We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that t...

Given a commutative Noetherian local ring R, the linearity defect of a finitely generated R-module M , denoted ldR(M), is an invariant that measures how far M and its syzygies are from having a linear resolution. Motivated by a positive known answer in the graded case, we study the question of whether ldR(k) < ∞ implies ldR(k) = 0. We give answers in special cases, and we discuss several interp...

Let (R,m) be a commutative Noetherian local ring with m = (0). We give a condition for R to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over R is not necessarily a...

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A ⋉M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra. Let A be a noetherian local ...

Let (R,m) be a commutative Noetherian local ring. Suppose that M and N are finitely generated modules over R such that M has finite projective dimension and such that TorRi (M,N) = 0 for all i > 0. The main result of this note gives a condition on M which is necessary and sufficient for the tensor product of M and N to be a Cohen–Macaulay module over R, provided N is itself a Cohen–Macaulay mod...

Let (R, m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If GC -dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d − t then either GC -dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to...

It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable subrings A = A1 ⊕ · · · ⊕ An such that the Ai-Mod are indecomposable module categories. Similarly any coalgebra C over a field can be written as a direct sum of indecomposable subcoalgebras C = ⊕ I Ci such that the categories of Ci-comodules are indecomposable. In this paper a decomposit...

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...

we define and studyco-noetherian dimension of rings for which the injective envelopeof simple modules have finite krull-dimension. this  is a moritainvariant dimension that measures how far the ring is from beingco-noetherian. the co-noetherian dimension of certain rings,including commutative rings, are determined. it is shown that the class ${mathcal w}_n$ of rings with co-noetherian dimension...