Let $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\mathfrak{p})$ the residue field of the local ring $R_\mathfrak{p}$. We prove that if an R-module M satisfies $\operatorname{Ext}_R^{n}(k(\mathfrak{p}),M)=0$ for some $n\geqslant\dim R$, then $\operatorname{Ext}_R^i(k(\mathfrak{p}),M)=0$ holds for all $i \geqslant n$. This improves a result of Christensen, Iyengar and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

This paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories $\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$ with $\mathcal{C}$ an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory $\mathcal{T}$ of $\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$ is constructed. Some applications of these two results include the equivalence of Grothendieck groups $K_0(\mathcal{C})$ and $K_0(\mathcal{T})$, the existences of a new abelian model structure on the category of complexes $\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$, and a t-structure on the derived category $\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$.

We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

Let R be a Cohen–Macaulay local ring. It is shown that under some mild conditions, the Cohen–Macaulay property is preserved under linkage. We also study the connection of the (Sn) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module.

We apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module ${{\Gamma }_{\alpha }}\left( I \right)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian $\text{DG}$-rings: let $A$ be a commutative non-positive $\text{DG}$-ring such that ${{\text{H}}^{0}}\left( A \right)$ is a noetherian ring and for each $i\,<\,0,\,\text{the}\,{{\text{H}}^{0}}\left( A \right)$-module ${{\text{H}}^{i}}\left( A \right)$ is finitely generated. Given an ideal $\overline{\mathfrak{a}}\,\subseteq \,{{\text{H}}^{0}}\left( A \right)$, we show that the local cohomology functor $\text{R}{{\Gamma }_{\overline{\mathfrak{a}}}}$ associated with $\overline{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\overline{\mathfrak{a}}$-adic completion functor $\text{L}{{\Lambda }_{\overline{\mathfrak{a}}}}$ does not increase flat dimension.

Let 𝔭 be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k (𝔭), M) = 0 for some n ⩾ R𝔭, where k(𝔭) is the residue field at 𝔭, then ${\rm Tor}_i^R $(k (𝔭), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k (𝔭), M) are proved, and applications to the theory of homological dimensions are explored.

This work concerns the linearity defect of a module $M$ over a Noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted $\text{ld}_{R}M$. Roughly speaking, $\text{ld}_{R}M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. It is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$-module $M$, each of the sequences $(\text{ld}_{R}(I^{n}M))_{n}$ and $(\text{ld}_{R}(M/I^{n}M))_{n}$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence $(\text{ld}_{R}C_{n})_{n}$ where $C$ is a finitely generated graded module over a standard graded algebra over $R$.

Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\text{f}{{\text{d}}_{R}}X,\,\text{Gf}{{\text{d}}_{R}}X$ and ${{\text{G}}_{\text{C}}}-\text{f}{{\text{d}}_{R}}\,X\,\text{by}\,\text{T}\left( X \right)$. Let $M$ be an $R$-module such that $\text{H}_{\mathfrak{a}}^{i}\left( M \right)\,=\,0$ for all $i\,\ne \,n$. It is proved that if $T\left( M \right)\,<\,\infty$, then $\text{T}\left( \text{H}_{\mathfrak{a}}^{n}\left( M \right) \right)\,\le \,\text{T}\left( M \right)\,+\,n$, and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen–Macaulay modules, dualizing modules, and Gorenstein rings.

Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$, and $X$ is an $R$-module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\text{H}_{\mathfrak{a}}^{i}\left( X \right)$. Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\text{H}_{\mathfrak{a}}^{i}\left( X \right)$ in terms of the flat dimensions of the modules $\text{H}_{\mathfrak{a}}^{j}\left( X \right),j\ne i$, and that of $X$.

Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field. In positive characteristic, the Frobenius endomorphism (and, more generally, any contracting endomorphism) can also be used for these characterizations. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.

In this paper, we further study Tate cohomology of modules over a commutative ring with respect to semidualizing modules using the ideals of Sather-Wagstaff et al. [‘Tate cohomology with respect to semidualizing modules’, J. Algebra 324 (2010), 2336–2368]. In particular, we prove a balance result for the Tate cohomology ${\widehat{\mathrm{Ext} }}^{n} $ for any $n\in \mathbb{Z} $. This result complements the work of Sather-Wagstaff et al., who proved that the result holds for any $n\geq 1$. We also discuss some vanishing properties of Tate cohomology.

In this paper we compute the weak Gorenstein global dimension of a coherent power series ring. It is shown that the weak Gorenstein global dimension of R[[x]] is equal to the weak Gorenstein global dimension of R plus one, provided R[[x]] is coherent.

This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite GC-dimension, finite C-projective dimension, locally finite C-injective dimension or locally finite C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.

Sather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

Let $R$ be a commutative Noetherian ring and $M$ a finite $R$-module. In this paper, we consider Zariski-openness of the FID-locus of $M$, namely, the subset of $\mathrm{spec}\,R$ consisting of all prime ideals ${\mathfrak p}$ such that $M_{\mathfrak p}$ has finite injective dimension as an $R_{\mathfrak p}$-module. We prove that the FID-locus of $M$ is an open subset of $\mathrm{spec}\,R$ whenever $R$ is excellent.