} = T_{d}^{Gr_{X}(d,V)} of rank d.

1.3. Proposition. The scheme Gr_{X}(d, V) is smooth over X of relative dimension d(n − d). For 0 < d < n, the Picard group of Gr_{X}(d, V) is given by

Pic(X) ⊕ ℤ ≅ Pic(Gr_{X}(d, V))
(ℓ, m)  ↦ π^{∗}(ℓ) · [det(T_{d})]^{m}.

In case d = 0 or d = n, the morphism π : Gr_{X}(d, V) → X is the identity.

Proof. The Picard group of a regular scheme coincides with its Chow group CH^{1}, which is computed in [15] for Grassmannians : see Theorem 16 for the case where X is a field, and §13 to work over a regular base X. Using the Plücker embedding, one checks that the generator in loc. cit. is indeed [det(T_{d})]. □

Let Δ_{d} denote the class of det(T_{d}) in Pic ( Gr_{X}(d, V))/2.

1.4. Corollary. If 0 < d < n, we have a natural identification

Pic ( Gr_{X}(d, V))/2 ≅ Pic(X)/2 ⊕ ℤ/2 · Δ_{d}.

1.5. Proposition. The class of the relative canonical bundle ω_{Gr_{X}(d,V)/X} of the projection π : Gr_{X}(d, V) → X is [ω_{Gr_{X}(d,V)/X}] = [det V]^{−d} · Δ_{d}^{n} in Pic(Gr_{X}(d, V)). In particular, if V = O_{X}^{n} is trivial, [ω_{Gr_{X}(d,V)/X}] = Δ_{d}^{n}.

Proof. The morphism π is smooth, so ω_{Gr_{X}(d,V)/X} is the determinant (highest exterior power) of the relative cotangent bundle of π. This cotangent bundle is the tautological bundle tensored by the dual of the tautological quotient bundle (see [10, Appendix B.5.8]). Taking the determinant, we get the result. □

* * *

We now extend the previous results from Grassmannians to some flag varieties.

1.6. Definition. Let k ⩾ 1 and (\underline{d}, \underline{e}) be a pair of k-tuples of non-negative integers \underline{d} = (d_{1}, …, d_{k}) and \underline{e} = (e_{1}, …, e_{k}) satisfying

0 < d_{1} < ⋯ < d_{k}  and  e_{1} + d_{1} ⩽ ⋯ ⩽ e_{k} + d_{k} .  (2)

(The second condition holds in particular if we have e_{1} ⩽ ⋯ ⩽ e_{k}.) Consider a flag

V_{d_{1}+e_{1}} ⊲ ⋯ ⊲ V_{d_{i}+e_{i}} ⊲ ⋯ ⊲ V_{d_{k}+e_{k}}  (3)

of vector bundles over X, where ⊲ indicates subbundles in the strong sense of Definition 1.1 and where the rank is given by the index: rk_{X}(V_{r}) = r.

We associate to this data the scheme Fl_{X}(\underline{d}, \underline{e}, V_{•}) over X, which parameterizes the flags of vector bundles P_{d_{1}} ⊲ P_{d_{2}} ⊲ ⋯ ⊲ P_{d_{k}} such that rk P_{d_{j}} = d_{j} and P_{d_{j}} ⊲ V_{d_{j}+e_{j}}. As a functor of points, this gives for any morphism f : Y → X

Fl_{X}(\underline{d}, \underline{e}, V_{•})(Y) := \{ 0 ^{⊲} P_{d_{1}} ⊲_{e_{1}}  ^{⊲} P_{d_{2}} ⊲_{e_{2}} ^{⊲} ⋯ ^{⊲} P_{d_{k}} ⊲_{e_{k}} 0 _{⊲} f^{∗}V_{d_{1}+e_{1}} _{⊲} f^{∗}V_{d_{2}+e_{2}} _{⊲} ⋯ _{⊲} f^{∗}V_{d_{k}+e_{k}} \} , (4)

where all P_{d_{i}} are vector bundles over Y of rank d_{i} such that all inclusions are subbundles in the sense of Definition 1.1. The integers along inclusions indicate codimensions. Following general practice, we shall drop the mention of f^{∗} in the sequel. Moreover, to avoid cumbersome notation, unless the original flag (3) varies, we drop the mention of V_{•} from the notation : Fl_{X}(\underline{d}, \underline{e}) = Fl_{X}(\underline{d}, \underline{e}, V_{•}).

1.7. Example. For k = 1, the scheme Fl_{X}(\underline{d}, \underline{e}) is simply Gr_{X}(d_{1}, V_{d_{1}+e_{1}}).

1.8. Remark. For any choice J of k′ indices among {1, …, k}, one can consider the pair of k′-tuples (\underline{d}′, \underline{e}′) obtained from (\underline{d}, \underline{e}) by keeping d_{i} and e_{i} only for indices i ∈ J. There is a natural morphism Fl_{X}(\underline{d}, \underline{e}) → Fl_{X}(\underline{d}′, \underline{e}′) over X, obtained by dropping the P_{d_{j}} for those indices j which are not in the chosen J.

Furthermore, for any vector bundle V such that V_{d_{k}+e_{k}} ⊲ V, there is a natural morphism f_{\underline{d},\underline{e},V} of schemes over X as follows :

\begin{aligned}f_{\underline{d},\underline{e},V} : & Fl_{X}(\underline{d}, \underline{e}) & ⟶ & Gr_{X}(d, V_{d_{k}+e_{k}}) & ↪ & Gr_{X}(d, V) \\ & (P_{d_{1}}, …, P_{d_{k}}) & ⟼ & P_{d_{k}} & ⟼ & P_{d_{k}} , \end{aligned}  (5)

where the first morphism is as above and the second is a closed immersion.

1.9. Definition. The scheme Fl_{X}(\underline{d}, \underline{e}) is equipped with tautological bundles T_{d_{i}}, 1 ⩽ i ⩽ k, of rank d_{i}, whose determinant classes are denoted by Δ_{d_{i}} := det(T_{d_{i}}). The stalk of T_{d_{i}} at a point (P_{d_{1}}, …, P_{d_{k}}) is P_{d_{i}}. In ambiguous cases, the full notation for T_{d_{i}} would be T_{d_{i}}^{Fl_{X}(\underline{d},\underline{e},V_{•})}.

1.10. Remark. If e_{i} = 0 then the vector bundles T_{d_{i}} = V_{d_{i}} and Δ_{d_{i}} = [det V_{d_{i}}] are both extended from X.

1.11. Lemma. Let k ⩾ 2 and let (\underline{d}, \underline{e}) be a pair of k-tuples satisfying (2). Let V_{•} be a flag as in (3). Define the (k − 1)-tuples \underline{d}_{|k−1} and \underline{e}_{|k−1} as the restrictions of \underline{d} and \underline{e} to the first k − 1 entries. Consider the scheme

Y := Fl_{X}( \underline{d}_{|k−1} , \underline{e}_{|k−1} , V_{•}) ,

which only “uses” the first k − 1 bundles V_{d_{1}+e_{1}} ⊲ ⋯ ⊲ V_{d_{k−1}+e_{k−1}}. Consider the pull-back to Y of the remaining bundle, still denoted V_{d_{k}+e_{k}} . Observe that T_{d_{k−1}}^{Y} ⊲ V_{d_{k}+e_{k}} and consider the quotient bundle

Ṽ := V_{d_{k}+e_{k}}/T_{d_{k−1}}^{Y}

over Y. It has rank d_{k} − d_{k−1} + e_{k}. We then have a canonical isomorphism of schemes over Y (hence over X) :

Fl_{X}(\underline{d}, \underline{e}, V_{•}) ≅ Gr_{Y} (d_{k} − d_{k−1}, Ṽ) .  (6)

Under this identification, we have T_{d_{i}}^{Fl(\underline{d},\underline{e},V_{•})} = T_{d_{i}}^{Y} for all 1 ⩽ i ⩽ k − 1 and

T_{d_{k}}^{Fl_{X}(\underline{d},\underline{e},V_{•})}/T_{d_{k−1}}^{Fl_{X}(\underline{d},\underline{e},V_{•})} = T_{d_{k}−d_{k−1}}^{Gr_{Y} (d_{k}−d_{k−1},Ṽ)} .  (7)

Proof. This simply amounts to the bijective correspondence between a flag P_{d_{1}} ⊲ ⋯ ⊲ P_{d_{k−1}} ⊲ P_{d_{k}} satisfying P_{d_{i}} ⊲ V_{d_{i}+e_{i}} for all 1 ⩽ i ⩽ k and the following data :

(a) the beginning of this flag P_{d_{1}} ⊲ ⋯ ⊲ P_{d_{k−1}} satisfying P_{d_{i}} ⊲ V_{d_{i}+e_{i}} for all 1 ⩽ i ⩽ k − 1,

(b) the bundle P_{d_{k}} such that P_{d_{k−1}} ⊲ P_{d_{k}} ⊲ V_{d_{k}+e_{k}}

and to observe that (b) is equivalent to a subbundle P̃ ⊲ V_{d_{k}+e_{k}}/P_{d_{k−1}} of rank d_{k} − d_{k−1}, where P̃ := P_{d_{k}}/P_{d_{k−1}}. Details are left to the reader. □

1.12. Convention. When using k-tuples \underline{d} = (d_{1}, …, d_{k}), it will unify several formulas to simply define d_{0} = 0.

1.13. Proposition. Let \underline{d} and \underline{e} be two k-tuples as in (2) and V_{•} be a flag as in (3). Then Fl_{X}(\underline{d}, \underline{e}) is smooth over X of relative dimension ∑_{i=1}^{k}(d_{i} − d_{i−1}) e_{i}. The Picard group of Fl_{X}(\underline{d}, \underline{e}) is generated by Pic(X) and the “new” classes Δ_{d_{i}} :

Pic(Fl_{X}(\underline{d}, \underline{e})) ≅ Pic(X) ⊕ ⨁_{{1⩽i⩽k \atop s.t. e_{i}≠0}} ℤ Δ_{d_{i}} .  (8)

The class of the relative canonical bundle ω_{Fl_{X}(\underline{d},\underline{e})/X} is given by the formula

[ω_{Fl_{X}(\underline{d},\underline{e})/X}] = ∏_{i=1}^{k} [det V_{d_{i}+e_{i}}]^{−d_{i}+d_{i−1}} · ∏_{i=1}^{k−1} Δ_{d_{i}}^{d_{i}−d_{i−1}+e_{i}−e_{i+1}} · Δ_{d_{k}}^{d_{k}−d_{k−1}+e_{k}} , (9)

where Δ_{d_{i}} = [det V_{d_{i}}] if e_{i} = 0 by Remark 1.10 and where we use Convention 1.12. In particular, for k = 1, this reads [ω_{Fl_{X}(\underline{d},\underline{e})/X}] = [det V_{d_{1}+e_{1}}]^{−d_{1}} · Δ_{d_{1}}^{d_{1}+e_{1}}.

Proof. By induction on k. The case k = 1 is that of a Grassmannian over X (Example 1.7) so the result follows from Propositions 1.3 and 1.5.

Let now k ⩾ 2. Consider Y = Fl_{X}(\underline{d}|_{k−1}, \underline{e}|_{k−1}, V_{•}) and the bundle Ṽ = V_{d_{k}+e_{k}}/T_{d_{k−1}} over Y, as in Lemma 1.11. Recall that rk_{Y} (Ṽ) = d_{k} − d_{k−1} + e_{k}, which is always strictly positive (d_{k} > d_{k−1}) and which is bigger than or equal to d_{k} − d_{k−1} with equality if and only if e_{k} = 0. Equation (6) and Propositions 1.3 and 1.5 immediately give smoothness, the formula for the relative dimension and that for the Picard group (8). Finally, to prove (9), observe that

[ω_{Fl_{X}(\underline{d},\underline{e})/Y}] = [ω_{Gr_{Y} (d_{k}−d_{k−1},Ṽ) / Y}]
= [det Ṽ]^{−d_{k}+d_{k−1}} · (Δ_{d_{k}−d_{k−1}}^{Gr_{Y} (d_{k}−d_{k−1},Ṽ)})^{rk(Ṽ)}
= [det Ṽ]^{−d_{k}+d_{k−1}} · (Δ_{d_{k}}^{Fl_{X}(\underline{d},\underline{e})})^{rk(Ṽ)} · (Δ_{d_{k−1}}^{Fl_{X}(\underline{d},\underline{e})})^{− rk(Ṽ)}
= [det V_{d_{k}+e_{k}}]^{−d_{k}+d_{k−1}} · Δ_{d_{k−1}}^{−e_{k}} · Δ_{d_{k}}^{d_{k}−d_{k−1}+e_{k}} .

