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Under semi-weak and weak compatibility conditions of bimodules, we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts. Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules. [ABSTRACT FROM AUTHOR]

Let k be a commutative ring, and let Λ and Γ be two k-algebras. In this paper, we give upper and lower bounds of Gorenstein global dimension and Gorenstein weak dimension over the tensor product Λ ⊗ k Γ . As its applications, the Gorenstein dimensions of several special algebras such as group algebras, matrix algebras, triangular matrix algebras and polynomial algebras can be computed. Moreover, we compare the Gorenstein global dimension of the enveloping algebra Λ e of Λ with the Gorenstein projective dimension of Λ . Some well-known results are also extended. [ABSTRACT FROM AUTHOR]

AbstractWe provide necessary and/or sufficient conditions for the stable Hom functors to be finitely presented. When the covariant Hom functor modulo projectives is finitely presented, its defect is isomorphic to the Bass torsion of the fixed argument. When the contravariant Hom functor modulo injectives is finitely presented, its defect is isomorphic to the cotorsion of the fixed argument. We also give a sufficient condition for the sub-stabilization of the tensor product to be finitely presented. A finite presentation of the tensor product leads to an unexpected application to injective modules. [ABSTRACT FROM AUTHOR]

Let A and B be abelian categories with enough projective and injective objects, and T : A → B a left exact additive functor. Then one has a comma category ( B ↓ T ). It is shown that if T : A → B is X -exact, then is a (hereditary) cotorsion pair in A and is a (hereditary) cotorsion pair in B if and only if is a (hereditary) cotorsion pair in ( B ↓ T ) and X and Y are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories A and B can induce special preenveloping classes in ( B ↓ T ). [ABSTRACT FROM AUTHOR]

It is well-established that weak n-tilting modules serve as generalizations of both n-tilting and n-cotilting modules. The primary objective of this paper is to delineate the characterizations of weak n-silting modules and elaborate on their applications. Specifically, we aim to establish the "triangular relation" within the framework of silting theory in a module category, and provide novel characterizations of weak n-tilting modules. Furthermore, we delve into the properties of n-(co)silting modules and their interrelations with some other types of modules. Additionally, we explore the conditions under which a weak n-silting module can be classified as partial n-silting, weak n-tilting, or partial n-tilting. Notably, we establish and prove that Bazzoni's renowned characterization of pure-injectivity for cotilting modules remains valid for weak n-silting modules with respect to F T . Lastly, we investigate weak n-silting and weak n-tilting objects in a morphism category. [ABSTRACT FROM AUTHOR]

Let T = ( R 0 C S ) be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule. It is proven that (1) A left S-module M in Bass class is C-torsionless (resp. C-reflexive) if and only if ( Hom S (C , M) M ) is a torsionless (resp. reflexive) left T-module; (2) A left S-module M in Bass class is C-Gorenstein projective if and only if ( Hom S (C , M) M ) is a Gorenstein projective left T-module; (3) If C is a faithfully semidualizing (S, R)-bimodule, then a left S-module M is C-n-tilting if and only if ( Hom S (C , M) S ⊕ M ) is an n-tilting left T-module. [ABSTRACT FROM AUTHOR]

Assume that B is a finite dimensional algebra, and A = B [ P 0 ] is the one-point extension algebra of B using a finitely generated projective B-module P0. The categories of B-modules and A-modules are connected via two adjoint functors known as the restriction and extension functors, denoted by R and E , respectively. These functors have nice homological properties and have been studied in the category mod- A of finitely presented modules that we extend to the category Mod- A of all A-modules. Our main focus is to investigate the behavior of important subcategories (tilting and τ-tilting subcategories) and objects (finendo quasi-tilting modules, silting modules, and cosilting modules) under these functors. [ABSTRACT FROM AUTHOR]

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. Kawai's result is a tool to calculate the cohomology variety of a module in a block B of a finite group algebra kG restricted to subgroups of a defect group P, provided that P is abelian. Kawai's result coincides with a Theorem of Avrunin and Scott specialized to modules in the principal block and their restrictions to p-subgroups. J. Rickard raised the question whether Kawai's result can be extended to modules in blocks with arbitrary defect groups. We show that this is indeed the case for modules whose corresponding module over some almost source algebra is fusion stable. We show that this fusion stability hypothesis is automatically satisfied for principal blocks and blocks with abelian defect groups. [ABSTRACT FROM AUTHOR]

