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Typical sheaves of generalized CM-modules. (English) Zbl 0771.13005

Let \((R,m)\) be a noetherian local ring of dimension \(d>0\), and let \(M\) be a finitely generated generalized Cohen-Macaulay module over \(R\). Let \(x_ 1,\dots,x_ d\) be a standard system of parameters with respect to \(M\). Let \(I:=(x_ 1,\dots,x_ d)R\), \(T:=\text{Proj}(\bigoplus_{n\geq 0}I^ n/I^{n+1})\), and \(\mathbb{P}^{d-1}=\mathbb{P}^{d-1}_{R/I}\), so that \(T\) is a closed subscheme of \(\mathbb{P}^{d-1}\) (with closed immersion \(\lambda:T\to\mathbb{P}^{d-1})\). The author introduces a sheaf \({\mathcal E}\) (the exceptional sheaf of \(M\) with respect to \(I)\) on \(T\) and a sheaf \({\mathcal K}\) (the typical sheaf of \(M\) with respect to \(x_ 1,\dots,x_ d)\) on \(\mathbb{P}^{d-1}\), both of which are sheaves of Cohen-Macaulay modules. These are related by the following short exact sequence on \(\mathbb{P}^{d-1}:\) \(0\to{\mathcal K}\to(M/IM)\otimes{\mathcal O}_{\mathbb{P}^{d- 1}}\to\lambda_ *{\mathcal E}\to 0\). The author calculates the cohomological Hilbert functions of \({\mathcal E}\) and \({\mathcal K}\), obtaining in particular the degree of each. Furthermore \(M\) is Cohen-Macaulay if and only if \({\mathcal K}=0\). (For a coherent sheaf \({\mathcal F}\) on a projective space \(X\) the \(i\)-th cohomological Hilbert function is the function \(n\mapsto\text{length }H^ i(X,{\mathcal F}(n))\) \((i\geq 0,n\in\mathbb{Z})\).

MSC:

13C14 Cohen-Macaulay modules
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

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