ON NEGATIVE ALGEBRAIC K-GROUPS

Abstract

We sketch a proof of Weibel's conjecture on the vanishing of negative algebraic K-groups and we explain an analog of this result for continuous K-theory of nonarchimedean algebras. Negative K-groups of schemesFor a scheme X Grothendieck introduced the K-group K 0 (X) in his study of the generalized Riemann-Roch theorem Théorie des intersections et théorème de Riemann-Roch [1971, Def. IV.2.2]. In case X has an ample family of line bundles one can describe K 0 (X) as the free abelian group generated by the locally free O X -modules V of finite type modulo the relationsee Théorie des intersections et théorème de Riemann-Roch [ibid., Sec. IV.2.9]. We denote by X[t] resp. X[t 1 ] the scheme X A 1 with parameter t resp. t 1 for the affine line A 1 , and we denote by X[t; t 1 ] the scheme X G m , where G m = A 1 n f0g. Bass successively defined negative algebraic K-groups of the scheme X (at least in the affine case) in degree i < 0 to beThe two classical key properties, essentially due to Bass [1968], satisfied by these algebraic K-groups are the Fundamental Theorem and Excision. The author is supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg). MSC2010: primary 19D35; secondary 13D15. 163 164 MORITZ KERZ Proposition 1 (Fundamental Theorem). For a quasi-compact, quasi-separated scheme X and i Ä 0 there exists an exact sequenceFurthermore, for a noetherian, regular scheme X we have K i (X) = 0 for i < 0.Proposition 2 (Excision). For a ring homomorphism A ! A 0 and an ideal I A which maps isomorphically onto an ideal I 0 of A 0 the map K i (A; I ) ! K i (A 0 ; I 0 ) of relative K-groups is an isomorphism for i Ä 0.Combining Proposition 2 with the Artin-Rees Lemma we get the following more geometric reformulation:Corollary 3. For a finite morphism of affine noetherian schemes f : X 0 ! X and a closed immersion Y ,! X such that f is an isomorphism over X nY the mapIn general it is a hard problem to actually calculate the negative K-groups in concrete examples. One of the examples calculated in C. Weibel [2001, Sec. 6] reads:Example 4. For a field k and the normal surface X = Spec k[x; y; z]/(z 2 x 3 y 7 ) we have K 1 (X) = k and K i (X) = 0 for i < 1.

Categorical Milnor squares and K-theory of algebraic stacks

“…Recall that in the nonsingular case, all the groups K −n (X) vanish for n > 0. See [Ker18] for some discussion about the relationship between negative K-groups and singularities in the setting of schemes.…”

Continuous K-Theory and Cohomology of Rigid Spaces

“…In the general case where X ′ is an aribtrary model, we have do proceed differently. For n < 0 and α ∈ K n (X ′ on π) there exists by Raynaud-Gruson's platification par éclatement an admissible blow-up X ′′ → X ′ such that the pullback of α vanishes in K n (X ′′ on π) [Ker18,7]. In the colimit over all models this yields that K cont…”

“…This notion was recently studied by Kerz-Saito-Tamme [KST19] and they showed that it coincides in non-positive degrees with the groups studied by Karoubi-Villamayor and Calvo. For an affinoid algebra A over a discretely valued field, Kerz proved the corresponding analytical statements to (i) and (ii); that is replacing algebraic K-theory by continuous K-theory and the polynomial ring by the ring of power series converging on a unit disc [Ker18]. Morrow showed that continuous K-theory extends to a sheaf of pro-spectra on rigid k-spaces for any discretely valued field k. The main result of this article provides analogous statements of (i)-(iii) above for continuous K-theory of rigid k-spaces; the statements (i) and (ii) extend Kerz' result to the global case and statement (iii) is entirely new.…”

Continuous K-theory and cohomology of rigid spaces

“…1 This notion was recently studied by Kerz-Saito-Tamme [32] and they showed that it coincides in non-positive degrees with the groups studied by Karoubi-Villamayor and Calvo. For an affinoid algebra A over a discretely valued field, Kerz proved the corresponding analytical statements to (i) and (ii); that is replacing algebraic K-theory by continuous Ktheory and the polynomial ring by the ring of power series converging on a unit disc [28]. Morrow showed that continuous K-theory extends to a sheaf of prospectra on rigid k-spaces for any discretely valued field k. The main result of this article provides analogous statements of (i)-(iii) above for continuous K-theory of rigid k-spaces; the statements (i) and (ii) extend Kerz' result to the global case and statement (iii) is entirely new.…”

“…Our proof uses rigid analytic spaces in the sense of Tate [44] and adic spaces introduced by Huber [27]. Another approach is the one of Berkovich spaces [6] for which there is also a version of our main result as conjectured in the affinoid case by Kerz [28,Conj. 14].…”