In this article, we prove Grothendieck's Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let k be a field of characteristic zero and S k be a strongly locally noetherian k-linear Grothendieck category. For a commutative noetherian k-algebra R, let S R denote the category of R-objects in S k obtained through a non-commutative base change by R of the abelian category S k . First, we establish Grothendieck's Vanishing Theorem for any object M in S R . Further, if R is local and S k is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object M in S R . [ABSTRACT FROM AUTHOR]

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an arbitrary Auslander–Reiten component is Dynkin or Euclidean or one of the infinite trees A ∞ , A ∞ ∞ or D ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we study the relative tilting theory in extriangulated categories. We introduce the notion of relative tilting classes in an extriangulated category and then give some equivalent characterizations. On the above basis, we discuss the relations between relative homological dimensions and relative (co)resolution dimensions in the context of relative tilting classes. We also introduce the notion of relative tilting triples, and then discuss the relations between them. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the structure of tame symmetric algebras Λ of period four (TSP4 algebras for short). For a tame algebra, the number of arrows starting or ending at a given vertex cannot be large. Here we will mostly focus on the case when the Gabriel quiver of Λ is biserial, that is, there are at most two arrows ending and at most two arrows starting at each vertex. We present a range of properties (with relatively short proofs) which must hold for the Gabriel quiver of such an algebra. In particular, we show that triangles (and squares) appear naturally, so as for weighted surface algebras (Erdmann and Skowroński in J Algebra 505:490–558, 2018, J Algebra 544:170–227, 2020, J Algebra 569:875–889, 2021). Furthermore, we prove results on the minimal relations defining the ideal I for an admissible presentation of Λ in the form KQ/I. This will be the input for the classification of all TSP4 algebras with biserial Gabriel quiver. [ABSTRACT FROM AUTHOR]

Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M. We first give some homological formulas over R ⋉ M . Then we investigate how to construct (hereditary, complete) cotorsion pairs and (hereditary, cofibrantly generated) Hovey triples over R ⋉ M . Finally, some applications are given in some Morita context rings. [ABSTRACT FROM AUTHOR]

We give a sufficient condition that a certain homological epimorphism between two finite dimensional algebras induces a triangle equivalence between their Gorenstein defect categories as well as between their stable categories of Gorenstein projective modules. [ABSTRACT FROM AUTHOR]

For a left Noetherian ring R, if the supremum of flat dimensions of all injective left R-modules is finite, we prove that strongly cotorsion left R-modules coincide with Gorenstein injective modules. Furthermore, if id R R < ∞ , we describe these two classes of modules as certain right perpendiculars D ⊥ , where D are the classes of injective modules, modules with finite injective and flat dimension, respectively. [ABSTRACT FROM AUTHOR]

Let P < ∞ (Λ -mod) be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra Λ. We develop a widely applicable criterion that reduces the test for contravariant finiteness of P < ∞ (Λ -mod) in Λ-mod to corner algebras eΛe for suitable idempotents e ∈Λ. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of P < ∞ (Λ -mod) . The consequences pursued here hinge on the fact that this finiteness condition is known to be equivalent to the existence of a strong tilting object in Λ-mod. We moreover characterize the situation in which the process of strongly tilting Λ-mod allows for unlimited iteration: This occurs precisely when, in the category mod- Λ ~ of right modules over the strongly tilted algebra Λ ~ , the subcategory of modules of finite projective dimension is in turn contravariantly finite; the latter condition can, once again, be tested on suitable corners eΛe of the original algebra Λ. In the (frequently occurring) positive case, the sequence of consecutive strong tilts, Λ ~ , Λ ~ ~ , Λ ~ ~ ~ , ... , is shown to be periodic with period 2 (up to Morita equivalence); moreover, any two adjacent categories in the sequence P < ∞ (mod- Λ ~) , P < ∞ ( Λ ~ ~ -mod) , P < ∞ (mod- Λ ~ ~ ~) , ... , alternating between right and left modules, are dual via contravariant Hom-functors induced by tilting bimodules which are strong on both sides. Our methods rely on comparisons of right P < ∞ -approximations in the categories Λ-mod, eΛe-mod and the Giraud subcategory of Λ-mod determined by e; these interactions hold interest in their own right. In particular, they underlie our analysis of the indecomposable direct summands of strong tilting modules. [ABSTRACT FROM AUTHOR]

In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. Among the applications are precise connections between Ringel duality, split quasi-hereditary covers and double centralizer properties, constructions of split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of q-Schur algebras and a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$ . These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring. They are shown to behave nicely under change of rings techniques.

