NIVEAU SPECTRAL SEQUENCES ON SINGULAR SCHEMES AND FAILURE OF GENERALIZED GERSTEN CONJECTURE

PAUL BALMER

Abstract. We construct a new local-global spectral sequence for Thomason’s non-connective K-theory, generalizing the Quillen spectral sequence to possibly non-regular schemes. Our spectral sequence starts at the E_{1}-page where it displays Gersten-type complexes. It agrees with Thomason’s hypercohomology spectral sequence exactly when these Gersten-type complexes are locally exact, a condition which fails for general singular schemes, as we indicate.

Our main result is the following application of abstract triangular geometry [2].

Theorem 1. Let X be a (topologically) noetherian scheme of finite Krull dimension. Then there exists a spectral sequence whose first page is

(1)  E_{1}^{p,q} = ⨁_{x∈X^{(p)}} K_{−p−q}(O_{X,x} on {x})  for p, q ∈ ℤ,

converging toward K_{−n}(X) for n ∈ ℤ, “along n = p + q”, that is, the indexing in the spectral sequence is such that d_{r} : E_{r}^{p,q} → E_{r}^{p+r , q−r+1} for r ≥ 1. Here, K_{∗}(X on Y) stands for Thomason’s non-connective (or “Bass”) K-theory of those perfect complexes of O_{X}-modules which are acyclic on X − Y, see [12, § 6].

Note the presence of negative K-groups, a crucial fact throughout the paper. Negative K-theory roots back to work of Bass and of Karoubi, independently.

For X regular, this coniveau spectral sequence is due to Quillen [10, Thm. 5.4], who also used dévissage to replace the local terms K_{∗}(O_{X,x} on {x}) by K-groups of residue fields, K_{∗}(κ(x)). Although Theorem 1 can also be proved by starting with (10.3.6) in Thomason [12, proof of Thm. 10.3], the conceptual proof given here relies on deep geometric facts and easily transposes to other theories, see Remark 3. Important progress appeared even before [12], e.g. in Levine [8] or Weibel [15, 16, 17], but always under restrictions on the singularities. The above Theorem seems to provide the most general coniveau spectral sequence one could wish for. Our proof is a direct application of two recent results : First, Schlichting’s localization long exact sequence involving negative K-groups, see [11], and secondly, the author’s decomposition in local terms of the idempotent completion of the successive quotients of the coniveau filtration of D^{perf}(X), see [2] or (7) below. These two recent papers, [2] and [11], grew as new branches of the same trunk, namely Thomason’s masterpiece [12] mentioned above. Actually, in [12, Thm. 10.3], Thomason constructs another spectral sequence, the hypercohomology spectral sequence :

(2)  E_{2}^{p,q} = H_{Zar}^{p}(X, K_{−q}) ⟹^{p+q=n} K_{−n}(X) ,

2000 Mathematics Subject Classification. 19E08, 19D35, 18E30.

Key words and phrases. spectral sequence, K-theory, singular schemes.

Research supported by NSF grant 0654397.

where K_{m} is the sheafification of the presheaf U ↦ K_{m}(U), for m ∈ ℤ. It is legitimate to ask whether the E_{2}-page of our spectral sequence (1) coincides with that of (2). We shall see in Remark 6 that this fails in general. Note that our spectral sequence (1) is potentially interesting even for local schemes, where the vanishing of cohomology forces Thomason’s spectral sequence (2) to be trivial.

Actually, we can also construct a niveau spectral sequence, involving dimension instead of codimension. To synthesize, let us consider a general dimension function

(3)  dim : Irred(X) ⟶ ℤ

on the set Irred(X) of irreducible closed subsets of X; i.e. we assume dim(Y) < dim(Y′) for every two Y, Y′ ∈ Irred(X) such that Y ⊂ Y′ and Y ≠ Y′. The Krull dimension dim(Y) := dim_{Krull}(Y) is an example, and so is the opposite of the Krull codimension dim(Y) := − codim_{Krull}(Y) but there are other dimension functions, e.g. in intersection theory. As usual, we extend our dimension function to any nonempty closed Z ⊂ X by dim(Z) := max { dim(Y) | Y ∈ Irred(X) and Y ⊂ Z } ∈ ℤ (and dim(∅) := −∞). We moreover assume that there exists d ∈ ℕ such that

(4)  −d ≤ dim(Z) ≤ d

for every non-empty closed subset Z ⊂ X. This obviously holds for dim_{Krull} and for − codim_{Krull} since we assume X to have finite Krull dimension.