The first equality uses (6), the second comes from Proposition 1.5 and the third from (7). The last equality is a direct computation (in which we drop the mention of Fl_{X}(\underline{d}, \underline{e}) for readability). By induction hypothesis, we get [ω_{Y/X}] from Equation (9) for k − 1, that is, for the flag variety Y. Since [ω_{Fl_{X}(\underline{d},\underline{e})/X}] = [ω_{Fl_{X}(\underline{d},\underline{e})/Y}] · [ω_{Y/X}] over Fl_{X}(\underline{d}, \underline{e}), we get (9) for k by adding the above. □

1.14. Corollary. Let \underline{d} and \underline{e} be two k-tuples as in (2) and V_{•} be a flag as in (3). Let V be a vector bundle of rank d+e such that V_{d_{k}+e_{k}} ⊲V . The class in Pic(Fl_{X}(\underline{d}, \underline{e})) of the relative canonical bundle for the morphism f_{\underline{d},\underline{e},V} : Fl_{X}(\underline{d}, \underline{e}) → Gr_{X}(d, V) of (5) is given by

[ω_{Fl_{X}(\underline{d},\underline{e})/ Gr_{X}(d,V)}] = ∏_{i=1}^{k} [det V_{d_{i}+e_{i}}]^{−d_{i}+d_{i−1}} · [det V]^{d_{k}}·
· ∏_{i=1}^{k−1} Δ_{d_{i}}^{d_{i}−d_{i−1}+e_{i}−e_{i+1}} · Δ_{d_{k}}^{−d_{k−1}+e_{k}−e} ,

(10)

where Δ_{d_{i}} = [det V_{d_{i}}] if e_{i} = 0 by Remark 1.10 and where we use Convention 1.12. For k = 1, this reads [ω_{Fl_{X}(\underline{d},\underline{e})/ Gr_{X}(d,V)}] = [det(V/V_{d_{1}+e_{1}})]^{d_{1}} · Δ_{d_{1}}^{e_{1}−e}.

Proof. Subtract (f_{\underline{d},\underline{e},V})^{∗} [ω_{Gr_{X}(d,V)/X}] = [det V]^{−d_{k}} · Δ_{d_{k}}^{d_{k}+e} (Prop. 1.5) from the bundle [ω_{Fl_{X}(\underline{d},\underline{e})/X}] given in (9). □

1.15. Remark. When V_{•} = OX^{•}, all the formulas are simpler, since all the [det V_{i}] are trivial. This applies in particular when X = Spec(R) for a local ring R.

2. Even Young diagrams

We introduce even Young diagrams that will parameterize the basis of the total Witt group of the Grassmann variety, to be constructed in Section 4.

2.1. Definition. Let d, e ⩾ 1. We shall call Young diagram in (d×e)-frame, or simply (d, e)-diagram, any d-tuple Λ = (Λ_{1}, Λ_{2}, …, Λ_{d}) of integers such that:

e ⩾ Λ_{1} ⩾ Λ_{2} ⩾ … ⩾ Λ_{d} ⩾ 0.

See Figure 1 in the Introduction. The area of Λ is |Λ| = Λ_{1} + Λ_{2} + … + Λ_{d}. These (d, e)-diagrams are just ordinary Young diagrams displayed in the upper left corner of a rectangle with d rows and e columns, possibly leaving empty rows below and empty columns to the right of the Young diagram. So, an ordinary Young diagram with ρ rows and γ columns defines a (d, e)-diagram for any d ⩾ ρ and e ⩾ γ.

2.2. Notation. The empty diagram (0, …, 0) ∈ ℕ^{d} is denoted by / □ and the full (d × e)-rectangle (e, …, e) ∈ ℕ^{d} by [d × e].

2.3. Definition. Let d, e ⩾ 1 and let Λ be a Young diagram in (d×e)-frame. The decreasing sequence Λ_{1} ⩾ Λ_{2} ⩾ … ⩾ Λ_{d} can be written in a unique way as a series of equalities and strict inequalities :

\underbrace{Λ_{1} = ⋯ = Λ_{d_{1}}}_{d_{1} terms} > \underbrace{Λ_{d_{1}+1} = … = Λ_{d_{2}}}_{d_{2}−d_{1} terms} > ⋯ > \underbrace{Λ_{d_{k−1}+1} = ⋯ = Λ_{d_{k}}}_{d_{k}−d_{k−1} terms} = Λ_{d} . (11)

Note that d_{k} = d. The integers k ⩾ 1 and 0 < d_{1} < … < d_{k} depend on Λ. If we need to stress this we shall write k = k(Λ) and d_{i} = d_{i}(Λ) for 1 ⩽ i ⩽ k(Λ).

For fixed d and e, there is a bijection (pictured on Figure 3) between the Young diagrams Λ in (d×e)-frame and pairs of k-tuples of integers

\underline{d} = (d_{1}, …, d_{k}) such that 0 < d_{1} < ⋯ < d_{k} = d
\underline{e} = (e_{1}, …, e_{k}) such that 0 ⩽ e_{1} < ⋯ < e_{k} ⩽ e (12)

with 1 ⩽ k ⩽ d. The integers k = k(Λ) and d_{i} = d_{i}(Λ) are the above ones and we set e_{i} = e_{i}(Λ) := e − Λ_{d_{i}} for all i = 1, …, k. The converse construction is obvious.

\begin{matrix}d \\ k = 4 \\ d_{4} \\ d_{3} \\ d_{2} \\ d_{1} \\ e \\ d \\ e_{3} \\ d_{1} \\ e_{1} \\ e_{2} \\ k = 3 \\ d_{2} \\ d_{3} \\ e \\ Λ \\ Λ \\ e_{1} =0 \\ e_{3} \\ e_{4} \\ e_{2} \end{matrix}

Figure 3. Two examples of the two k-tuples (d_{1}, …, d_{k}) and (e_{1}, …, e_{k}) corresponding to a Young diagram Λ in (d×e)-frame.

2.4. Definition. Let d, e ⩾ 1. Fix a complete flag of vector bundles over X

0 = V_{0} ⊲ V_{1} ⊲ ⋯ ⊲ V_{i} ⊲ ⋯ ⊲ V_{d+e} =: V .  (13)

Note that we baptize V the bundle of dimension d + e, to lighten notation.

Let Λ be a Young diagram in (d×e)-frame. By Definition 2.3, this amounts to a pair (\underline{d}, \underline{e}) of k-tuples of integers satisfying (12), and hence satisfying (2). We can now apply Definition 1.6 to \underline{d} and \underline{e} and the flag (3) taken from (13) above :

Fl_{X}(d, e, V_{•}; Λ) := Fl_{X}(\underline{d}, \underline{e}, V_{•}) = \{ 0 ^{⊲} P_{d_{1}} ⊲_{e_{1}} ^{⊲} P_{d_{2}} ⊲_{e_{2}}^{⊲} ⋯ ^{⊲} P_{d_{k}} ⊲_{e_{k}} 0 _{⊲} V_{d_{1}+e_{1}} _{⊲} V_{d_{2}+e_{2}} _{⊲} ⋯ _{⊲} V_{d_{k}+e_{k}} \} .

(14)

As usual, instead of Fl_{X}(d, e, V_{•}; Λ), we might simply write Fl(Λ) or anything “in between” depending on what is obvious from the context.

As in (5), there is a natural morphism f_{Λ} from Fl_{X}(d, e; Λ) to Gr(d, V)

\begin{aligned}f_{Λ} = f_{d,e;Λ} := f_{\underline{d},\underline{e},V} : & Fl_{X}(d, e; Λ) & ⟶ & Gr(d, V) \\ & (P_{d_{1}}, …, P_{d_{k}}) & ⟼ & P_{d_{k}} . \end{aligned}  (15)

When X = Spec(F) is a field, one can understand the image of f_{Λ} as the subset of those subspaces P_{d} ⊲ V whose intersection with each V_{d_{i}+e_{i}} is of dimension at least d_{i}. This is the classical Schubert cell associated to the diagram Λ. It is pretty clear that f_{Λ} is a birational morphism. The advantage of Fl_{X}(d, e; Λ) over the Schubert cell is that Fl_{X}(d, e; Λ) is not singular by Proposition 1.13.

2.5. Example. Following up on Example 1.7, when Λ = / □ is the empty diagram, that is for k = 1 and e_{1} = e, we have Fl_{X}( / □) = Gr_{X}(d, V) and f\begin{matrix}/ \\ □ \end{matrix} is the identity. At the other end, for Λ = [d × e] the whole (d × e)-rectangle, that is for k = 1 and e_{1} = 0, we have Fl_{X}(d, e; Λ) = Gr_{X}(d, V_{d}) = X and f_{Λ} is a closed immersion.

2.6. Definition. Let Λ be a Young diagram in (d×e)-frame. We define ρ(Λ) ∈ {0, …, d} to be the number of non-zero rows of Λ. Complementarily, we define ζ(Λ) = d − ρ(Λ) to be the number of zero rows at the end of Λ, that is

ρ(Λ) = d  and ζ(Λ) = 0  if Λ_{d} > 0 ,
ρ(Λ) = d_{k−1} and ζ(Λ) = d − d_{k−1} if Λ_{d} = 0 .

For the empty diagram, we have ρ( / □) = 0 and ζ( / □) = d.

We are going to use a certain class of (d, e)-diagrams, that we call the even (d, e)- diagrams. Defining them by a picture is very easy. The condition to be even is that any segment of the (d, e)-diagram which does not belong to the outer (d×e)-frame must have even length. See Figure 2. The formal definition is the following.

2.7. Definition. Let Λ be a Young diagram in (d×e)-frame and let \underline{d} and \underline{e} be the associated k-tuples as in Definition 2.3. We say that Λ is even if all the following conditions are satisfied:

(i) d_{i+1} − d_{i} is even for all i = 1, …, k − 2 (for k ⩾ 3 otherwise no condition), (ii) e_{i+1} − e_{i} is even, for all i = 1, …, k − 1 (for k ⩾ 2), (iii) when 0 < e_{1} < e we also require d_{1} to be even, (iv) when 0 < e_{k} < e we also require d_{k} − d_{k−1} to be even.

2.8. Example. For any d, e ⩾ 1, both the empty diagram / □ and the full-rectangle [d × e] are even (d, e)-diagrams (see Not. 2.2). Indeed, in both cases, k = 1 and \underline{d} = (d), whereas \underline{e}( / □) = (e) and \underline{e}([d × e]) = (0); so there is no condition to check.

When d = 1 or e = 1, these are the only even Young diagram in (d×e)-frame. For more examples, the reader can find all even Young diagrams in the cases

(d, e) = (4, 4), (4, 5) and (5, 5) in Figures 14, 15 and 16, at the end of the paper.

2.9. Remark. Definition 2.7 depends on d and e as well as on the Young diagram Λ. For an even (d, e)-diagram Λ to remain even in a bigger frame, we might have one or two more conditions to check, namely (iii) or (iv) in Definition 2.7, in the case where Λ was touching the right border or the bottom border of its (d×e)-frame.

2.10. Remark. For each even (d, e)-diagram we will construct an element in one of the Witt groups of the Grassmannian Gr_{X}(d, V). The proof that these Witt classes actually form a total basis will proceed by induction on d + e = rk(V), using the long exact sequence of localization associated to a natural “cellular” decomposition of the Grassmannians. In that proof, we shall need the description in terms of Young diagrams of the various Witt group homomorphisms appearing in that long exact sequence. As we shall see, these are the ones of the next proposition. This explains why the following constructions are relevant here.

2.11. Definition. Let Λ′ be an even (d, e − 1)-diagram with d ≥ 1, e ≥ 2 and such that ζ(Λ′) is even. We define the (d, e)-diagram ῑ(Λ′) as

ῑ(Λ′) = (Λ′_{1} + 1, …, Λ′_{d} + 1).  (16)

Let Λ be an even (d, e)-diagram with d ≥ 2, e ≥ 1 and such that Λ_{d} = 0. We define the (d − 1, e)-diagram ῡ(Λ) as

ῡ(Λ) = Λ_{|d−1,e}.  (17)

Let Λ′′ be an even (d − 1, e)-diagram with d, e ≥ 2 and such that Λ′′_{d−1} is odd. We define the (d, e − 1)-diagram ¯∂(Λ′′) as

¯∂(Λ′′) = (Λ′′_{1} − 1, …, Λ′′_{d−1} − 1, 0).  (18)

See Figures 4, 5 and 6.