As generalizations of silting modules and star modules, respectively, the notions and basic properties of Gorenstein weak n-silting and weak n-star modules are given. We mainly show the "triangular relation" in tilting theory is also valid in silting and star theories in the context of Gorenstein homological algebras. We establish more closed connections among silting and tilting, star theories, that is, we investigate relations between Gorenstein weak n-silting, Gorenstein weak n-tilting and Gorenstein weak n-star modules. Based on these relations, we obtain new characterizations of Gorenstein weak n-tilting modules. [ABSTRACT FROM AUTHOR]

It is well known that the concept of cotilting modules generalizes injective cogenerators and in turn, the concept of cosilting modules generalizes cotilting modules. In this paper, we further investigate the close connections among injective cogenerators, cotilting modules and cosilting modules from the viewpoint of morphism categories. Some applications are also given. [ABSTRACT FROM AUTHOR]

A successful theme in the development of triangulated categories has been the study of compact objects. A weak dual notion called 0-cocompact objects was introduced in Oppermann et al. (Adv Math 350:190–241, 2019), motivated by the fact that sets of such objects cogenerate co-t-structures, dual to the t-structures generated by sets of compact objects. In the present paper, we show that the notion of 0-cocompact objects also appears naturally in the presence of certain dualities. We introduce "partial Serre duality", which is shown to link compact to 0-cocompact objects. We show that partial Serre duality gives rise to an Auslander–Reiten theory, which in turn implies a weaker notion of duality which we call "non-degenerate composition", and throughout this entire hierarchy of dualities the objects involved are 0 -(co)compact. Furthermore, we produce explicit partial Serre functors for multiple flavors of homotopy categories, thus illustrating that this type of duality, as well as the resulting 0 -cocompact objects, are abundant in prevalent triangulated categories. [ABSTRACT FROM AUTHOR]

Let R be a ring and X a complex of R-modules. We give some characterizations of the dimension of X related to special Gorenstein projective precovers. Let R be a ring such that (G P , G P ⊥) forms a cotorsion pair cogenerated by a set, where G P denotes the category consisting of all Gorenstein projective R-modules. Then the dimension related to special Gorenstein projective precovers Gppd (X) is equal to the infimum of the set { n ∈ Z | there exists a complete Gorenstein projective resolution of X T → σ G → X such that σi is bijective for each i ≥ n }. Forthmore, if Gppd (X) is finite and n an integer then Gppd (X) ≤ n if and only if Ext G P i (X , Q) = 0 for any i > n and any projective R-module Q, where Ext G P i (− , −) is the relative cohomology functor. [ABSTRACT FROM AUTHOR]

Let R be a ring and n be a non-negative integer. In this paper, we introduce and study the notions of Gorenstein n-weak injective and Gorenstein n-weak flat modules by using the notion of special super finitely presented modules. On an arbitrary ring, we investigate the relationships between Gorenstein n-weak injective and Gorenstein n-weak flat modules. Among other results, we prove that any R-module admits a Gorenstein n-weak injective (resp. Gorenstein n-weak flat) cover and pre-envelope. Then, we deduce that the class of Gorenstein n-weak injective (resp. Gorenstein n-weak flat) R-modules coincides with that of two-degree Gorenstein n-weak injective (resp. two-degree Gorenstein n-weak flat) R-modules. [ABSTRACT FROM AUTHOR]

For a recollement (A , B , C) of abelian categories, we show that n-tilting (resp. n-cotilting) subcategories in A and C can be glued to get n-tilting (resp. n-cotilting) subcategories in B under certain conditions. [ABSTRACT FROM AUTHOR]