For every i ∈ ℤ, we define X_{(i)} := \{ x ∈ X | dim (\overline{\{x\}}) = i \}. Note that the set X^{(i)} = { x ∈ X | dim_{Krull}(O_{X,x}) = i } of Theorem 1 is nothing but X_{(−i)} for the dimension function − codim_{Krull}. With this general notion of dimension, we prove :

Theorem 2. Let X be a topologically noetherian scheme with a dimension function (3) satisfying (4). Then there exists a converging spectral sequence

E_{1}^{p,q} = ⨁_{x∈X_{(−p)}} K_{−p−q}(O_{X,x} on {x}) ⟹^{p+q=n} K_{−n}(X) .

Proof. We define for each p ∈ ℤ the following subcategory of D^{perf}(X)

(5)  D_{p} = D_{(≤p)}^{perf}(X) := { E ∈ D^{perf}(X) | dim(supph(E)) ≤ p } ,

where the closed subset supph(E) ⊂ X is the support of the total homology of the perfect complex E. Because of (4), we have a finite filtration

(6)  0 = D_{−d−1} ⊂ D_{−d} ⊂ ⋯ ⊂ D_{p−1} ⊂ D_{p} ⊂ ⋯ ⊂ D_{d} = D^{perf}(X) .

Since this filtration can be realized at the level of Waldhausen models, there is a “naive” spectral sequence starting with the Waldhausen K-theory of (some model of) D_{p} / D_{p−1} and converging to Thomason’s connective K-theory of X. This would lead to Quillen’s classical spectral sequence for X regular. Nevertheless there is no good description of the quotient D_{p} / D_{p−1} when X is singular. Still, following Thomason’s insight of [12, § 5], we can instead describe the idempotent completion of this quotient. Recall from [3] that the idempotent completion 𝒦̃ of a triangulated category 𝒦 inherits a unique triangulation such that 𝒦 ↪ 𝒦̃ is exact. Then, by [2, Thm. 3.24 and § 4.1], we know that for every p ∈ ℤ, localization induces an equivalence

(7)  { \atop D_{p} / D_{p−1}} ⟶^{∼} ∐_{x∈X_{(p)}} D_{\{x\}}^{perf}(O_{X,x})

between the idempotent completion of the quotient D_{p} / D_{p−1} and the coproduct over x ∈ X_{(p)} of the derived categories of perfect complexes of O_{X,x}-modules with homology supported on the closed point x ∈ Spec(O_{X,x}).

The K-theoretic price to pay for introducing idempotent completions is the appearance of negative K-groups in the localization long exact sequence. To explain this, we introduce some notations and results from Schlichting [11]. Let 𝔐 be a collection of “models” and let D(𝒜) be a triangulated category functorially associated to 𝒜 ∈ 𝔐. Here 𝔐 will be the category of complicial bi-Waldhausen categories and D(𝒜) the derived category of 𝒜. We loosely speak of the non-connective K-theory of D(𝒜) to mean the non-connective K-theory of 𝒜, as defined in [11].

We say that a sequence 𝒜 → ℬ → 𝒞 is exact in 𝔐 if D(𝒜) → D(ℬ) → D(𝒞) is exact up to direct summands, meaning that D(𝒜) is a thick triangulated subcategory of D(ℬ), that the composite D(𝒜) → D(𝒞) is zero and that the induced functor D(ℬ)/D(𝒜) → D(𝒞) is fully faithful and cofinal, i.e. is an equivalence after idempotent completion, which means in particular that any object of D(𝒞) is a direct summand of an object of D(ℬ)/D(𝒜). Given such an exact sequence 𝒜 → ℬ → 𝒞, we have by [11, Thm. 1, Thm. 6 and § 6.5] a long exact sequence of K-groups :