2.12. Proposition. (a) When d ≥ 1 and e ≥ 2, the map ῑ defines a bijection

[figure: \left\{{even Young (d, e − 1)-diagrams \atop Λ′ such that ζ(Λ′) is even}\right\}; ≃; \left\{{even Young (d, e)-diagrams \atop Λ such that Λ_{d} > 0}\right\}; Λ′; ῑ(Λ′) = (Λ′_{1} + 1, …, Λ′_{d} + 1); (Λ_{1} − 1, …, Λ_{d} − 1); Λ]

(b) When d ≥ 2 and e ≥ 2, the map ῡ defines a bijection

[figure: \left\{{even Young (d, e)-diagrams \atop Λ such that Λ_{d} = 0}\right\}; ≃; \left\{{even Young (d − 1, e)-diagrams \atop Λ′′ such that Λ′′_{d−1} is even}\right\}; Λ; ῡ(Λ) = Λ_{|d−1,e}; (Λ′′_{1}, …, Λ′′_{d−1}, 0); Λ′′]

(c) When d, e ≥ 2, the map ¯∂ defines a bijection

[figure: \left\{{even Young (d − 1, e)-diagrams \atop Λ′′ such that Λ′′_{d−1} is odd}\right\}; ≃; \left\{{even Young (d, e − 1)-diagrams \atop Λ′ such that ζ(Λ′) is odd}\right\}; Λ′′; ¯∂(Λ′′) = (Λ′′_{1} − 1, …, Λ′′_{d−1} − 1, 0); (1 + Λ′_{1}, …, 1 + Λ′_{d−1}); Λ′]

Proof. The proof essentially consists in checking that the announced constructions are well-defined and that they preserve even diagrams. Checking that they are mutually inverse constructions is straightforward. The notation Λ_{|d−1,e} is the obvious one : we view a diagram with empty last row in a smaller frame. All this is most easily performed and followed on pictures. For instance, the maps from left to right are pictured in the upper parts of Figures 4, 5 and 6 below. □

\begin{matrix}e \\ d \\ d \\ 0 \\ even \\ odd \\ e − 1 \end{matrix}

Figure 4. Morphism ῑ on various (d, e − 1)-diagrams Λ′.

\begin{matrix}d − 1 \\ d \\ 0 \\ e \\ e \end{matrix}

Figure 5. Morphism ῡ on various (d, e)-diagrams Λ.

\begin{matrix}0 \\ d − 1 \\ odd \\ even \\ d \\ e − 1 \\ e \end{matrix}

Figure 6. Morphism ¯∂ on various (d − 1, e)-diagrams Λ′′. 3. Total bases and lax-similitude

3.1. Convention. From now on, X denotes a regular noetherian ℤ[\frac{1}{2}]-scheme of finite Krull dimension.

For precise statements and proofs of our results, it is convenient to use the language of total bases and lax similitude, as developed in [7]. Here is a brief list of the relevant facts.

We restrict to the subcategory S_{X} of X-schemes π_{Y} : Y → X that are smooth over X and that satisfy the following assumptions on Picard groups and global sections [7, Definition 4.1]:

(I) The map π_{Y}^{∗} : Pic(X) → Pic(Y) is injective.

(II) The abelian group Pic_{X}(Y) := Pic(Y)/π_{Y}^{∗} ( Pic(X)) has no 2-torsion. (III) The map π_{Y}^{∗} : 𝔾_{m}(X) → 𝔾_{m}(Y)/𝔾_{m}(Y)^{2} is surjective.

This ensures that the notions considered below are well-behaved.

3.2. Remark. All schemes considered in the computations of the remaining Sections 4 to 6 are in the category S_{X}. Indeed, most are flag varieties constructed iteratively from X as towers of Grassmann bundles (see Lemma 1.11), so the Picard group assumptions follow from Proposition 1.13. Similarly, Property (III) follows from [12, Thm. 2.3.1], i.e. in all our examples, 𝔾_{m}(X) → 𝔾_{m}(Y) is already surjective. The remaining schemes are vector bundles over these flag varieties, so each of them has the same Picard group and invertible global sections as its base.

3.3. Definition ([7, Definition 2.3]). Let L_{1} and L_{2} be line bundles over a scheme Y. An alignment A : L_{1} ⇝ L_{2} is a pair A = (M, ψ) consisting of a line bundle M over Y together with an isomorphism ψ : M^{⊗2} ⊗L_{1} →^{∼} L_{2}. Of course, such an alignment exists if and only if [L_{1}] = [L_{2}] in Pic(Y)/2. It induces an isomorphism on Witt groups

A^{⟲} : W^{∗}(Y, L_{1}) →^{∼} W^{∗}(Y, L_{2})

defined as the composition of multiplication by the form M →^{∼} M^{∨} ⊗ M^{⊗2} (square periodicity) and of the identification of the dualities with values respectively in M^{⊗2} ⊗ L_{1} and in L_{2} using ψ.

When Y is a scheme in S_{X}, we also use a relative notion. An X-alignment from L_{1} to L_{2} is an alignment A : π_{Y}^{∗} K ⊗ L_{1} ⇝ L_{2} for some line bundle K over X. We denote this by A : L_{1} ⇝_{K} L_{2}.

3.4. Definition ([7, Definition 2.5]). Two Witt classes w_{1} ∈ W^{j}(Y, L_{1}) and w_{2} ∈ W^{j}(Y, L_{2}) are lax-similar if there exists an alignment A such that w_{2} = A^{⟲}(w_{1}). This is an equivalence relation written w_{1} ↭ w_{2}. Note also that w_{1} ↭ w_{2} forces [L_{1}] = [L_{2}] in Pic(Y)/2.

3.5. Definition ([7, Definition 3.4]). Let f : Ȳ → Y, let L be a line bundle on Y and let L̄ be a line bundle on Ȳ. If an alignment Ā : f^{∗}L ⇝ L̄ exists, we define a lax pull-back

Ā^{⟲} ∘ f^{∗} : W_{Z}^{∗}(Y, L) ⟶ W_{f^{−1}Z}^{∗}(Ȳ , f^{∗}L) ⟶^{∼} W_{f^{−1}Z}^{∗}(Ȳ , L̄) .

3.6. Remark. It is easy to see that two lax pull-backs (along the same morphism) of lax-similar elements are lax-similar.

3.7. Definition ([7, Definition 3.5]). Similarly, if an alignment Ā : L̄ ⇝ ω_{f} ⊗ f^{∗}L exists, we define a lax push-forward

f_{∗} ∘ Ā^{⟲} : W∗+d ; Z̄(Ȳ , L̄) ⟶^{∼} W∗+d ; Z̄(Ȳ , ω_{f} ⊗ f^{∗}L) ⟶ W_{fZ̄}^{∗}(Y, L) .

The freedom in the use of lax push-forwards is summarized in the following

3.8. Theorem. Let f : Ȳ → Y be a morphism of schemes in S_{X} and let L̄ be a line bundle over Ȳ.

(a) A lax push-forward starting from W^{∗}(Ȳ , L̄) exists if and only if

[L̄] ∈ Im([ω_{f}] ⊗ f^{∗} : Pic(Y)/2 → Pic(Ȳ)/2)  (19)

or equivalently replacing Pic(−)/2 by Pic_{X}(−)/2.

(b) Assuming (19) holds, the lax push-forward can be chosen to land in W^{∗}(Y, L) for a line bundle L on Y if and only if

[ω_{f} ⊗ f^{∗}L] = [L̄] ∈ Pic(Ȳ)/2.  (20)

(c) Given two line bundles L_{1} and L_{2} on Y both satisfying (20), lax pushforwards from W^{∗}(Ȳ , L̄) to W^{∗}(Y, L_{1}) and to W^{∗}(Y, L_{2}) are lax-similar on Y (i.e. there exists an alignment A^{⟲} : W^{∗}(Y, L_{1}) → W^{∗}(Y, L_{2}) turning one push-forward into the other) if and only if

[L_{1}] = [L_{2}] ∈ Pic(Y)/2  (21)

or equivalently replacing Pic(−)/2 by Pic_{X}(−)/2. This condition is automatically satisfied if f^{∗} : Pic_{X}(Y)/2 → Pic_{X}(Ȳ)/2 is injective, so in that case, there is no need to specify the specific target of the lax push-forward if one is only interested in lax-similitude classes of the images.

Proof. This is detailed in § 4 of [7]. The first two parts are straightforward from (1). Use [7, Lemma 4.3 (d)] to replace Pic(−)/2 by Pic_{X}(−)/2 in (a). To replace Pic(−)/2 by Pic_{X}(−)/2 in the last statement, use [7, Lemma 4.3 (c)], in which case [7, Proposition 4.7] gives that the images are lax-similar. □

3.9. Remark. Remark 3.6 and Theorem 3.8 mean that as long as one is only interested in elements up to lax-similarity, there is no need to be specific about where lax pull-backs and push-forwards start and land, as long as they exist. One only needs to keep track of classes of line bundles in Pic_{X}(−)/2. See also [7, Remark 2.10] for connecting homomorphisms.

Let Y be a scheme in S_{X} and let Z be a closed subset of Y. Let I be a set. Given a family of line bundles (L_{i})_{i∈I} and a class p ∈ Pic_{X}(Y)/2, let I_{p} denote the subset of those i ∈ I such that [L_{i}] = p. Let (w_{i})_{i∈I} be a family of Witt classes, for various shifts and twists : w_{i} ∈ W^{j_{i}}(Y, L_{i}).

3.10. Definition. Let L be a line bundle on Y and let k ∈ ℤ be an integer. Let J ⊂ I_{[L]} be a finite subset. Given a family of X-alignments (A_{i} : L_{i} ⇝_{K_{i}} L)_{i∈J} and

a family of coefficients (λ_{i})_{i∈J} with λ_{i} ∈ W^{k−j_{i}}(X, K_{i}), we can form the linear combination

∑_{i∈J} λ_{i} ·_{A_{i}} w_{i} := ∑_{i∈J} A_{i}^{⟲}(π_{Y}^{∗} (λ_{i}) · w_{i}),

which is an element of W^{k}(Y, L). See details in [7, § 6].

3.11. Definition. Let P ⊆ Pic_{X}(Y)/2 – typically P is the whole Pic_{X}(Y)/2. The family (w_{i})_{i∈I} is called a total basis of the P -part of the Witt groups of Y over X with support in Z, if [L_{i}] ∈ P for all i and if for every line bundle L with [L] ∈ P, we have the following two properties :

(a) Total generation : Any element in W^{k}(Y, L) can be obtained as a linear combination of a finite subfamily (w_{i})_{i∈J}, i.e. alignments and coefficients as in Definition 3.10 can be found yielding the element as the linear combination.

(b) Total independence : Any linear combination of the w_{i} yielding the zero element in W^{k}(Y, L) has zero coefficients.

A total basis yields the following result, expressed in classical terms.

3.12. Theorem ([7, Proposition 6.9]). For every line bundle L with [L] ∈ P, every k ∈ ℤ and for every choice, for those i ∈ I_{[L]}, of a line bundle K_{i} over X and a K_{i}-alignment C_{i} : L_{i} ⇝_{K_{i}} L, the following map is an isomorphism

\begin{matrix}θ = θ(C_{•}) : ⨁_{i∈I_{[L]}}W^{k−j_{i}}(X, K_{i}) & ⟶^{∼} & W_{Z}^{k}(Y, L) \\ (x_{i})_{i∈I_{[L]}} & ⟼ & ∑ x_{i} ·_{C_{i}} w_{i} . \end{matrix}  (22)

3.13. Remark. In the same spirit, if one replaces elements of a total basis by lax-similar ones (Def. 3.5), they still form a total basis. See [7, Cor. 6.14].

Finally, here is a way to keep track of total bases along localization sequences. This will be our main tool to construct inductively total bases for Grassmann varieties, together with homotopy invariance and dévissage, under which total bases are naturally preserved (see [7, Corollaries 6.13 and 6.16] for precise statements).

Let U be the open complement of a closed subset Z ⊂ Y, and let υ : U ↪ Y be the corresponding open embedding. Assume U ∈ S_{X}. Let e : W_{Z}^{∗}(Y, L) → W^{∗}(Y, L) be the extension of support map. Recall from [3] that there is a long exact sequence of localization

⋯ ⟶ W_{Z}^{i}(Y, L) ⟶^{e} W^{i}(Y, L) ⟶^{υ^{∗}} W^{i}(U, υ^{∗}L) ⟶^{∂} W_{Z}^{i+1}(Y, L) → ⋯ (23)

3.14. Theorem ([7, Thm. 7.1]). Let P be a subset of Pic_{X}(Y)/2. Assume that the restriction υ_{|P}^{∗} : P → Pic_{X}(U)/2 is injective and let P_{U} = υ^{∗}(P) ⊂ Pic_{X}(U)/2.

Let I, J and K be sets and let (w′_{i})_{i∈I} and (w_{j})_{j∈J} be Witt classes on Y, let (v_{i})_{i∈I} and (v′_{k})_{k∈K} be Witt classes on Y with support in Z and let (u_{k})_{k∈K} and (u′_{j})_{j∈J} be Witt classes on U, whose line bundles are restricted from Y. Suppose the following conditions hold (see Figure 7) :

(a) for every i ∈ I, we have lax-similitude e(v_{i}) ↭ w′_{i}

(b) for every j ∈ J, we have lax-similitude υ^{∗}(w_{j}) ↭ u′_{j}

(c) for every k ∈ K, we have lax-similitude ∂(u_{k}) ↭ v′_{k}. Then, the following properties are satisfied :

(1) for every i ∈ I, we have υ^{∗}(w′_{i}) = 0;

(2) for every j ∈ J, we have ∂(u′_{j}) = 0;

(3) for every k ∈ K, we have e(v′_{k}) = 0.

