The \(K\)-groups of a blow-up scheme and an excess intersection formula
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 \(K_ 0\).
scientific article; zbMATH DE number 447098