⋯ → K_{1}(ℬ) → K_{1}(𝒞) → K_{0}(𝒜) → K_{0}(ℬ) → K_{0}(𝒞) → K_{−1}(𝒜) → K_{−1}(ℬ) → ⋯

In our situation, since the filtration (6) can be defined on the level of models, and using our abuse of notation K_{∗}(D(𝒜)) = K_{∗}(𝒜), we get a long exact sequence

⋯ → K_{n}(D_{p−1}) → K_{n}(D_{p}) → K_{n}({ \atop D_{p} / D_{p−1}}) → K_{n−1}(D_{p−1}) → ⋯ (n ∈ ℤ) .

So, we obtain as usual an exact couple and the associated spectral sequence, which looks as follows in cohomological indexing : E_{1}^{p,q} = K_{−p−q}({ \atop D_{−p} / D_{−p−1}}). This spectral sequence has the wanted form by (7) and by agreement of the negative K-theory of [11] with Thomason’s, see [11, Thm. 5]. Convergence is clear since E_{1}^{p,q} is concentrated in the band −d ≤ p ≤ d by (3). □

Remark 3. We can replace everywhere K-theory by any cohomology theory which associates a long exact sequence to every short sequence of triangulated categories 𝒥 → 𝒦 → ℒ which is exact up to direct summands (see above). This is true in particular for triangular Witt groups tensored with ℤ[\frac{1}{2}] by the localization theorem [1, 6.2 & 6.8] and the cofinality theorem of Hornbostel–Schlichting [7, App. A].

Definition 4. Let X be a (topologically) noetherian scheme of finite Krull dimension d. For any q ∈ ℤ, we define the q^{th} augmented weak Gersten complex

0 →K_{q}(X) → ⨁_{x∈X^{(0)}}K_{q}(O_{X,x}) → ⨁_{x∈X^{(1)}}K_{q−1}(O_{X,x} on {x}) → ⋯

⋯ → ⨁_{x∈X^{(p)}}K_{q−p}(O_{X,x} on {x}) → ⋯ → ⨁_{x∈X^{(d)}}K_{q−d}(O_{X,x} on {x}) → 0 .

to be the −q^{th} line of the E_{1}-page of our coniveau spectral sequence (1), augmented by the edge homomorphism K_{q}(X) → ⊕_{x∈X^{(0)}}K_{q}(O_{X,x}), which is just localization. We can of course drop the “on {x}” when x ∈ X^{(0)}, since there {x} = Spec(O_{X,x}). A similar complex exists with dimension instead of codimension and differs from the above in general. We call these complexes weak Gersten complexes because they differ from Gersten complexes for regular schemes in that we cannot replace the local terms by the K-groups of the residue fields, since we do not have dévissage in this context.

Remark 5. After Mochizuki [9], one might say that a local noetherian ring R satisfies the generalized Gersten conjecture if the augmented weak Gersten complexes are exact for X = Spec(R), for all q ∈ ℤ. This is equivalent to the vanishing of K_{n}(D^{p+1}) → K_{n}(D^{p}) for all n ∈ ℤ and p ≥ 0, where the subcategory D^{p} := { E ∈ D^{perf}(X) | codim(supph(E)) ≥ p } is again nothing but D_{−p} in notation (5) applied to the dimension function − codim_{Krull}.

Remark 6. For q ≥ 2 this conjecture is known to fail in general. For instance, for the local domain R obtained by localization of a cusp at its singularity, say R = k[t^{2}, t^{3}]_{(t^{2},t^{3})}, the (edge) homomorphism K_{2}(R) → K_{2}(k(t)) is not injective as proved in Dennis–Sherman [6, Appendix], where other examples are provided.

Proposition 7. If O_{X,x} satisfies the above generalized Gersten conjecture for every x ∈ X then the E_{2}-page of our spectral sequence (1) agrees with that of Thomason (2). Conversely, for R local, if the E_{2}-page of our spectral sequence agrees with Thomason’s on X = Spec(R) then the generalized Gersten conjecture holds for R.