(4) If, furthermore,

(i) the (v_{i})_{i∈I} and (v′_{k})_{k∈K} form together a total basis of the P -part of the Witt groups of Y with support in Z, over X,

(ii) the (u_{k})_{k∈K} and (u′_{j})_{j∈J} form together a total basis of the P_{U}-part

of the Witt groups of U, over X, then the (w′_{i})_{i∈I} and (w_{j})_{j∈J} form together a total basis of the P -part of the Witt groups of Y, over X.

Finally, we will need the following fact about push-forwards along blow-up :

3.15. Proposition. Let X be a quasi-compact and quasi-separated scheme (e.g. an affine or a noetherian scheme) and let Z ↪ X be a regular immersion of pure codimension d. Let π : B → X be the blow-up of X along Z. Then

(a) There is a natural isomorphism Rπ_{∗}(O_{B}) ≅ O_{X} in the derived category of X.

(b) Assume further that X is regular and that ω_{B/X} is a square, which happens exactly when d is odd by [6, Proposition A.11 (iii)]. Then a lax push-forward (in the sense of Definition 3.7) W^{0}(B, O_{B}) → W^{0}(X, O_{X}) maps the unit class 1_{B} to an element lax-similar to the unit class 1_{X}.

\begin{matrix}v_{i} & w′_{i} & u_{k} \\ v′_{k} & w_{j} & u′_{j} \end{matrix} eυ^{∗} ∂ P -part of W_{Z}(Y) P -part of W(Y) P_{U}-part of W(U)

Figure 7. Families mapping to each other up to lax-similitude in Theorem 3.14. No arrow means mapped to zero.

Proof. Part (a) can be found in SGA 6, see [2, Lemme VII.3.5, p. 441] or the more recent account in [19, Lemme 2.3 (a)].

Part (b) follows from (a) and holds at the level of symmetric forms already, before taking Witt classes. Indeed, when d = 1, we have B = X and there is nothing to prove. When d ⩾ 3 then line bundles over X, and homomorphisms between them, are determined by their restriction to the open complement U = X ∖ Z of Z since Z is of codimension at least 2. The result follows by the base-change formula for push-forwards [8, Thm. 5.5] and by Theorem 3.8, since π_{|π^{−1}(U)} : π^{−1}(U) → U is an isomorphism. □

4. Construction of the total basis

For this section, let Λ be a Young diagram in (d×e)-frame and recall the k-tuples \underline{d} and \underline{e} associated to Λ in Definition 2.3.

4.1. Remark. Our goal is to construct classes in the total Witt group of Gr_{X}(d, V) by lax-pushing-forward the unit form 1 ∈ W(Fl_{X}(Λ)) = W^{0}(Fl_{X}(Λ), O) along the morphism f_{Λ} : Fl_{X}(Λ) → Gr_{X}(d, V) of (15). As recalled in Theorem 3.8 (a), this lax push-forward only exists conditionally, namely only when the class of the relative canonical bundle ω_{Fl_{X}(Λ)/ Gr_{X}(d,V)} in Pic_{X}(Fl_{X}(Λ))/2 belongs to the image of

(f_{Λ})^{∗} : Pic_{X}(Gr_{X}(d, V))/2 ⟶ Pic_{X}(Fl_{X}(Λ))/2 .

This is true if and only if the following conditions are satisfied:

(a) d_{i} − d_{i−1} + e_{i+1} − e_{i} is even for every i = 2, …, k − 1 (for k ⩾ 3)

(b) when 0 < e_{1} < e and k ≥ 2, require moreover d_{1} + e_{2} − e_{1} even.

We shall be more precise in Proposition 4.8 below but the reader can verify our claim using (10) in Corollary 1.14. For this, note that Δ_{d_{1}} = [det V_{d_{1}}] comes from X when e_{1} = 0 and that Δ_{d} always comes from Gr_{X}(d, V) since (f_{Λ})^{∗}(Δ_{d}^{Gr_{X}(d,V)}) = Δ_{d}^{Fl_{X}(Λ)}, as can be checked on the tautological bundles already.

Conditions (a) and (b) hold in particular when Λ is even in the sense of Definition 2.7. Indeed, for such Λ not only the sum d_{i} − d_{i−1} + e_{i+1} − e_{i} is even but actually both terms d_{i} − d_{i−1} and e_{i+1} − e_{i} are. Compare Figure 8.

When (a) and (b) hold (e.g. for Λ even), there exists a line bundle L on Gr_{X}(d, V) such that [ω_{f_{Λ}}] · f_{Λ}^{∗}[L] = 1 in Pic(Fl_{X}(Λ))/2. Therefore, there is a lax pushforward 3.7 along f = f_{Λ} i.e. a homomorphism :

W^{0} (Fl(Λ), O) ≃ W^{0} (Fl(Λ), ω_{f_{Λ}} ⊗ f_{Λ}^{∗}L) →^{(f_{Λ})_{∗}} W^{|Λ|} ( Gr_{X}(d, V), L) .

2 3 4

Figure 8. Framed Young diagram satisfying Conditions (a) and (b) of Remark 4.1 but which is not even (at all).

(Use that dim(f_{Λ}) = dim Fl_{X}(Λ) − dim Gr_{X}(d, V) = −|Λ| by Propositions 1.3 and 1.13.) Consequently, we can produce a Witt class over Gr_{X}(d, V), by pushing the unit form in the first group. This is what we are going to do below for Λ even, making the class of L in Pic(Gr_{X}(d, V))/2 more explicit in terms of the diagram Λ. Once this class of L in Pic(Gr_{X}(d, V))/2 or in Pic_{X}(Gr_{X}(d, V))/2 is fixed, the choices involved in the lax push-forward of Definition 3.7 will be irrelevant. A nice fact is that this class in the Picard group can be read very easily on the diagram, as we now explain.

4.2. Remark. The perimeter of a Young diagram Λ is an even integer. Indeed, from the lower-left corner of Λ to its upper-right corner, there are two paths which follow the boundary (the upper path and the lower path) and they have the same length, namely the lattice distance between these two corners.

4.3. Definition. Let Λ be a Young diagram. We define t(Λ) ∈ ℤ/2 to be the class of half the perimeter of Λ. From the above remark, t(Λ) is also the class of the (lattice) distance from the lower-left corner of Λ to its upper-right corner. That is :

t(Λ) = [Λ_{1} + ρ(Λ)] ∈ ℤ/2

where ρ(Λ) is the number of non-zero rows of Λ (Def. 2.6). Note that this Definition does not depend on an ambient frame.

4.4. Remark. On an even Young diagram Λ in (d×e)-frame, there is another way to read t(Λ) ∈ ℤ/2 on the diagram. Add the (parity of) the length of the segments where Λ touches the right and the bottom of the frame. See Figure 9. This is justified and generalized in Proposition 4.5.

\begin{matrix}1 \\ 2 \\ 1 \\ 1 \\ 5 \\ 4 \\ t(Λ) = 0 \\ t(Λ) = 1 \\ t(Λ) = 0 \\ t(Λ) = 0 \\ t(Λ) = 1 \end{matrix}

Figure 9. Class t(Λ) ∈ ℤ/2, for different Λ.

4.5. Proposition. Let Λ be an even Young diagram in (d×e)-frame and let \underline{d}, \underline{e} be the associated k-tuples (Def. 2.3). Then t(Λ) = [d_{i} + (e − e_{j})] ∈ ℤ/2 for any i, j ∈ {1, …, k} such that e_{i} < e (only i = k should be avoided when e_{k} = e).

Proof. Measure the half-perimeter of Λ as the length of the lower boundary of Λ, from the lower-left corner of Λ to its upper-right corner (see Rem. 4.2). Since Λ is even, all segments on this lower half-perimeter which are not on the outside frame have even length. So, the only two segments to contribute to the lower half-perimeter with possible odd length, are on the outside (d×e)-frame, i.e. :

• the vertical segment most to the right, which has length d_{1}, and

• the lowest horizontal segment, which has length e − e_{k} when Λ touches the lower part of the (d×e)-frame (otherwise e_{k} = e and this length is even).

In any case, this shows that t(Λ) = [d_{1} + (e − e_{k})] ∈ ℤ/2, that is, the announced formula for i = 1 and j = k. The other formulas follow from this one since by Definition 2.7 the successive differences d_{i} − d_{i−1} and e_{j+1} − e_{j} are even for all i = 2, …, k − 1, for all j = 1, …, k − 1, and also for i = k when e_{k} < e. □

4.6. Definition. Let Λ be an even Young diagram in (d×e)-frame. We define the twist T (Λ) of Λ as the following class in Pic(Gr_{X}(d, V))/2 = Pic(X)/2 ⊕ ℤ/2 · Δ_{d} (see Corollary 1.4 and recall that Δ_{d} is the determinant of the tautological bundle) :

T (Λ) = T (Λ, d, e) := [det V]^{ρ(Λ)} · Δ_{d}^{t(Λ)} ,

where we recall that t(Λ) is the half-perimeter of Λ modulo 2 (Def. 4.3) and that ρ(Λ) is the number of non-zero rows of Λ (Def. 2.6).

4.7. Remark. The important part of T (Λ) is of course Δ_{d}^{t(Λ)}, which is not coming from the base X. Also, when V is trivial, the other term disappears anyway; this holds in particular with X = Spec(R) for a local ring R (e.g. a field).

4.8. Proposition. Let Λ be an even Young diagram in (d×e)-frame. Then

[ω_{f_{Λ}}] · f_{Λ}^{∗}(T (Λ)) = 1 in Pic (Fl_{X}(Λ))/2 .

Proof. Suppose first that k = k(Λ) is at least 2. Remove from (10) all even exponents coming from the fact that Λ is even and use Proposition 4.5 for i = k − 1 and j = k. This gives in Pic (Fl_{X}(Λ))/2 :

[ω_{f_{Λ}}] = [det V_{d_{1}+e_{1}}]^{d_{1}} · [det V_{d+e_{k}}]^{d+d_{k−1}} · [det V]^{d} · Δ_{d_{1}}^{d_{1}} · Δ_{d}^{t(Λ)} .

Now observe that [det V_{d_{1}+e_{1}}]^{d_{1}} · Δ_{d_{1}}^{d_{1}} = 1 in Pic (Fl_{X}(Λ))/2. Indeed, either e_{1} > 0 hence d_{1} is even, or e_{1} = 0 hence Δ_{d_{1}} = [det V_{d_{1}}] by Remark 1.10. So, we can simplify the above equation in Pic (Fl_{X}(Λ))/2 :

[ω_{f_{Λ}}] = [det V_{d+e_{k}}]^{d+d_{k−1}} · [det V]^{d} · Δ_{d}^{t(Λ)} .

Now, if e_{k} < e then d − d_{k−1} = d_{k} − d_{k−1} is even and ρ(Λ) = d; on the other hand, if e_{k} = e, then ρ(Λ) = d_{k−1}. In both cases, the above expression becomes [det V]^{ρ(Λ)} · Δ_{d}^{t(Λ)}, which is T (Λ) by Definition 4.6.

Similarly, the case k = 1 is an easy consequence of Corollary 1.14. □

Let Λ be an even Young diagram in (d×e)-frame. Choose L_{Λ} a line bundle of class T (Λ) in Pic ( Gr_{X}(d, V))/2. (For simplicity, choose L\begin{matrix}/ \\ □ \end{matrix} = O.) By the equality of Proposition 4.8, there exists an isomorphism ψ_{Λ} : M_{Λ}^{⊗2} →^{∼} ω_{f_{Λ}} ⊗f_{Λ}^{∗}L_{Λ} over Fl_{X}(Λ), i.e. an alignment A_{Λ} = (M_{Λ}, ψ_{Λ}) : O_{Fl_{X}(Λ)} ⇝ ω_{f_{Λ}} ⊗ f_{Λ}^{∗}L_{Λ}, which induces a lax push-forward along the morphism f_{Λ} : Fl_{X}(Λ) ⟶ Gr_{X}(d, V), as in Definition 3.7 :

(f_{Λ})_{∗} ∘ A_{Λ}^{⟲} : W^{0} (Fl_{X}(Λ), O) ⟶ W^{|Λ|} ( Gr_{X}(d, V), L_{Λ}) .  (24)

4.9. Definition. For any even Young diagram Λ in (d×e)-frame, we define

ϕ_{d,e}(Λ) ∈ W^{|Λ|} ( Gr_{X}(d, V) , L_{Λ})  (25)

as the image of the unit form 1 ∈ W^{0}(Fl_{X}(Λ), O) by the lax push-forward (24).

4.10. Remark. The definition of ϕ_{d,e}(Λ) involves the choice of L_{Λ} in the class T (Λ) and the choices of M_{Λ} and of the isomorphism ψ_{Λ} : M_{Λ}^{⊗2} →^{∼} ω_{f_{Λ}} ⊗ f_{Λ}^{∗}L_{Λ}. By Theorem 3.8 (c), the lax-similitude class of ϕ_{d,e}(Λ) is independent of these choices. By Remark 3.13, our Main Theorem 6.1 (that they form a total basis) will hold regardless of these choices. For these reasons, and also to lighten notation, we do not incorporate these choices in the notation ϕ_{d,e}(Λ).

