Invariants of linkage of modules

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$, $N$ be two Cohen-Macaulay $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $\mathrm{Ext}^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Tor}^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is a Cohen-Macaulay module then we show that $\mathrm{Ext}^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(D^\dagger , N) = 0$ for all $i \gg 0$, where $D^\dagger = \mathrm{Ext}^r_A(D,A)$ and $r = \mathrm{codim}(D)$. As a consequence we get that $\mathrm{Ext}^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $\mathrm{End}_A(M)/\mathrm{rad}\,\mathrm{End}_A(M) \cong (\mathrm{End}_A(N)/\mathrm{rad}\,\mathrm{End}_A(N))^{\mathrm{op}}$. We also give a negative answer to a question of Martsinkovsky and Strooker.

A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\operatorname{\underline{CM}}(A)$ be the stable category of maximal Cohen-Macauley $A$-modules and let $\operatorname{\underline{ICM}}(A)$ denote the set of isomorphism classes in $\operatorname{\underline{CM}}(A)$. We define a function $\xi \colon \operatorname{\underline{ICM}}(A) \to \mathbb{Z}$ which behaves well with respect to exact triangles in $\operatorname{\underline{CM}}(A)$. We then apply this to (Gorenstein) liaison theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liaison classes of $\{\,\mathfrak{m}^n \mid n\geq 1 \,\}$ is an infinite set. We also prove that if $A$ is Henselian with finite representation type with $A/\mathfrak{m}$ uncountable then for each $m \geq 1$ the set $\mathcal {C}_m = \{\, I \mid I \text { is a codim $2$ CM-ideal with } e_0(A/I) \leq m \,\}$ is contained in finitely many even liaison classes $L_1,\dots ,L_r$ (here $r$ may depend on $m$).

On the cofiniteness of Artinian local cohomology modules

Let [Formula: see text] be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and [Formula: see text] an ideal of [Formula: see text]. Let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module and [Formula: see text] be an integer. In this paper we show that, the [Formula: see text]-module [Formula: see text] is nonzero and [Formula: see text]-cofinite if and only if [Formula: see text]. Also, several applications of this result will be included.

Endomorphism ring of local cohomology modules with respect to a pair of ideals

Let (R, 𝔪, k) be a complete Gorenstein local ring of dimension n. Let [Formula: see text] be the local cohomology module with respect to a pair of ideals I, J and [Formula: see text]. In this paper we will show that the endomorphism ring [Formula: see text] is a commutative ring. In particular if [Formula: see text] for all i ≠ t, then B is isomorphic to R. Also we prove that, B is a finite R-module if and only if [Formula: see text] is an Artinian R-module, where d = n - t. Moreover we will show that in the case that [Formula: see text] for all i ≠ t the natural homomorphism [Formula: see text] is nonzero which gives a positive answer to a conjecture due to Hellus–Schenzel (see [On cohomologically complete intersections, J. Algebra 320 (2008) 3733–3748]).

Some loci related to Cohen–Macaulayness

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. The pseudo Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness) was introduced by Cuong–Nhan [Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules, J. Algebra267 (2003) 156–177] as an extension of the Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness). In this paper, we describe the pseudo Cohen–Macaulay (pseudo CM) locus and pseudo generalized Cohen–Macaulay (pseudo generalized CM) locus of M. We also study the non-Cohen–Macaulay locus and the non-generalized Cohen–Macaulay locus of the canonical module K(M) of M in case where R is a quotient of a Gorenstein local ring.

On Weakly Laskerian and Weakly Cofinite Modules

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

Linkage and Strongly Generalized Cohen–Macaulay Ideals

The notion of SGCM ideals is introduced as an extension of the known notion of SCM ideals. Some properties of SGCM ideals, which are closely related to SCM ideals in a Gorenstein local ring, are given. It turns out that the SGCM property is retained through the even linkage class in a Gorenstein local ring. Also, we establish the equality of the finiteness dimension of the residue class rings of two evenly linked ideals. Finally, some slight relations between SGCM ideals and CM d-sequences are obtained.

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.