Niveau spectral sequences on singular schemes and failure of generalized Gersten conjecture

Abstract

We construct a new local-global spectral sequence for Thomason's non-connective K-theory, generalizing the Quillen spectral sequence to possi bly non-regular schemes. Our spectral sequence starts at the Ei-page where it displays Gersten-type complexes. It agrees with Thomason's hypercohomol ogy 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 dimen sion. Then there exists a spectral sequence whose first page is (1) E`q= p K-p-q(Ox,x on {x}) for p,q E 2, xEX(P) converging toward Kn (X) for n E 2, along n = p + q ; that is, the indexing in the spectral sequence is such that dr : EPq _ EP+r q-,r+l for r > 1. Here, K*(X on Y) stands for Thomason's non-connective (or Bass) K-theory of those perfect complexes of Ox-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 devissage to replace the local terms K. (Ox,x on {x}) by K-groups of residue fields, K, (i'(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. Im portant 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 pro vide 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 second, the author's decom position in local terms of the idempotent completion of the successive quotients of the coniveau filtration of Dperf(X); see [2] or (7) below. These two recent papers, Received by the editors September 17, 2007, and, in revised form, January 9, 2008. 2000 Mathematics Subject Classification. Primary 19E08, 19D35, 18E30.