4.11. Example. Let Λ = / □ be the empty Young diagram in (d×e)-frame, which is even, as we know. Then T (Λ) is trivial and f\begin{matrix}/ \\ □ \end{matrix} : Fl_{X}(d, e; / □) → Gr_{X}(d, V) is the identity, so ω_{f_{Λ}} = O and ϕ_{d,e}( / □) = 1 ∈ W^{0}(Gr_{X}(d, V), O) is the unit form on the Grassmannian if L_{Λ} = M_{Λ} = O and the isomorphism M_{Λ}^{⊗2} ≃ ω_{f_{Λ}} ⊗ f_{Λ}^{∗}L_{Λ} is the trivial one. (Otherwise, ϕ_{d,e}( / □) is only lax-similar to the unit form.)

4.12. Remark. By Proposition 3.15, the lax-similitude class of our Witt classes ϕ_{d,e}(Λ) does not really depend on the chosen desingularization Fl(Λ) of the Schubert subvariety corresponding to the Young diagram Λ. Indeed the unit will remain the unit when pushed-forward between two such desingularizations, if one is obtained from the other by blow-up. (The condition on the relative bundle being even in Proposition 3.15 is automatically satisfied if both desingularizations have even relative bundle with respect to the Grassmannian.)

5. Cellular decomposition

We describe the usual relative cellular decomposition of Grassmannians. Fix d, e ⩾ 2 for the whole section.

5.1. Notation. Fix a complete flag V_{•} of vector bundles on X

0 = V_{0} ⊲ V_{1} ⊲ ⋯ ⊲ V_{d+e} = V ,

as in (13). We set V^{1} = V_{d+e−1} to be the chosen codimension one subbundle of V. We have an obvious closed immersion Gr_{X}(d, V^{1}) ↪ Gr_{X}(d, V), of codimension d, whose open complement is denoted by U_{X}(d, V_{•}).

5.2. Notation. Let P_{d} ⊲ V be a subbundle of rank d. We write P_{d} ˙⋪V^{1} to express that P_{d} is not a subbundle of V^{1} but moreover satisfies the equivalent conditions :

(a) The natural map from P_{d}/(P_{d} ∩ V^{1}) = (P_{d} + V^{1})/V^{1} into V/V^{1} is an isomorphism.

(b) P_{d} ∩ V^{1} is a subbundle of P_{d} (in the strong sense of Definition 1.1). Over a field, this amounts to P_{d} ⊄ V^{1} but this is not sufficient in general.

5.3. Definition. Using notation of Section 1, we have a commutative diagram

[figure: Gr_{X}(d, V^{1}); ι; |; Gr_{X}(d, V); U_{X}(d, V_{•}); υ; ∘; α; υ̃; ∘; Fl_{X}((d − 1, d), (e, e − 1)); π̃; ι̃; |; Fl_{X}((d − 1, d), (e, e)); π; α̃; Fl_{X}(d − 1, V^{1}) .] (26)

which looks as follows on points :

[figure: {P_{d} ⊲ V^{1}}; ι; |; {P_{d} ⊲ V}; {P_{d} ˙⋪V^{1}}; υ; ∘; υ̃; ∘; α; {P_{d−1} ⊲ P_{d} ⊲ V^{1}}; π̃; ι̃; |; {P_{d−1} ⊲ P_{d} ⊲ V | P_{d−1} ⊲ V^{1}}; π; α̃; {P_{d−1} ⊲ V^{1}} .] (27)

Here ι, ι̃, π, π̃ and α̃ are the obvious morphisms. The morphism υ̃ maps P_{d} to the flag P_{d−1} ⊲ P_{d} with P_{d−1} := P_{d} ∩ V^{1} (see Not. 5.2). Finally α is defined as α̃ ∘ υ̃.

5.4. Proposition. In Diagram (26), the scheme Fl_{X}((d−1, d), (e, e)) is the blow-up of Gr_{X}(d, V) along Gr_{X}(d, V^{1}) with exceptional fiber Fl_{X}((d − 1, d), (e, e − 1)).

Proof. This is probably folklore to algebraic geometers. By compatibility of blowups with pull-backs, we can reduce to the case where X is affine (even X = Spec(ℤ)) and suppose that V is free and that V^{1} = ker(V ↠ O) is the kernel of a (split) epimorphism to O. We omit X in the notation for the rest of the proof.

Let us check that B := Fl((d−1, d), (e, e)) has the universal property of the blowup (see [17, § 8.1.2, Corollary 1.16]), i.e. it is final among schemes over Gr(d, V) in which the preimage of Z := Gr(d, V^{1}) is an effective Cartier divisor (i.e. a codimension one closed subscheme locally given by a principal ideal). Let us first check that B indeed has this property. Note that the left-hand square of (26) is cartesian. Moreover, we have an identification B = Fl((d − 1, d), (e, e)) = ℙ_{Y} (V/T_{d−1}) where Y = Gr(d − 1, V^{1}) as in Lemma 1.11. Under this identification, the inverse image Fl((d − 1, d), (e, e − 1)) of Z becomes ℙ_{Y} (V^{1}/T_{d−1}). So this inverse image is locally ℙ_{Y}^{e−1} ⊂ ℙ_{Y}^{e} hence an effective Cartier divisor, as wanted.

Suppose now that f : W → Gr_{X}(d, V) is a morphism for which f^{−1}(Gr_{X}(d, V^{1})) is an effective Cartier divisor. Let us consider W as a functor of points and show that there exists a unique morphism g : W → B such that π̃ ∘ g = f. Consider on Gr(d, V), the morphism s : T_{d} → O obtained by composing the inclusion T_{d} ↣ V and the projection V ↠ V/V^{1} = O. By definition, Z is the zero locus of this morphism, i.e. it is defined by the ideal im(s) ⊂ O_{Gr(d,V)}. By assumption on f : W → Gr_{X}(d, V), the ideal im(f^{∗}(s)) = f^{−1}(im(s)) ⊂ O_{W} is invertible. Hence, over each point Spec(R) → W of W, ker(s_{|R}) is a codimension one subbundle of T_{d}, which is contained in V^{1} by construction. This defines the wanted morphism g : W → B sending each Spec(R) → W to ker(s_{|R}) ⊲ P_{d}. Uniqueness is easy. If g : W → B satisfies g ∘ π̃ = f then g maps a point w : Spec(R) → W to P_{d−1} ⊲ P_{d} with P_{d} = f(w) forced. On the other hand P_{d−1} ⊂ V^{1} forces P_{d−1} to be in ker(s_{|R}), hence to be equal to it by dimension counting. □

5.5. Definition. Let B_{X}(d, V_{•}) = Fl_{X}((d−1, d), (e, e)) be the blow-up of Gr_{X}(d, V) along Gr_{X}(d, V^{1}) and let E_{X}(d, V_{•}) = Fl_{X}((d − 1, d), (e, e − 1)) be the exceptional fiber. By (26), Gr_{X}(d, V) now has a decomposition like in [6, Hypothesis 1.2], namely there exists an auxiliary morphism α̃ : B_{X}(d, V) → Y := Gr_{X}(d − 1, V^{1}) from the blow-up to another scheme Y, such that α := α̃ ∘ υ̃ is an 𝔸^{∗}-bundle :

[figure: Gr_{X}(d, V^{1}); ι; |; Gr_{X}(d, V); U_{X}(d, V_{•}); υ; ∘; υ̃; ∘; α; E_{X}(d, V_{•}); π̃; ι̃; |; B_{X}(d, V_{•}); π; α̃; Gr_{X}(d − 1, V^{1}) .]  (28)

Indeed, α is an 𝔸^{e}-bundle because of the canonical isomorphism between U_{X}(d, V_{•}) and ℙ_{Y} (V/T_{d−1}^{Y}) ∖ ℙ_{Y} (V^{1}/T_{d−1}^{Y}), under which α corresponds to the structure morphism to Y.

5.6. Remark. We compute the relevant Picard groups and canonical bundles, via the methods of Section 1. Let us start with Picard groups, using (8). Since Pic(X) is a direct summand of the Picard group of all schemes in (28), we focus on the relative Picard groups Pic_{X}(−) := Pic(−)/ Pic(X). Then “Pic_{X}(−) of (28)” equals

[figure: ℤΔ_{d}; π̃^{∗}=( \begin{matrix}0 \\ 1 \end{matrix} ); ℤΔ_{d}; ι^{∗}=1; π^{∗}=( \begin{matrix}0 \\ 1 \end{matrix} ); υ^{∗}=1; ℤυ^{∗}(Δ_{d}) = ℤα^{∗}(Δ_{d−1}); ℤΔ_{d−1} ⊕ ℤΔ_{d}; ℤΔ_{d−1} ⊕ ℤΔ_{d}; ι̃^{∗}=( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); υ̃^{∗}=( 1 1 ); ℤΔ_{d−1}; α̃^{∗}=( \begin{matrix}1 \\ 0 \end{matrix} ); α^{∗}=1] (29)

(In the case X = Spec(R) for a local ring R, the Picard groups are exactly as above.) Here we used that the closed subscheme Gr_{X}(d, V^{1}) is of codimension d ⩾ 2 in Gr_{X}(d, V) to see that υ^{∗} : Pic(Gr_{X}(d, V)) ≅ Pic(U_{X}(d, V_{•})). We also used that e ⩾ 2, otherwise e − 1 = 0 and Δ_{d−1} ∈ Pic(X) by Remark 1.10. (When d = 1, resp. e = 1, we loose all components ℤΔ_{d−1}, resp. all components ℤΔ_{d} in the left column, in the previous diagram.) Alternatively, “Pic_{X}((28)) = (29)” follows from the computation of the Picard groups provided in [6, Proposition A.6]. Finally, the maps into the upper right corner of Diagram (29) are deduced from

υ^{∗}(Δ_{d}) · α^{∗}(Δ_{d−1})^{−1} = [V/V^{1}] ,  (30)

which itself follows from Condition (a) in Notation 5.2.

We shall use push-forwards along some morphisms of (28). The classes of the relevant relative canonical bundles in the respective (plain) Picard groups are :

[ω_{ι}] = [V/V^{1}]^{d} · Δ_{d}^{−1}  (31)
[ω_{π}] = [V/V^{1}]^{d−1} · Δ_{d−1}^{d−1} · Δ_{d}^{1−d}  (32)
[ωι̃] = [V/V^{1}] · Δ_{d−1} · Δ_{d}^{−1}  (33)
[ωπ̃] = Δ_{d−1}^{d} · Δ_{d}^{1−d} .  (34)

Indeed, Corollary 1.14 gives (31), (32) and [ω_{ιπ̃}] = [ω_{πι̃}] = [V/V^{1}]^{d} · Δ_{d−1}^{d} · Δ_{d}^{−d}, out of which the other two follow by multiplicativity of ω_{−}. Again, when the fixed complete flag V_{•} = OX^{•} is trivial, the “noise” V/V^{1} vanishes.

We end this section with two geometric lemmas which will be useful in the proof of the main theorem.

5.7. Lemma. Let d, e ⩾ 2 and let Λ be an even Young (d, e)-diagram with empty last row (i.e. Λ_{k} = 0, i.e. ζ(Λ) > 0). Hence Λ_{|d−1,e} is an even (d − 1, e)-diagram. Then the base-changes to U_{X}(d, V_{•}) of the morphisms f_{d,e;Λ} and f_{d−1,e;Λ_{|_{d−1,e}}} coincide, that is, we have two cartesian squares

[figure: Gr_{X}(d, V); □; U_{X}(d, V_{•}); υ; α; □; Gr_{X}(d − 1, V^{1}); Fl_{X}(d, e; Λ); f_{d,e;Λ}; U′; α′; Fl_{X}(d − 1, e; Λ_{|d−1,e}) .; f_{d−1,e;Λ_{|_{d−1,e}}}]  (35)

Proof. Let us check this on points. Let \underline{d} and \underline{e} be the k-tuples associated to Λ as usual (Def. 2.3). We need to distinguish two cases, namely d_{k} − d_{k−1} > 1 and d_{k} − d_{k−1} = 1.