We introduce the concepts of FP-cosilting and FP-cotilting modules, which extend finitely cosilting modules and finitely cotilting modules (Colby-modules) from the ∏ -coherent case to the coherent case. Some properties of FP-cosilting and FP-cotilting modules, including a Cotilting Theorem induced by these modules, are obtained. [ABSTRACT FROM AUTHOR]

Let RR be a ring, CC be a left RR-module and S=EndR(C)S={{\rm{End}}}_{R}\left(C). When CC is semidualizing, the Auslander class AC(S){{\mathcal{A}}}_{C}\left(S) and the Bass class ℬC(R){{\mathcal{ {\mathcal B} }}}_{C}\left(R) associated with CC have been the subject of extensive investigations. It has been proved that these classes, also known as Foxby classes, are one of the central concepts of (relative) Gorenstein homological algebra. In this paper, we answer several natural questions which arise when we weaken the condition of CC being semidualizing: if we let CC be w-tilting (see Definition 2.1), we establish the conditions for the pair (AC(S),AC(S)⊥1)\left({{\mathcal{A}}}_{C}\left(S),{{\mathcal{A}}}_{C}{\left(S)}^{{\perp }_{1}}) to be a perfect cotorsion theory and for the pair (BC⊥1(R),BC(R))\left({}^{{\perp }_{1}}B_{C}\left(R),{B}_{C}\left(R)) to be a complete hereditary cotorsion theory. This tells us when the classes of Auslander and Bass are preenveloping and precovering, which generalizes a number of results disseminated in the literature. We investigate Gorenstein flat modules relative to a not necessarily semidualizing module CC and we find conditions for the class of GC{G}_{C}-projective modules to be special precovering, the class of GC{G}_{C}-flat modules to be covering, the one of Gorenstein CC-projective modules to be precovering and that of Gorenstein CC-injective modules to be preenveloping. We also find how to recover Foxby classes from AddR(C){{\rm{Add}}}_{R}\left(C)-resolutions of RR.

We introduce and study the concepts of weak n-injective and weak n-flat modules in terms of super finitely presented modules whose projective dimension is at most n, which generalize the n-FP-injective and n-flat modules. We show that the class of all weak n-injective R-modules is injectively resolving, whereas that of weak n-flat right R-modules is projectively resolving and the class of weak n-injective (or weak n-flat) modules together with its left (or right) orthogonal class forms a hereditary (or perfect hereditary) cotorsion theory. [ABSTRACT FROM AUTHOR]

We introduce the concepts of n-silting and n-cosilting modules, which are not only generalizations of n-tilting and n-cotilting modules, but also generalizations of silting and cosilting modules. Some characterizations and properties of n-silting and n-cosilting modules are obtained. [ABSTRACT FROM AUTHOR]

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz' support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras. [ABSTRACT FROM AUTHOR]

We introduce the concept of weak silting modules, which is a generalization of both silting modules and Tor-tilting modules. It is shown that W is a weak silting module if and only if its character module W+ is cosilting. Some properties of weak silting modules are given. [ABSTRACT FROM AUTHOR]

Let T be a right exact functor from an abelian category ℬ into another abelian category A . Then there exists a functor p from the product category A × ℬ to the comma category ( (T ↓ A) ). In this paper, we study the property of the extension closure of some classes of objects in ( (T ↓ A) ), the exactness of the functor p and the detailed description of orthogonal classes of a given class P (X , Y) in ( (T ↓ A) ). Moreover, we characterize when special precovering classes in abelian categories A and ℬ can induce special precovering classes in ( (T ↓ A) ). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring Λ = ( R M 0 S ) is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them. [ABSTRACT FROM AUTHOR]

Let R be a graded ring and n ⩾ 1 be an integer. We introduce and study the notions of Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules by using the notion of special finitely presented graded modules. On n-gr-coherent rings, we investigate the relationships between Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules. Among other results, we prove that any graded module in R-gr (or gr-R) admits a Gorenstein n-FP-gr-injective (or Gorenstein n-gr-flat) cover and preenvelope, respectively. [ABSTRACT FROM AUTHOR]

