LetR be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality limn→∞ `(F n R(M)) pnd = `(M) holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for wh...

We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an R-modul...

In this article we investigate the relations between Gorenstein projective dimensions of [Formula: see text]-modules and their socles for text]-minimal Auslander–Gorenstein algebras text]. First give a description projective-injective in terms socles. Then prove that text]-module text] has dimension at most if only its socle is cogenerated by text]-module. Furthermore, show can be characterised...

In this paper, we study the relation between m-strongly Gorenstein projective (resp. injective) modules and n-strongly Gorenstein projective (resp. injective) modules whenever m 6= n, and the homological behavior of n-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Goren...

R ring (always commutative and Noetherian) (R,m,k) local ring with maximal ideal m and k = R/m L,M,N, . . . R-modules (always finitely generated) M HomR(M,R), the dual of M D(M) the Auslander dual of M (Definition 2) σM : M wM∗∗ the natural evaluation map; KM = Ker(σM ), CM = Coker(σM ) G-dimR(M),G-dim(M) Gorenstein dimension of M (Definition 16) G-dim(M) <loc ∞ M has locally finite Gorenstein ...

We will introduce the notion of Gorenstein category as the convenient setup for doing Gorenstein Homological Algebra in categories of sheaves, or in general in categories without enough projective objects. We will illustrate this notion by showing that the category of Qcoh(X) of quasi-coherent sheaves on a locally Gorenstein projective scheme fits into this setup. Then we will focus on the cate...

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed...

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely-generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely-generated Gorenstein-projective modules.

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the onl...