Modules with minimax Cousin cohomologies

Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X, the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

COMPARISON OF MULTIGRADED AND UNGRADED COUSIN COMPLEXES

This paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a $\mathbb{Z}$-graded module $M$ over a $\mathbb{Z}$-graded commutative Noetherian ring $R$, the graded Cousin complex for $M$ introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.AMS 2000 Mathematics subject classification: Primary 13A02; 13E05; 13D25; 13D45

Smooth Formal Embeddings and the Residue Complex

Let π:X → S be a finite type morphism of noetherian schemes. A smooth formal embedding of X (over S) is a bijective closed immersion X ⊂ 𝖃 , where 𝖃 is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion 𝖃 = Y/X where X ⊂ Y is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham(co)homology.Our main application is an explicit construction of the Grothendieck residue complex when S is a regular scheme. By definition the residue complex is the Cousin complex of π!OS, as in [RD]. We start with I-C. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf .We then use smooth formal embeddings to obtain the coboundary operator . We exhibit a canonical isomorphism between the complex (K·x/s, δ ) and the residue complex of [RD]. When π is equidimensional of dimension n and generically smooth we show that H-nK·x/s is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi [KW].Another issue we discuss is Grothendieck Duality on a noetherian formal scheme 1d583 . Our results on duality are used in the construction of K·x/s.

Comparison of graded and ungraded Cousin complexes

Let R = ⊕n∈zRn be a ℤ-graded commutative Noetherian ring and let M be a ℤ-graded R-module. S. Goto and K. Watanabe introduced the graded Cousin complex *C(M)* for M, a complex of graded R-modules. Also one can ignore the grading on M and construct the Cousin complex C(M)* for M, discussed in earlier papers by the second author. The main results in this paper are that *C(M)* can be considered as a subcomplex of C(M)* and that the resulting quotient complex is always exact. This sheds new light on the known facts that, when M is non-zero and finitely generated, C(M)* is exact if and only if *C(M)* is (and this is the case precisely when M is Cohen-Macaulay).

A characterization of generalized Hughes complexes

While trying to place the grade-theoretic analogue of the Cousin complex (for a commutative Noetherian ring A) of K. R. Hughes [7], and the complexes of modules of generalized fractions studied by the first author and H. Zakeri[18], and by L. O'Carroll [10, p. 420], on a similar footing, Sharp and M. Yassi[15] introduced the concept of generalized Hughes complex. Such complexes are described as follows.

Generalized fractions and Hughes' gradetheoretic analogue of the Cousin complex

Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [19, Section 2]: it is a complex of A-modules and A-homomorphismswith the property that, for each n ∈ N0 (we use N0 to denote the set of non-negative integers),Cohen–Macaulay rings can be characterized in terms of the Cousin complex: A is a Cohen–Macaulay ring if and only if C(A) is exact [19, (4.7)]. Also, the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring (see [19,(5.4)]).