When d_{k} > d_{k−1} + 1, that is, when there is more than one zero line at the end of Λ (i.e. ζ(Λ) > 1), we then have k(Λ_{|d−1,e}) = k(Λ) = k and the k-tuples \underline{d}(Λ_{|d−1,e}) and \underline{e}(Λ_{|d−1,e}) are almost the same as \underline{d} and \underline{e} except for the last entry of \underline{d}(Λ_{|d−1,e}) which becomes d − 1. Diagram (35) then looks as follows on points (as usual the P_{i} and P′_{j} are “variables” whereas the V_{i} belong to the fixed complete flag):

[figure: ^{{P_{d}^{⊲} V}}; ^{{P_{d}^{˙⋪}V^{1}}}; ^{{P′_{d−1}^{⊲} V^{1}}}; ⋯ ^{⊲} P_{d_{k−1}} ⊲^{e_{k−1}}  ^{⊲} P_{d} ⊲^{e} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V; { ; |; } ; |; ⋯ ^{⊲} P_{d_{k−1}} ⊲^{e_{k−1}}  ^{⊲} P_{d} ⊲^{e} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V; { ; |; } ; |; V^{1} ; ˙⋪; ⋯ ^{⊲} P_{d_{k−1}} ⊲^{e_{k−1}}  ^{⊲} P′_{d−1} ⊲^{e} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V^{1}; { ; |; } ; |; α; f_{d,e;Λ}; f_{d−1,e;Λ_{|_{d−1,e}}}; α′; □; □]

where the morphisms α sends P_{d} to P′_{d−1} := P_{d} ∩ V^{1} and similarly for α′ .

On the other hand, when d_{k} = d_{k−1} + 1, that is, when Λ has only one zero line (i.e. ζ(Λ) = 1), then we have k(Λ_{|d−1,e}) = k(Λ) − 1 = k − 1 and the (k − 1)-tuples \underline{d}(Λ_{|d−1,e}) and \underline{e}(Λ_{|d−1,e}) are respectively \underline{d} and \underline{e} truncated from their last entry. Diagram (35) then looks as follows on points :

[figure: ^{{P_{d}^{⊲} V}}; ^{{P_{d}^{˙⋪}V^{1}}}; ^{{P′_{d−1}^{⊲} V^{1}}}; ⋯ ^{⊲} P_{d_{k−1}} ⊲^{e_{k−1}}  ^{⊲} P_{d} ⊲^{e} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V; { ; |; } ; |; ⋯ ^{⊲} P_{d_{k−1}} ⊲^{e_{k−1}}  ^{⊲} P_{d} ⊲^{e} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V; { ; |; } ; |; V^{1} ; ˙⋪; ⋯ ^{⊲} P_{d_{k−1}} ⊲^{e_{k−1}} ⋯_{⊲} V_{d_{k−1}+e_{k−1}}; { ; |; } ; |; α; f_{d,e;Λ}; f_{d−1,e;Λ_{|_{d−1,e}}}; α′; □; □]

where α still sends P_{d} to P′_{d−1} := P_{d} ∩ V^{1} and where f_{d−1,e;Λ_{|_{d−1,e}}} sends a flag to P_{d_{k−1}}. Note that in this case, α′ drops the last subspace P_{d} in the flag.

In both cases, it is easy to check that the two squares are cartesian. □

5.8. Lemma. Let d, e ⩾ 2 and let Λ′′ be an even (d−1, e)-diagram such that Λ′′_{d−1} is odd. Hence we can consider the even (d, e−1)-diagram Λ′ = (Λ′′_{1}−1, …, Λ′′_{d−1}−1, 0). Then, there exists a commutative diagram

[figure: Gr_{X}(d, V^{1}); E_{X}(d, V_{•}); α̃ ι̃; π̃; Gr_{X}(d − 1, V^{1}); Fl_{X}(d, e − 1; Λ′); f_{d,e−1;Λ′}; F′; f′; π′; □; Fl_{X}(d − 1, e; Λ′′); f_{d−1,e;Λ′′}]  (36)

where E_{X}(d, V_{•}) is the exceptional fiber of Diagram (28) and where the right-hand square is cartesian. Moreover, either π′ is an isomorphism or the scheme F′ (with the morphism π′) identifies with the blow-up of Fl_{X}(d, e − 1; Λ′) along a closed regular subscheme of odd codimension.

Proof. Let k = k(Λ′′), \underline{d} = \underline{d}(Λ′′) and \underline{e} = \underline{e}(Λ′′) as usual (Def. 2.3). We need to distinguish two cases, namely Λ′′_{d−1} > 1 and Λ′′_{d−1} = 1.

Suppose first that Λ′′_{d−1} > 1. Then k(Λ′) = k + 1 and \underline{d}(Λ′) and \underline{e}(Λ′) are just \underline{d}(Λ′′) and \underline{e}(Λ′′) with one more entry at the end, namely d and e − 1 respectively. We can describe the pull-back in Diagram (36) as follows (on points):

[figure: ^{{P_{d−1}^{⊲} P_{d}^{⊲} V^{1}}}; ^{{P_{d−1}^{⊲} V^{1}}}; P_{d_{1}} ⊲ ^{⊲} ⋯ ^{⊲} P_{d−1} ⊲ ^{⊲} P_{d} ⊲ V_{d_{1}+e_{1}} _{⊲} ⋯_{⊲} V_{d−1+e_{k}} _{⊲} V^{1}; { ; |; } ; |; P_{d_{1}} ⊲ ^{⊲} ⋯ ^{⊲} P_{d−1} ⊲ V_{d_{1}+e_{1}} _{⊲} ⋯_{⊲} V_{d−1+e_{k}}; { ; |; } ; |; f′; □]

This pull-back F′ is Fl_{X}(d, e − 1; Λ′). So, take π′ = id. The morphism f′ composed with π̃ sends a flag P_{d_{1}} ⊲ ⋯ ⊲ P_{d−1} ⊲ P_{d} to P_{d} and so does f_{d,e−1;Λ′}.

Suppose now that Λ′′_{d−1} = 1. Then k(Λ′) = k and \underline{e}(Λ′) = \underline{e}(Λ′′), whereas \underline{d}(Λ′) is obtained from \underline{d}(Λ′′) by replacing its last entry by d. We can describe the pull-back in Diagram (36) as follows :

[figure: ^{{P_{d−1}^{⊲} P_{d}^{⊲} V^{1}}}; ^{{P_{d−1}^{⊲} V^{1}}}; P_{d_{1}} ⊲ ^{⊲} ⋯ ^{⊲} P_{d_{k−1}} ⊲ ^{⊲} P_{d−1} ⊲ ^{⊲} P_{d} ⊲ V_{d_{1}+e_{1}} _{⊲} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V_{d+e−2}_{⊲} V^{1}; { ; |; } ; |; P_{d_{1}} ⊲ ^{⊲} ⋯ ^{⊲} P_{d−1} ⊲ V_{d_{1}+e_{1}} _{⊲} ⋯_{⊲} V_{d+e−2}; { ; |; } ; |; f′; □]

where f′ is the obvious morphism. The left-hand square of (36) is defined by :

[figure: ^{{P_{d}^{⊲} V^{1}}}; ^{{P_{d−1}^{⊲} P_{d}^{⊲} V^{1}}}; P_{d_{1}} ⊲ ^{⊲} ⋯ ^{⊲} P_{d_{k−1}} ⊲ ^{⊲} P_{d} ⊲ V_{d_{1}+e_{1}} _{⊲} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V^{1}; { ; |; } ; |; P_{d_{1}} ⊲ ^{⊲} ⋯ ^{⊲} P_{d_{k−1}} ⊲ ^{⊲} P_{d−1} ⊲ ^{⊲} P_{d} ⊲ V_{d_{1}+e_{1}} _{⊲} ⋯_{⊲} V_{d_{k−1}+e_{k−1}} _{⊲} V_{d+e−2}_{⊲} V^{1}; { ; |; } ; |; f′; π̃; π′]

The morphism π′ : F′ → Fl_{X}(d, e − 1; Λ′) simply drops P_{d−1} in this case.

Let V^{2} := V_{d+e−2} and Y := Fl_{X}((d_{1}, …, d_{k−1}), (e_{1}, …, e_{k−1}), V_{•}). We have Fl_{X}(d, e−1; Λ′) = Gr_{Y} (d−d_{k−1}, V^{1}/T_{d_{k−1}}) by Lemma 1.11. As in Definition 5.5, we consider the blow-up B_{Y} (d−d_{k−1} , V^{2}/T_{d_{k}−1}⊲V^{1}/T_{d_{k}−1}) of Gr_{Y} (d−d_{k−1}, V^{1}/T_{d_{k−1}}) along the closed regular immersion of Gr_{Y} (d−d_{k−1}, V^{2}/T_{d_{k−1}}). By Proposition 5.4, this blow-up coincides with F′ and the morphism π of (27) here becomes the above morphism π′. In other words, the following diagram commutes :

[figure: Fl_{X}(d, e − 1; Λ′); F′; π′; Gr_{Y} (d − d_{k−1}, V^{1}/T_{d_{k−1}}); B_{Y} (d − d_{k−1}, V^{2}/T_{d_{k}−1} ⊲ V^{1}/T_{d_{k−1}}) .; “π”]

The closed immersion Gr_{Y} (d − d_{k−1}, V^{2}/T_{d_{k−1}}) ↪ Gr_{Y} (d − d_{k−1}, V^{1}/T_{d_{k−1}}) is of odd codimension equal to d − d_{k−1}. □

6. Main result

We are now ready to state and prove the main result of the paper.

6.1. Theorem. Let d, e ⩾ 1. Let X be a regular scheme over ℤ[\frac{1}{2}]. Let V be a vector bundle of rank d + e which admits a complete flag (13) of subbundles (for instance, V free). Then, the elements ϕ_{d,e}(Λ) of Definition 4.9, for Λ even, form a total basis of the Witt groups of Gr_{X}(d, V) over X.

Theorem 6.1 is unchanged when the ϕ_{d,e}(Λ) are changed to lax-similar elements, so it holds independently of all choices in the construction of ϕ_{d,e}(Λ), as mentioned in Remark 4.10. Furthermore, lax pull-backs, push-forwards and connecting homomorphisms are compatible with lax-similarity by Remark 3.9. We therefore use the following convention :

6.2. Convention. All pull-backs, push-forwards and connecting homomorphisms in the rest of Section 6 are lax.

This precisely means that they are classical pull-backs, push-forwards and connecting homomorphisms, preceded or followed by alignment morphisms in order to start and end in Witt groups with specific line bundles, chosen for the various statements to make sense. These alignments exist and the freedom in their choices has no influence on the lax-similarity classes considered, by the results of [7], as summarized in Section 3.

6.3. Remark. The case d = 1 or e = 1 is that of the projective bundle ℙ(V). In that case, Walter [20] has proved such a result (see [20, Proposition 8.1] when V has a complete flag). One can check that his generator ξ is lax-similar in the sense of Definition 3.4 to our ϕ(Λ) when Λ = [d × e] is the only non-empty even diagram in (d×e)-frame. The same result was also proved in the case V = O_{X}^{d+e} in [5], using methods closer to the present geometric philosophy. We could therefore assume the starting point of the induction (i.e. the case d = 1 or e = 1). However our proof of the induction step covers this “fringe” case as well.

Proof of Theorem 6.1. The proof occupies the rest of Section 6. We proceed by induction on rk(V) = d + e. The initial case d = e = 1 will follow from the same proof, except that we shall not use any induction hypothesis in that case. In other words, from now on, when d + e ≥ 2, we can assume the result for vector bundles V of rank less than d + e. We fix a complete flag

0 = V_{0} ⊲ V_{1} ⊲ ⋯ ⊲ V_{d+e} = V .

We set as before V^{1} := V_{d+e−1}.

The idea of the proof is to use Theorem 3.14 in the localization setting of Section 5, i.e. for the regular closed immersion ι : Z = Gr_{X}(d, V^{1}) ↪ Gr_{X}(d, V) with open complement U = U_{X}(d, V_{•}). For every line bundle L on Gr_{X}(d, V), the localization long exact sequence (23) reads here

(37) [figure: ···; W_{Z}^{i}(Gr_{X}(d, V), L); e; W^{i}(Gr_{X}(d, V), L); υ^{∗}; W^{i}(U_{X}(d, V_{•}), υ^{∗}L); ∂; W_{Z}^{i}(Gr_{X}(d, V), L); e; W^{i+1}(Gr_{X}(d, V), L); υ^{∗}; W^{i+1}(U_{X}(d, V_{•}), υ^{∗}L); ···]

where e : W_{Z}^{i}(Gr_{X}(d, V), L) → W^{i}(Gr_{X}(d, V), L) is extension of support.

6.4. Remark. The reader will notice along the proof that when d = 1 (resp. e = 1), the situation is somewhat particular. This is because the map υ^{∗} (resp. ι^{∗}) on relative Picard groups modulo 2 is no longer injective. So, some adjustments need to be made. In a first reading, it might be a good idea to skip these special cases to follow more easily the structure of the proof.

Before applying Theorem 3.14, we first need to use dévissage to obtain a total basis of the Witt groups of Gr_{X}(d, V) with support in Gr_{X}(d, V^{1}), from a total basis of the Witt groups of Gr_{X}(d, V^{1}). Let (ι_{Z})_{∗} be the push-forward along ι from the Witt groups of Z to the Witt groups with support in Z. We therefore have e ∘(ι_{Z})_{∗} = ι_{∗}, where e is the extension of support from Z to the whole Gr_{X}(d, V).