Proof. The argument is standard. Assume the weak Gersten complex locally exact on X. Since the unaugmented complex (i.e. without K_{q}(X) at the beginning) is already a complex of skyscraper Zariski sheaves on X, it defines a flasque resolution of the sheafification K_{q} of the presheaf U ↦ K_{q}(U). Hence, the homology of this complex, which sits in our E_{2}-page, equals the Zariski cohomology of K_{q}. Conversely, for X = Spec(R) local, suppose the E_{2}-page of our spectral sequence consists of Zariski cohomology, then it vanishes except for p = 0, which is equivalent to the exactness of the augmented weak Gersten complex. □

Remark 8. By Proposition 7 and Remark 6, our coniveau spectral sequence (1) differs from Thomason’s (2) in general. As already mentioned, an advantage of our spectral sequence is that it can be non-trivial even for local schemes, where there is no non-trivial cohomology.

We now give an example of a local ring R for which the inclusion D^{p+1} ↪ D^{p} does not even induce the zero map on K_{0} (see Remark 5).

Let k be an infinite field and define the ring R as the localization of k[x, y, z, w] = k[X, Y, Z, W]/⟨XW −Y Z⟩ at the point 𝔪 = ⟨x, y, z, w⟩. In [5], Dutta, Hochster and McLaughlin provide examples of finite length R-modules M which admit a finite projective resolution (that is M ∈ D_{{𝔪}}^{perf}(R) = D^{3}) but with the property that

(8)  χ(M, R/𝔭) ≠ 0  (and even χ(M, R/𝔭) < 0)

for some 𝔭 ∈ Spec(R)^{(1)}, for instance for 𝔭 = ⟨x, y⟩, see [5, middle of p. 254]. Of course, χ(M_{1}, M_{2}) = ∑(−1)^{i} length( Tor_{i}^{R}(M_{1}, M_{2})) is Serre’s intersection multiplicity. Note that this χ(M_{1}, M_{2}) ∈ ℤ is well-defined for any perfect complex M_{1} ∈ D^{perf}(R) = K^{b}(R–free) and any finitely generated R-module M_{2} satisfying supph(M_{1}) ∩ supp(M_{2}) ⊂ {𝔪}; in this case Tor_{i}^{R}(M_{1}, M_{2}) simply is the homology in degree i of the (naive) tensor product M_{1} ⊗M_{2} ; these homology groups are finite length R-modules since supph(M_{1} ⊗ M_{2}) ⊂ supph(M_{1}) ∩ supp(M_{2}) ⊂ {𝔪}.

Proposition 9 (Levine). With the above notations, [M] ∈ K_{0}(D^{3}) does not map to zero under the canonical homomorphism K_{0}(D^{3}) → K_{0}(D^{2}). In particular, this local ring R does not satisfy the generalized Gersten Conjecture of Remark 5.

Proof. For each a ∈ k consider the prime 𝔭_{a} := ⟨x − az, y − aw⟩ ∈ Spec(R)^{(1)}. Note that for a ≠ b, we have 𝔭_{a} + 𝔭_{b} = ⟨x, y, z, w⟩ = 𝔪. Hence, for each 𝔮 ∈ Spec(R) with 𝔮 ≠ 𝔪, we have at most one a ∈ k such that 𝔭_{a} ⊂ 𝔮. Consider now a closed subset C ⊂ Spec(R) such that codim(C) = 2. Then C = V (𝔮_{1}) ∪ … ∪ V (𝔮_{n}) for prime ideals 𝔮_{1} , …, 𝔮_{n} ∈ Spec(R)^{(2)}. By the above considerations, since the field k is infinite, there exists an element a ∈ k such that 𝔭_{a} ⊄ 𝔮_{i} for all i = 1, …, n. Therefore, C ∩ V (𝔭_{a}) = V (𝔮_{1} + 𝔭_{a}) ∪ ⋯ ∪ V (𝔮_{n} + 𝔭_{a}) = {𝔪}.

