The relative form of Gersten's conjecture over a discrete valuation ring: The smooth case
type: Item
instance of: scholarly article
title: The relative form of Gersten's conjecture over a discrete valuation ring: The smooth case
zbMATH Open document ID: 0685.14008
DOI: 10.1016/0022-4049(87)90043-0
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: The authors state that ``the purpose of this note is to prove the following `relative' version of Gersten's conjecture: Theorem. Let \(X=Spec(R)\) be an affine scheme, flat and of finite type over a discrete valuation ring \(\Lambda\): suppose that there is a finite set of points \(S\subset X\) such that X is smooth over Spec(\(\Lambda\)) at the points of S. Let us write \(A={\mathcal O}_{X,S}\). If \(Y=Spec(R/tR)\to X\) is a principal effective relative (i.e. flat over \(\Lambda\)) divisor, then the exact functors \(M^{(i)}(Y)\to M^{(i)}(Spec(A))\) and \(MF^{(i)}(Y)\to MF^{(i)}(Spec(A))\) [...] both induce the zero map on K-theory.'' In the statement of this result, \(M^{(i)}(Y)\) denotes the category of coherent sheaves of modules over Y supported in codimension i, and \(MF^{(i)}(Y)\) denotes the category of coherent sheaves of modules over Y that are flat over \(\Lambda\). Several applications of this result are given. \{Reviewers remark: Corollary 5 is very similar to proposition 3.2 in a paper by \textit{L. Claborn} and the reviewer, Ill. J. Math. 12, 228-253 (1968; Zbl 0159.049), where other special cases of this theorem are demonstrated.\}
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 4122110
zbMATH Keywords: Gersten's conjecture
zbMATH Keywords: zero map on K-theory
zbMATH Keywords: coherent sheaves
Wikidata QID: Q123135674
author: Henri Gillet
author: Marc Levine
reviewed by: Robert M. Fossum
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90043-0
OpenAlex ID: W2008203804
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description: scientific article; zbMATH DE number 4122110