6.5. Lemma. Assume e ≥ 2 (and recall Convention 6.2). The elements

(ι_{Z})_{∗}(ϕ_{d,e−1}(Λ′′)) with Λ′′ even (d, e − 1)-diagram

form a total basis of the Witt groups of Gr_{X}(d, V) with support in Z = Gr_{X}(d, V^{1}).

Proof. Since e ≥ 2 the map ι^{∗} : Pic_{X}(Gr_{X}(d, V))/2 → Pic_{X}(Gr_{X}(d, V^{1}))/2 is an isomorphism (see Remark 5.6). We can therefore apply [7, Corollary 6.16] with P = Pic_{X}(Gr_{X}(d, V)), using that the ϕ_{d,e−1}(Λ′′) form a total basis of the Witt groups of Gr_{X}(d, V^{1}) by the induction assumption. □

When e = 1, the situation is slightly different. Since Gr_{X}(d, V^{1}) = X, the map ι^{∗} is no longer injective on Picard groups modulo 2, and there are two essentially different lax push-forwards. Indeed, in that case Pic_{X}(Gr_{X}(d, V))/2 = ℤ/2 · Δ_{d} and Pic_{X}(X)/2 = 1. Observe the convenient Pic_{X}(Gr_{X}(d, V))/2 = {Δ_{d}^{d} , Δ_{d}^{d+1}}.

6.6. Definition. Assume e = 1 and therefore Gr_{X}(d, V^{1}) = X. Let L_{d} (resp. L_{d+1}) be a line bundle on Gr_{X}(d, V) such that [L_{d}] = Δ_{d}^{d} ∈ Pic_{X}(Gr_{X}(d, V))/2 (resp. [L_{d+1}] = Δ_{d}^{d+1}). Let δ_{d} (resp. δ_{d+1}) be a lax push-forward of the unit form 1_{X} to W_{X}^{∗}(Gr_{X}(d, V), L_{d}) (resp. to W_{X}^{∗}(Gr_{X}(d, V), L_{d+1})) along ι : X ↪ Gr_{X}(d, V).

6.7. Lemma (Analogue of Lemma 6.5 in special case). Assume e = 1. Then

(a) The element δ_{d} (resp. δ_{d+1}) forms a total basis of the {Δ_{d}^{d}}-part (resp. the {Δ_{d}^{d+1}}-part) of the Witt groups of Gr_{X}(d, V) with support in X = Gr_{X}(d, V^{1}).

(b) Together, δ_{d} and δ_{d+1} form a total basis of the Witt groups of Gr_{X}(d, V) with support in X = Gr_{X}(d, V^{1}).

Proof. Part (a) follows from [7, Corollary 6.15], applied to the subset P = {Δ_{d}^{d}} (resp. P = {Δ_{d}^{d+1}}). Part (b) follows from (a) since the union of total bases of disjoint parts P_{1} and P_{2} forms a total basis of the part P = P_{1} ∪ P_{2}. See [7, Lemma 6.10]. Here we use Pic_{X}(Gr_{X}(d, V))/2 = {Δ_{d}^{d}} ⊔ {Δ_{d}^{d+1}}. □

We need to track our classes ϕ_{d,e}(Λ) along the morphisms involved in the long exact sequence of localization. Recall the constructions ῑ, ῡ and ¯∂ on even Young diagrams (Def. 2.11) and the lax-similarity equivalence relation ↭ (Def. 3.4).

6.8. Proposition. With the above notation (and with Convention 6.2), we have

(a) Let Λ′ be an even Young diagram in (d×(e − 1))-frame, with d ≥ 1, e ≥ 2. The push-forward ι_{∗} satisfies (see Figure 4):

ι_{∗}(ϕ_{d,e−1}(Λ′)) ↭ ϕ_{d,e}(ῑ(Λ′)) if ζ(Λ′) is even.

(b) Let Λ be an even Young diagram in (d×e)-frame, with d ≥ 2 and e ≥ 1. The restriction morphism υ^{∗} satisfies (see Figure 5):

υ^{∗}(ϕ_{d,e}(Λ)) ↭ α^{∗}(ϕ_{d−1,e}(ῡ(Λ))) if Λ_{d} = 0.

(c) Let Λ′′ be an even Young diagram ((d − 1) × e)-frame, with d, e ≥ 2. The connecting homomorphism ∂ satisfies (see Figure 6):

∂(α^{∗}(ϕ_{d−1,e}(Λ′′))) ↭ (ι_{Z})_{∗}(ϕ_{d,e−1}(¯∂(Λ′′))) if Λ′′_{d−1} is odd.

Proof. (a). Let Λ′ be an even (d, e − 1)-diagram such that ζ(Λ′) is even. Let \underline{d} and \underline{e} be the corresponding k-tuples (Def. 2.3). By assumption we can consider the even (d, e)-diagram ῑ(Λ′) := (Λ′_{1} + 1, …, Λ′_{d} + 1). Observe that it has the same associated k-tuples (with respect to the (d×e)-frame). From (14), it is then easy to see that Fl_{X}(d, e − 1; Λ′) = Fl_{X}(d, e; ῑ(Λ′)) and that the diagram

[figure: Gr_{X}(d, V^{1}); ι; Gr_{X}(d, V); Fl_{X}(d, e − 1; Λ′); f_{d,e−1;Λ′}; Fl_{X}(d, e; ῑ(Λ′)); f_{d,e,ῑ(Λ′)}]

commutes. Thus, (a) follows by composition of push-forwards and by Theorem 3.8.

(b). Let Λ be an even (d, e)-diagram such that Λ_{d} = 0. Let \underline{d} and \underline{e} be the corresponding k-tuples (Def. 2.3). By assumption we can consider the even (d−1, e)- diagram ῡ(Λ) := Λ_{|d−1,e}. We then have two cartesian squares :

[figure: Gr_{X}(d, V); □; U_{X}(d, V_{•}); υ; α; □; Gr_{X}(d − 1, V^{1}); Fl_{X}(d, e; Λ); f_{d,e;Λ}; U′; Fl_{X}(d − 1, e; ῡ(Λ)); f_{d−1,e;ῡ(Λ)}]

by Lemma 5.7. The equality in (b) then follows by base change on the above two cartesian squares (see [8, Theorem 5.5], the horizontal morphisms are smooth, so flat, and the vertical maps are proper, including U′ → U_{X}(d, V_{•})). Using Remark 3.6 and Theorem 3.8, both sides of the desired equation are lax-similar to the push-forward of the same 1_{U′} ∈ W^{0}(U′) along U′ → U_{X}(d, V_{•}).

(c). Let Λ′′ be an even (d − 1, e)-diagram such that Λ′′_{d−1} is odd. Let \underline{d} and \underline{e} be the corresponding k-tuples (Def. 2.3). By assumption, we can consider the even (d, e − 1)-diagram

¯∂(Λ′′) := (Λ′′_{1} − 1, …, Λ′′_{d−1} − 1, 0) .

The key tool here is [6, Theorem 2.6] which allows us to compute ∂ geometrically, as the pull-back to the exceptional fiber along α̃ ∘ ι̃: E_{X}(d, V_{•}) → Gr_{X}(d − 1, V^{1}) followed by the push-forward along π̃ : E_{X}(d, V_{•}) → Gr_{X}(d, V^{1}), as long as the twist of the Witt class under consideration satisfies the assumptions of [6, Theorem 2.6].

Here, the twist Δ_{d−1}^{t(Λ′′)} of ϕ_{d−1,e}(Λ′′) is given by t(Λ′′) = [d + 1] ∈ ℤ/2. Using Diagram (29) and Equation (32), one can easily check that this is the twist for which of [6, Theorem 2.6] applies. By Lemma 5.8, we have the right-hand cartesian square in the following commutative diagram :

[figure: Gr_{X}(d, V^{1}); E_{X}(d, V_{•}); π̃; ^{α̃ι̃}; □; Gr_{X}(d − 1, V^{1}); Fl_{X}(d, e − 1; ¯∂(Λ′′)); f_{d,e−1; ¯∂(Λ′′)}; F′; π′; Fl_{X}(d − 1, e; Λ′′) .; f_{d−1,e;Λ′′}]

We can now compute ∂(α^{∗}(ϕ_{d−1,e}(Λ′′))) by starting with the unit form 1 in the lower right corner, pushing-forward along f_{d−1,e;Λ′′}, pulling back along α̃ι̃ and then pushing forward along π̃. By base-change on the right-hand square, this class ∂(α^{∗}(ϕ_{d−1,e}(Λ′′))) is also the (lax) push-forward of the unit form on F′ along π′ and then along f_{d,e−1; ¯∂(Λ′′)}. The key point is to check that π′_{∗} preserves the unit form, up to lax-similitude. By Lemma 5.8, we know that π′ is either an isomorphism or a blow-up along a closed regular immersion of odd codimension. In both cases π′_{∗}(1) ↭ 1 by Proposition 3.15 and we get the result. □

6.9. Proposition (Analogue of Proposition 6.8 in special cases). When e = 1, the push-forward ι_{∗} : W^{0}(X, O_{X}) → W^{d}(Gr_{X}(d, V), L_{d+1}) satisfies

ι_{∗}(1_{X}) = e(δ_{d+1}) ↭ ϕ_{d,1}([d × 1]).  (38)

When d = 1, the restriction morphism υ^{∗} satisfies

υ^{∗}(ϕ_{1,e}( / □)) ↭ α^{∗}(1_{X}).  (39)

When e = 1 and d ≥ 2, the connecting homomorphism of the localization sequence with twist L_{d} satisfies

∂(α^{∗}(ϕ_{d−1,1}([(d − 1) × e]))) ↭ δ_{d} .  (40)

When e = d = 1, the connecting homomorphism of the localization sequence with twist L_{d} = L_{1} satisfies

∂(α^{∗}(1_{X})) ↭ δ_{1} .  (41)

Proof. The proof is entirely analogous to the one of Proposition 6.8, so we leave the details to the reader. □

We now apply localization, using Theorem 3.14, in which a part P of the relative Picard group modulo 2 needs to be specified. Again, there is a general case, when d ≥ 2, and a particular case where d = 1.

Case d, e ≥ 2. We choose P to be the whole Pic_{X}(Gr_{X}(d, V))/2. Note that we indeed have υ^{∗} : Pic_{X}(Gr_{X}(d, V)/2 → Pic_{X}(U_{X}(d, V_{•}))/2 injective by (29). By induction and by Lemma 6.5, the ϕ_{d,e−1}(Λ′), Λ′ even, form a total basis of the Witt groups of Gr_{X}(d, V) with support in Gr_{X}(d, V^{1}). By induction, we also know that the ϕ_{d−1,e}(Λ′′), Λ′′ even, form a total basis of the Witt groups of Gr_{X}(d−1, V^{1}). Homotopy invariance and [7, Corollary 6.13] ensure that the α^{∗}(ϕ_{d−1,e}(Λ′′)) form a total basis of the Witt groups of U_{X}(d, V_{•}). The correspondence of Proposition 6.8 and Proposition 2.12 then show that assumptions (a), (b) and (c) of Theorem 3.14 are satisfied when choosing the following collections of Witt classes :

I = \xymatrix{\left\{{even Young (d, e − 1)-diagrams \atop Λ′ such that ζ(Λ′) is even}\right\} \ar[r]^{≃}_{ῑ} & \left\{{even Young (d, e)-diagrams \atop Λ such that Λ_{d} > 0}\right\}}

v_{Λ′} = (ι_{Z})_{∗}(ϕ_{d,e−1}(Λ′))  and  w′_{Λ′} = ϕ_{d,e}(ῑ(Λ′)) ↭ e(v_{Λ′})  for all Λ′ ∈ I

J = \xymatrix{\left\{{even Young (d, e)-diagrams \atop Λ such that Λ_{d} = 0}\right\} \ar[r]^{≃}_{ῡ} & \left\{{even Young (d − 1, e)-diagrams \atop Λ′′ such that Λ′′_{d−1} is even}\right\}}

w_{Λ} = ϕ_{d,e}(Λ)  and  u′_{Λ} = α^{∗}(ϕ_{d−1,e}(ῡ(Λ))) ↭ υ^{∗}(w_{Λ})  for all Λ ∈ J

K = \xymatrix{\left\{{even Young (d − 1, e)-diagrams \atop Λ′′ such that Λ′′_{d−1} is odd}\right\} \ar[r]^{≃}_{¯∂} & \left\{{even Young (d, e − 1)-diagrams \atop Λ′ such that ζ(Λ′) is odd}\right\}}

u_{Λ′′} = α^{∗}(ϕ_{d−1,e}(Λ′′)) and v′_{Λ′′} = (ι_{Z})_{∗}(ϕ_{d,e−1}(¯∂(Λ′′))) ↭ ∂(u_{Λ′′}) for all Λ′′ ∈ K We therefore conclude by part (4) of Theorem 3.14 that we have a total basis {ϕ_{d,e}(Λ)}_{Λ} indexed by Λ in {Λ even with Λ_{d} = 0} ⊔ {Λ even with Λ_{d} > 0}, i.e. by all even Young diagrams in (d×e)-frame. This is the result.

