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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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