On the vanishing of relative negative K-theory

Sadhu, Vivek

doi:10.1142/s0219498820501522

Cited by 3 publications

(1 citation statement)

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“…Almost verbatim the same argument as in the proof of Theorem 8 shows that the conclusion remains true with X replaced by a scheme X 0 which is smooth of finite type over a noetherian scheme of dimension d < 1. This was observed in Sadhu [2017].…”

Section: Weibel's Conjecturementioning

confidence: 64%

ON NEGATIVE ALGEBRAIC K-GROUPS

Kerz

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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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.

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Section: Weibel's Conjecturementioning

confidence: 64%