The \(K\)-groups of a blow-up scheme and an excess intersection formula
type: Item
instance of: scholarly article
title: The \(K\)-groups of a blow-up scheme and an excess intersection formula
zbMATH Open document ID: 0816.19004
DOI: 10.1007/BF01232430
published in: Inventiones Mathematicae
publication date: 1995-06-27T00:00:00Z
full work available at URL: https://eudml.org/doc/144100
review text: Let \(X\) be a noetherian scheme and \(Y\) a closed subscheme regularly immersed in \(X\) and everywhere of codimension \(d\). Let \(X'\) be the blow- up of \(X\) along \(Y\). Denote by \(K(X)\) the \(K\)-theory spectrum of \(X\); its homotopy groups are the algebraic \(K\)-groups of the category of perfect complexes on \(X\). If \(X\) has an ample line bundle these groups are isomorphic to the Quillen algebraic \(K\)-groups of the category of algebraic vector bundles on \(X\). Theorem 2.1: There is a natural homotopy equivalence to \(K(X')\) from the product of \(K(X)\) and \(d-1\) copies of \(K(Y)\). Grothendieck proved that \(K_ 0 (X')\) contains a subgroup isomorphic to the product of \(K_ 0 (X)\) and \(d-1\) copies of \(K_ 0 (Y)\). Theorem 2.1 shows this subgroup is in fact all of \(K_ 0 (X')\) and extends the result to higher and lower \(K\)-groups. The second main result of the paper is an excess intersection formula for algebraic \(K\)-theory. Preserving the above notations, denote by \(i\) the regular closed immersion from \(Y\) to \(X\). Let \(f\) be a proper map from \(Z\) to \(X\). Let \(W\) be the fiber product of \(Z\) and \(Y\) over \(X\), and denote by \(i'\) and \(f'\) the induced maps from \(W\) to \(Z\) and to \(Y\). Suppose that \(i'\) is also a regular immersion, and let \({\mathcal F}\) be the vector bundle on \(W\) which is the kernel of the canonical surjection of the pull-back of the conormal bundle of \(i\) to the conormal bundle of \(i'\). Then the push-forward maps \(i_ *\) and the usual pull-back maps \(f^*\) on \(K\)-theory are related by an equality up to homotopy: Theorem 3.1: \[ f^* \circ i_ * = i_ *' \circ \bigl( [\lambda_{-1} {\mathcal F}]\cup f'{}^*(\;) \bigr) \] Setting \(f = i\), one has also \(f' = i' = i\) and \({\mathcal F} = {\mathcal N}\) is the conormal bundle of \(i\). This case of the formula generalizes the ``key formula'' of Grothendieck for \(K_ 0\).
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 19D99
zbMATH DE Number: 447098
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: blow-up
zbMATH Keywords: excess intersection
zbMATH Keywords: locally complete intersection morphism
zbMATH Keywords: perfect complex
zbMATH Keywords: regular immersion
author: Robert W. Thomason
MaRDI profile type: Publication
cites work: Higher algebraic K-theory: I
cites work: Q3701607
description: scientific article; zbMATH DE number 447098