Case d ≥ 2, e = 1. The proof is the same as in the case d, e ≥ 2, except that Proposition 6.9 is used instead of Proposition 6.8. We take (see Figure 10)

\begin{matrix}I = \{⋆\} , & v_{⋆} = δ_{d+1}, & w′_{⋆} = ϕ_{d,1}([d × 1]) ↭ e(v_{⋆}), \\ J = \{ / □\} , & w\begin{matrix}/ \\ □ \end{matrix} = ϕ_{d,1}( / □), & u′ ; / □ = α^{∗}(ϕ_{d−1,1}( / □)) ↭ υ^{∗}(w\begin{matrix}/ \\ □ \end{matrix}), \\ K = \{⋄\} , & u_{⋄} = α^{∗}(ϕ_{d−1,1}( / □)), & v′_{⋄} = δ_{d} ↭ ∂(u\begin{matrix}/ \\ □ \end{matrix}). \end{matrix}

Case d = 1, e ≥ 2. In that case, Gr_{X}(d − 1, V^{1}) = Gr_{X}(0, V^{1}) = X, so we can no longer choose P to be the whole relative Picard group modulo 2 because injectivity of υ^{∗} would not hold (see Remark 5.6). We therefore first choose P = P_{1} := {Δ_{d}^{d+1}} = {1} and take (see P_{1}-part of Figure 11)

I = {[1 × (e − 1)]} , v_{[1×(e−1)]} = (ι_{Z})_{∗}(ϕ_{1,e−1}([1 × (e − 1)])),

w′_{[1×(e−1)]} = ϕ_{1,e}([1 × e]) ↭ e(v_{[1×(e−1)]}),

J = \{ / □\} ,  w\begin{matrix}/ \\ □ \end{matrix} = ϕ_{1,e}( / □),

u′ ; / □ = α^{∗}(1_{X}) ↭ υ^{∗}(w\begin{matrix}/ \\ □ \end{matrix}),

K = ∅.

WITT GROUPS OF GRASSMANN VARIETIES 27

⋄ ⋆ ⋄ ⋆ Gr(d, V_{d+1}) X X Gr(d − 1, V_{d}) ῡ ¯∂ ῑ

Figure 10. Image of the diagrams by the morphisms ῑ, ῡ and ¯∂ when d ≥ 2 and e = 1. No arrow means the corresponding generator is mapped to zero.

As in previous cases, by part (a) of Proposition 6.8, by equation (39) and by induction, using Theorem 3.14, we get that ϕ_{1,e}([1 × e]) and ϕ_{1,e}( / □) form a basis of the {1}-part of the Witt groups of Gr_{X}(1, V).

We then choose P = P_{2} := {Δ_{d}}, and take (see P_{2}-part of Figure 11)

\begin{matrix}I = J = ∅, \\ K = \{⋆\} & u_{⋆} = α^{∗}(1_{X}), & v′_{⋆} = (ι_{Z})_{∗}(ϕ_{1,e}( / □)) ↭ ∂(u_{⋆}). \end{matrix}

By Theorem 3.14 using (39), we obtain that the empty set is a basis of the {Δ_{d}}- part of the Witt groups of Gr_{X}(1, V).

Putting together both bases (one of which is empty), we obtain a basis of the Witt groups of Gr_{X}(d, V) by [7, Lemma 6.10], since P_{1} ⊔ P_{2} = Pic_{X}(Gr_{X}(1, V)). This basis contains exactly the only two generators corresponding to even diagrams of size (1, e), i.e. / □ and [1 × e]. See Figure 11.

Gr(1, V_{e}) Gr(1, V_{e}) Gr(1, V_{e+1}) X ⋆ / □ P_{1}- part P_{2}- part ῡ ¯∂ ῑ

Figure 11. Image of the diagrams by the morphisms ῑ, ῡ and ¯∂ when d = 1 and e ≥ 2. No arrow means the corresponding generator is mapped to zero.

Case d = e = 1. It works as in the case d = 1, e ≥ 2 except that (ι_{Z})_{∗}(ϕ_{1,e−1}( / □)) is replaced by δ_{1} and that (ι_{Z})_{∗}(ϕ_{1,e}([1 × 1])) is replaced by δ_{0}. See Figure 12. Note that, as announced, this case does not use any induction assumption. This completes the induction and therefore the proof of Main Theorem 6.1. □

7. Corollaries and examples

Here is how to deduce a more classical result on Witt groups that does not involve “total” concepts. Recall that π is the structural morphism of Gr_{X}(d, V) and that T_{d} is its tautological bundle. Also recall that the generator ϕ_{d,e}(Λ) lives in W^{|Λ|} ( Gr_{X}(d, V), L_{Λ}) where |Λ| is the area of Λ. Any line bundle on Gr_{X}(d, V) is, up to isomorphism, of the form π^{∗}K ⊗ det(T_{d})^{⊗l}, for some line bundle K

28 PAUL BALMER AND BAPTISTE CALMÈS

/ □ ⋆ / □ ⋆ / □ ⋆ Gr(1, V_{2}) X X X P_{1}- part P_{2}- part ῡ ¯∂ ῑ

Figure 12. Image of the diagrams by the morphisms ῑ, ῡ and ¯∂ when d = e = 1. No arrow means the corresponding generator is mapped to zero.

on X and some l ∈ ℤ. A diagram Λ is such that [L_{Λ}] = [π^{∗}K ⊗ det(T_{d})^{⊗l}] ∈ Pic_{X}(Gr_{X}(d, V))/2 if and only if t(Λ) = l ∈ ℤ/2, where t(Λ) is the half-perimeter of Λ.

7.1. Corollary. Let K be a line bundle on X and let l be an integer. For each even (d, e)-diagram Λ such that l = t(Λ) ∈ ℤ/2, choose a line bundle N_{Λ} over Gr_{X}(d, V) and an isomorphism N_{Λ}^{⊗2} ⊗ π^{∗}(K ⊗ det(V)^{⊗−ρ(Λ)}) ⊗ L_{Λ} →^{∼} π^{∗}(K) ⊗ det(T_{d})^{l}. Then the morphism of W^{0}(X, O_{X})-modules

⨁_{Λ even s. t. ; t(Λ)=l∈ℤ/2} W^{k−|Λ|} (X, K ⊗ det(V)^{⊗−ρ(Λ)}) →^{∼} W^{k} ( Gr_{X}(d, V), π^{∗}(K) ⊗ det(T_{d})^{⊗l})

sending (x_{Λ}) to ∑ x_{Λ} · ϕ_{d,e}(Λ) (notation of Definition 3.10) is an isomorphism.

Proof. Apply Theorem 3.12 to the total basis formed by the ϕ_{d,e}(Λ). □

7.2. Corollary. Let d′ (resp. e′) be the integral part of d/2 (resp. e/2) and consider the binomial coefficient \binom{a+b}{a} = \frac{(a+b)!}{a!b!}. The cardinal of a total basis of the Witt groups of Gr_{X}(d, V) over X is 2\binom{d′+e′}{e′}. If we assume moreover that W^{i}(X, K) = 0 for i ≢ 0 mod 4 or [K] ≠ [O_{X}] ∈ Pic(X)/2, for instance for X local (e.g. a field), then, as a module over W^{0}(X, O_{X}),

• the classical Witt group of symmetric forms W^{0}(Gr_{X}(d, V), O) has rank \binom{d′+e′}{e′},

• the classical Witt group of antisymmetric forms W^{2}(Gr_{X}(d, V), O) is zero.

• the Witt groups W^{1}(Gr_{X}(d, V), L) and W^{3}(Gr_{X}(d, V), L) are zero when [L] = Δ_{d} ∈ Pic_{X}(Gr_{X}(d, V))/2.

Proof. Let “4-block” mean “2 × 2 square”. Every even Young diagram Λ is either (a) a union of 4-blocks and ϕ(Λ) counts in W^{0}(Gr_{X}(d, V), O),

(b) a single row plus 4-blocks (e even) and ϕ(Λ) counts in W^{e}(Gr_{X}(d, V), L_{Λ}),

(c) a single column plus 4-blocks (d even) and ϕ(Λ) counts in W^{d}(Gr_{X}(d, V), L_{Λ}),

(d) a single row and a single column plus 4-blocks (d and e odd) and ϕ(Λ) counts

in W^{d+e−1}(Gr_{X}(d, V), O). All possibilities (a)-(d) are exclusive and can be enumerated easily by counting the diagrams of 4-blocks, which amounts to counting the usual Young diagrams in (d′×e′)-frame. We get \binom{d′+e′}{e′} elements in case (a), and the other results depend on the parities of d and e but are also very easy to figure out in each case. □

7.3. Corollary. The connecting homomorphism ∂ is zero for all line bundles L (and thus the long exact sequence (37) decomposes as split short exact sequences as for Chow groups) if and only if both d and e are even.

Proof. Looking back at the proof of the main theorem, and at (18) (or Figure 6), we see that ∂ is zero if and only if there is no even (d−1, e)-diagram Λ′′ such that Λ′′_{d−1} is odd. This implies that e is even (otherwise Λ′′ = [(d−1)×e] would be such an even diagram) and that d − 1 is odd (otherwise Λ′′ = (1, …, 1) would be such an even diagram). Conversely, assume e even and the existence of an even (d−1, e)-diagram Λ′′ such that Λ′′_{d−1} is odd. Then e_{k} is odd (since e = Λ′′_{d−1} + e_{k} is even), hence all e_{i} are odd since Λ′′ is an even diagram. In particular, e_{1} is odd, hence e_{1} > 0 and therefore d_{1} is even. This implies that d−1 = d_{k} = (d_{k} −d_{k−1})+…+(d_{2} −d_{1})+d_{1} is even, i.e. d is odd, as was to be shown. □

7.4. Notation. For d, e ⩾ 1, we write 𝔾_{X}(d, e) for the split Grassmannian

𝔾_{X}(d, e) = Gr_{X}(d, O_{X}^{d+e}) .

7.5. Example. Figure 13 shows how the different generators are mapped to each other, up to lax-similitude, by ι_{∗}, υ^{∗} and ∂ in the long exact sequence (37) for 𝔾(3, 3). We use dévissage (resp. homotopy invariance) to identify the generators of the Witt groups of 𝔾(3, 2) (resp. of 𝔾(2, 3)) with generators of the Witt groups of 𝔾(3, 3) with support on 𝔾(3, 2) (resp. of the open complement of 𝔾(3, 2)).

ῡ ῑ¯∂ 𝔾(3, 2) 𝔾(3, 3) 𝔾(2, 3) 𝔾(3, 2)

Figure 13. Maps ῑ, ῡ and ¯∂ on diagrams, corresponding to the maps ι_{∗}, υ^{∗} and ∂ on generators (no arrow means mapped to zero).

7.6. Example. Figures 14, 15 and 16 give the even Young diagrams in (d×e)-frame and the corresponding shifts in ℤ/4 and twists Δ_{d} or O for the Grassmannians 𝔾(4, 4), 𝔾(4, 5) and 𝔾(5, 5).

We conclude with a few comments.

7.7. Remark. As mentioned in Remark 6.3, we could have considered a larger set of elements ϕ_{d,e}(Λ) using Remark 4.1 instead of assuming Λ even. This larger set is also stable by applying ι_{∗}, υ^{∗} and ∂. Some of these extra elements are then easily seen to be zero from the exact sequence, but some of them are nonzero.

7.8. Remark. Part of the multiplicative structure on Witt groups can be computed at each induction step using the projection formula. Unfortunately, this is not enough for the whole computation. Despite the results for the Grothendieck and Figure 14. Diagrams of generators for 𝔾(4, 4): first row in shift 0 and twist O, second row in shift 0 and twist Δ_{d}.

Figure 15. Diagrams of generators for 𝔾(4, 5): first row in shift 0 and twist O, second row in shift 0 and twist Δ_{d}.

Figure 16. Diagrams of generators for 𝔾(5, 5): first row in shift 0 and twist O, second row in shift 1 and twist Δ_{d}.

the Chow rings using basis of Schubert cells, it is unclear to the authors what kind of Littlewood-Richardson type rule one should expect.

Note however that all generators ϕ_{d,e}(Λ) are nilpotent except for Λ = / □. This can be checked using homotopy invariance and a discussion on the supports or simply the fact that these Witt classes are generically trivial.

7.9. Remark. Although we do not need it here, it is possible to show that, for V = O_{X}^{d+e}, the isomorphism 𝔾(d, e) = Gr(d, V) ≃ Gr(e, V^{∨}) = 𝔾(e, d) sends ϕ_{d,e}(Λ) to ϕ_{e,d}(Λ^{∨}), up to lax-similitude of course, where Λ^{∨} is the dual partition.

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Paul Balmer, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

URL: http://www.math.ucla.edu/~balmer

Baptiste Calmès, Université d’Artois, Laboratoire de Mathématiques de Lens, France

URL: http://www.math.uni-bielefeld.de/~bcalmes