Let us assume ab absurdo that [M] ↦ 0 ∈ K_{0}(D^{2}). For C ⊂ Spec(R), recall the standard notation D_{C}^{perf}(R) := { E ∈ D^{perf}(R) | supph(E) ⊂ C }. Since we have

K_{0}(D^{2}) = colim_{C⊂Spec(R), closed with codim(C)≥2} K_{0}( D_{C}^{perf}(R)) ,

we can assume that [M] ↦ 0 in K_{0}(D_{C}^{perf}(R)) for some closed subset C ⊂ Spec(R) with codim(C) ≥ 2, that is, codim(C) = 2 or C = {𝔪}. In any case, by the first part of the proof, there exists a ∈ k such that C ∩ V (𝔭_{a}) = {𝔪}. Let us fix such a C ⊂ Spec(R) and such an a ∈ k.

In this situation, the intersection multiplicity χ(E, R/𝔭_{a}) ∈ ℤ is defined for any E ∈ D_{C}^{perf}(R). Since it is additive on distinguished triangles in D_{C}^{perf}(R), the expression χ(E, R/𝔭_{a}) only depends on the class of E in K_{0}(D_{C}^{perf}(R)). Therefore, our assumption [M] = 0 in K_{0}(D_{C}^{perf}(R)) implies

(9)  χ(M, R/𝔭_{a}) = 0 .

Similarly, since M ∈ D_{{𝔪}}^{perf}(R), the number χ(M, M_{2}) is defined for any finitely generated R-module M_{2} and, as before, induces a well-defined homomorphism χ(M, −) on K_{0}(R–mod) = G_{0}(R). To get a contradiction between (8) and (9), hence to finish the proof, it now suffices to show that [R/𝔭_{a}] = [R/𝔭] in G_{0}(R) for all a ∈ k. To see this, it is enough to check that the following sequence is exact :

(10)  0 → Ker(β) → R^{2} ⟶^{β} R^{2} ⟶^{α_{a}} R → R/𝔭_{a} → 0

where α_{a} = (x − az y − aw) and β = (\begin{matrix}w & −y \\ −z & x \end{matrix}), for this will imply that

[R/𝔭_{a}] = [R] − [Ker(β)] in G_{0}(R) and the latter is independent of a ∈ k. In fact, only exactness at the middle R^{2} requires proof, and this can be given in the spirit of [5, Lemma 2.2]. Since localization is exact, we prove exactness of (10) with

R replaced by R = k[x, y, z, w] = k[X, Y, Z, W]/⟨XW − Y Z⟩. Let ( \begin{matrix}u \\ v \end{matrix} ) ∈ R^{2} in the

kernel of α_{a}. Let U, V ∈ k[X, Y, Z, W] such that U ↦ u and V ↦ v in R. Then

α_{a} · ( \begin{matrix}u \\ v \end{matrix} ) = 0 reads

(X − aZ) U + (Y − aW) V = (XW − Y Z) G

for some G ∈ k[X, Y, Z, W]. A direct computation from the above gives

(X − aZ) (U − W G) = −(Y − aW) (V + ZG) .

This implies that (Y − aW) | (U − W G), i.e. U = W G + (Y − aW) H for some H ∈ k[X, Y, Z, W]. Replacing U in the last equation and canceling by (Y − aW) gives V = −ZG − (X − aZ) H. Finally, defining g and h in R to be the classes of G and H respectively, we obtain ( \begin{matrix}u \\ v \end{matrix} ) = ( \begin{matrix}w & −y \\ −z & x \end{matrix} ) · ( \begin{matrix}g−ah \\ −h \end{matrix} ), as was to be shown. □

Remark 10. One can visualize the above proof by tracking the following objects

[figure: ([M] , [R/𝔭_{a}]); [R/𝔭_{a}]=[R/𝔭] in G_{0}(R); [M]=0 in K_{0}(D_{C}^{perf} (R)); ([M] , [R/𝔭]); χ; ( 0 , [R/𝔭_{a}]); χ; 0 ≠ χ(M, R/𝔭) ,]

in the commutative diagram

\xymatrix{K_{0}( D_{\{𝔪\}}^{perf}(R)) × K_{0}( D_{V (𝔭_{a})}^{b}(R – mod)) \ar[r] \ar[d] \ar[dr]^{χ} & K_{0}( D_{\{𝔪\}}^{perf}(R)) × K_{0}( D^{b}(R – mod)) \ar[d]^{χ} \\ K_{0}( D_{C}^{perf}(R)) × K_{0}( D_{V (𝔭_{a})}^{b}(R – mod)) \ar[r]_{χ} & ℤ ,}

whose lower homomorphism χ is well-defined because of the construction of a ∈ k such that C ∩ V (𝔭_{a}) = {𝔪}, made at the very beginning of the proof.

