Generalized local cohomology and regularity of Ext modules

Abstract

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg R (M, N ) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that reg R (M, N ) is finite in several cases. In the case that the base ring is a field, we show thatThis formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules (in the case R is Cohen-Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.

Regularity bounds for complexes and their homology

“…For example, we obtain the following important case of a result due to Chardin. As another application, we also obtain regularity bounds for Ext, broadly extending works of Caviglia [9], Chardin and Divaani-Aazar [12]; see Corollary 7.4 and Remark 7.8. A new and readily comprehensible consequence of our method is the following While our method appears to be new and yields improvements to various known results on the regularity of Tor and Ext, we would like to point out that the methods of Chardin [10] and Eisenbud-Huneke-Ulrich [19] have their own strength and produce significant results in other directions.…”

“…A new and readily comprehensible consequence of our method is the following While our method appears to be new and yields improvements to various known results on the regularity of Tor and Ext, we would like to point out that the methods of Chardin [10] and Eisenbud-Huneke-Ulrich [19] have their own strength and produce significant results in other directions. Some further results concerning the regularity of various functors may be found in [6], [10], [12], [13].…”

“…We prove the above formulas for regularity in Sections 3 and 4 (Theorems 3•9 and 4•3). The major concern in this paper is to unify various regularity bounds for Tor and Ext in the literature; see, e.g., [6,9,10,12,19]. Simple as they are, formulas (1•1) and (1•2) prove to be useful for this purpose.…”

A duality theorem for generalized local cohomology

“…We remark that the map defined in [CCHS,1.4 (b)] is induced by the map from Recall that there is a spectral sequence [CD,2.12] and [B,§6,Théorème 1,b]. As…”

“…(iii) Suzuki duality in the context of graded algebras over a field (see [CD,3.1]) corresponds to the case where q = 0, (iv) the Herzog-Rahimi spectral sequence corresponds to the case where M = S (see [HR] for the standrard bigraded case, and [CCHS] for the general case) using the spectral sequence…”