Let S and R be rings, S C R be a semidualizing bimodule and n ≥ 0 be an integer or n = ∞. We introduce and study C-FPn-injective and C-FPn-flat modules as a common generalization of some known modules such as C-injective (resp. C-FP-injective, C-weak injective) and C-flat (resp. C-projective, C-weak flat) modules. Suppose that S C R is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FPn-flat left S-modules and C-FPn-injective left R-modules. Then we show that the pairs (F F C n (S) , F I C n (S o p)) and (F I C n (R) , F F C n (R o p)) are coproduct-closed and product-closed duality pairs and both F F C n (S) and F I C n (R) are covering and preenveloping, where F F C n (S) and F I C n (R) denote the classes of C-FPn-flat left S-modules and C-FPn-injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FPn-injective and C-FPn-flat modules. [ABSTRACT FROM AUTHOR]

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński. [ABSTRACT FROM AUTHOR]

We construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to Π p + q A , with Π the parity changing functor. Finally, we show that, given an automorphism of V, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of V. [ABSTRACT FROM AUTHOR]

For an extension of associative algebras B⊂A over a field and an A‐bimodule X, we obtain a Jacobi–Zariski long nearly exact sequence relating the Hochschild homologies of A and B, and the relative Hochschild homology, all of them with coefficients in X. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness. [ABSTRACT FROM AUTHOR]

In this paper, the relative derived categories with respect to X -Gorenstein projective modules and Y -Gorenstein injective modules are introduced to unify Gorestein derived categories and Ding derived categories in certain sense. A triangle-equivalence and description of morphisms in such relative derived categories are given. We also discuss a generalized Tate cohomology and obtain Avramov–Martsinkovsky exact sequences. Finally, some applications are obtained. [ABSTRACT FROM AUTHOR]

Let R and S be arbitrary rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule, and let 𝒜 C ⁢ (R op) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ (R) {\mathcal{B}_{C}(R)} be the Auslander and Bass classes, respectively. Then both pairs (𝒜 C ⁢ (R op) , ℬ C ⁢ (R)) and (ℬ C ⁢ (R) , 𝒜 C ⁢ (R op)) (\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(% \mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}})) are coproduct-closed and product-closed duality pairs and both 𝒜 C ⁢ (R op) {\mathcal{A}_{C}(R^{\mathrm{op}})} and ℬ C ⁢ (R) {\mathcal{B}_{C}(R)} are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if ℬ C ⁢ (R) {\mathcal{B}_{C}(R)} is enveloping in Mod ⁡ R {\operatorname{Mod}R} , then 𝒜 C ⁢ (S) {\mathcal{A}_{C}(S)} is enveloping in Mod ⁡ S {\operatorname{Mod}S}. Also, some applications to the Auslander projective dimension of modules are given. [ABSTRACT FROM AUTHOR]

In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable and unbounded homology for complexes of modules. In the case of module arguments, we get some results that improve the known results. [ABSTRACT FROM AUTHOR]

Let R be a ring and n, k two non-negative integers. In this paper, we introduce the concepts of n-weak injective and n-weak flat modules and via the notion of special super finitely presented modules, we obtain some characterizations of these modules. We also investigate two classes of modules with richer contents, namely W I k n (R) and W F k n (R o p) which are larger than that of modules with weak injective and weak flat dimensions at most k. Then over any arbitrary ring, we study the existence of W I k n (R) and W F k n (R o p) covers and pre-envelopes. [ABSTRACT FROM AUTHOR]

Let R be a Gorenstein ring. We first study Gorenstein orthogonal classes and Gorenstein cotorsion pairs of R-modules. Then, we introduce the concept of Gorenstein weak tilting R-modules, which is a "Gorenstein analogue" of weak tilting modules. It is proven that a left R-module W is Gorenstein weak n-tilting if and only if its character module W+ is Gorenstein n-cotilting if and only if its Gorenstein flat dimension is at most n and its left Gorenstein Tor-orthogonal class consists precisely of all right R-modules with a G-exact right Prod (W +) -resolution of finite length. Finally, we explore the connections between Gorenstein weak tilting modules and Gorenstein tilting modules. [ABSTRACT FROM AUTHOR]

Let K be a field, and let G be a group. In the present paper, we investigate when the group ring K[G] has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre's theorem for the weak dimension and the Gorenstein weak dimension. [ABSTRACT FROM AUTHOR]