Finally, we discuss a condition for (1) to be a fourth quadrant spectral sequence.

Conjecture 11 (Weibel [14, 2.9]). For X a noetherian scheme of Krull dimension d, we have K_{m}(X) = 0 for m < −d.

Very recently, this Conjecture has been proved for X essentially of finite type over a field of characteristic zero by Cortiñas–Haesemeyer–Schlichting–Weibel [4]. As already observed by Vorst [13, Cor. 1.9 (iii)], at least in the reduced affine case, we now show that it suffices to prove Conjecture 11 for local rings. (This can also be proved with Thomason’s hypercohomology spectral sequence.)

Proposition 12. Let X be a noetherian scheme of Krull dimension d. Then the following are equivalent :

(i) Conjecture 11 holds for the local ring O_{X,x} for every x ∈ X.

(ii) For all p ≥ 0, all x ∈ X^{(p)} and all q > 0 we have K_{−p−q}(O_{X,x} on {x}) = 0. Moreover, when (i) and (ii) hold true, then X satisfies Conjecture 11 as well.

Proof. Of course E_{1}^{p,q} = 0 except for 0 ≤ p ≤ d since X^{(p)} is empty for other p ∈ ℤ. Claim : Suppose that K_{−p−q}(O_{X,x} on {x}) = 0 for all 0 ≤ p ≤ d − 1, all x ∈ X^{(p)} and for all q > 0, then for n > d, we have K_{−n}(X) ≅ ⨁_{x∈X^{(d)}} K_{−n}(O_{X,x} on {x}).

The hypothesis of the Claim means that in our spectral sequence (1), we have : E_{1}^{p,q} = 0 for p ≠ d and q > 0. So, for q > 0, we have E_{1}^{p,q} = E_{∞}^{p,q} and this group vanishes except for p = d where it gives the above direct sum. Hence the Claim.

Assume (i) and let us prove (ii) by induction on p ≥ 0. For p = 0, we have of course K_{∗}(O_{X,x} on {x}) = K_{∗}(O_{X,x}), so (ii) is true by assumption (i). For p ≥ 1 and x ∈ X^{(p)}, apply the Claim to the scheme Spec(O_{X,x}). The hypothesis of the Claim holds by induction hypothesis. Then, for q > 0, the conclusion of the Claim applied to n = p + q gives K_{−p−q}(O_{X,x}) = K_{−p−q}(O_{X,x} on {x}) and the former vanishes by (i). This proves (ii).

Conversely, assume (ii). Then the first page of our spectral sequence (1) vanishes for q > 0. Since it is concentrated in 0 ≤ p ≤ d and converges to K_{∗}(X), we get Conjecture 11 for X. This already shows the “moreover part” of the statement.

This implication “(ii)⇒Conj. 11” applied to X = Spec(O_{X,x}) also proves (ii)⇒(i), since (ii) is a local property. □

Corollary 13. Conjecture 11 is equivalent to : For every noetherian local ring (R, 𝔪) of Krull dimension d and all n > d, we have K_{−n}( Spec(R) on {𝔪}) = 0. This is also equivalent to our coniveau spectral sequence (1) being concentrated in the fourth quadrant (p ≥ 0 and q ≤ 0) for every finite dimensional noetherian scheme X. □

Acknowledgments : I thank Henri Gillet for useful discussions, Marc Levine for kindly providing Proposition 9, Marco Schlichting for precious comments and for the reference to [6] and Chuck Weibel for several references, among which [13]. Finally, I thank the referee for remarks and references.

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Version of January 9, 2008.

Paul Balmer, UCLA Math Deptartment, Box 951555, Los Angeles, CA 90095-1555.

E-mail address: balmer@math.ucla.edu
URL: http://www.math.ucla.edu/∼balmer