Extending coherent and quasi-coherent sheaves on generically closed spaces
Both the Picard group of a quasi-affine scheme and the class group of a normal domain may be described in a module theoretic setting as special cases of the so-called relative Picard group of a ring R associated to an idempotent kernel functor on R-mod. This paper aims to show that similar methods apply in a much more general context. Let us give some examples. It is well known that reflexive Modules (i.e., sheaves of modules) over an integral normal scheme X are locally free sheaves of rank one over \(X^{(1)}\), the subspace of points of codimension 1 of X. Moreover the class group of X is just the Picard group of \(X^{(1)}\). Another example is obtained by considering the local scheme at \(x\in X\) for some scheme X; i.e. consider the scheme Spe\(c({\mathcal O}_{X,x})\) canonically embedded in X. As a set this local scheme at x consists of all \(y\in X\) such that x is in the closure of \(\{\) \(y\}\) in X. - The general problem is the following: Given some subspace \(Y\subset X\), for example, an open subspace or one of the above mentioned, classify the extensions of a quasi-coherent sheaf on Y with structure sheaf \({\mathcal O}_ x| Y\) to a quasi-coherent sheaf on X. This problem arises in the description of relative invariants with respect to Y. If Y is open in X and \({\mathcal M}\) is quasi-coherent on Y then \(i_*{\mathcal M}\) is quasi-coherent on X, where i:\(Y\to X\) is the canonical inclusion. If X is Noetherian (we will assume this throughout) then a result of A. Grothendieck states that any coherent sheaf on Y may be extended to a coherent sheaf on X (note that \(i_*{\mathcal M}\) is not necessarily coherent itself). However, \(X^{(1)}\) is not even a subscheme of X, so in the situations we want to deal with Grothendieck's result cannot be applied. The subspaces Y we consider will be geometrically stable (generically closed), i.e. \(y\in Y\) and \(x\in X\), with \(y\in \{\bar x\}\), the closure of \(\{\) \(x\}\) in X, imply \(x\in Y\). The advantage of these spaces is that for a Noetherian ring the generically closed subsets of Spec(R) correspond bijectively to idempotent kernel functors in R-mod. We develop this correspondence further in a general geometric context, and we derive results similar to the extension result of A. Grothendieck. Along the way we show how some results of \textit{G. Horrocks} [Proc. Lond. Math. Soc., III. Ser. 14, 689- 713 (1964; Zbl 0126.168)] and \textit{R. Treger} [J. Algebra 54, 444-466 (1978; Zbl 0406.14001)] appear as applications of our constructions.
scientific article; zbMATH DE number 3867486
scientific article