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Intersection sheaves over normal schemes. (English) Zbl 1198.14006

For a flat projective surfective morphism \(\pi:X\rightarrow Y\) of noetherian schemes of relative dimension \(d\) and invertible sheaves \({\mathcal L}_1,\dots,{\mathcal L}_{d+1}\) on \(X\), one can construct an invertible sheaf \({\mathcal I}_{X/Y}({\mathcal L}_1,\dots,{\mathcal L}_{d+1})\) on \(Y\), called intersection sheaf. The name comes from the fact that, when \(\pi\) is a morphism of smooth algebraic \(k\)-schemes over a field \(k\), it satisfies the equality \(c_1({\mathcal I}_{X/Y}({\mathcal L}_1,\dots,{\mathcal L}_{d+1}))=\pi_{*}(c_1({\mathcal L}_1)\dots c_1({\mathcal L}_{d+1}))\) in the Chow group \(\text{CH}^1(Y)\). In this paper, the construction is generalized to the case of equi-dimensional projective surjective morphisms to normal separated noetherian schemes. The author constructs the sheaf by showing that for a Zariski open subset \(U\) of \(Y\) such that \(\text{codim}(Y-U)\geq 2\) and \(\pi\) is flat over \(U\), the intersection sheaf \({\mathcal I}_{\pi^{-1}(U)/U}({\mathcal L}_1|_{\pi^{-1}(U)},\dots,{\mathcal L}_{d+1} |_{\pi^{-1}(U)})\) naturally extends to an invertible sheaf on \(Y\). Several applications to the study of a family of effective algebraic cycles and that of polarized endomorphisms are also given.

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
14C25 Algebraic cycles
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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