arXiv subjects
Publications and source records attributed to Ehsan Tavanfar.
We show that any complete local (normal) domain admits a module-finite quasi-Gorenstein normal (complete local) domain extension. In the geometric vein, we show that any normal projective variety $X$ over a field admits a finite surjective morphism $Y\rightarrow X$ from a normal quasi-Gorenstein projective variety $Y$. Notably, our results resolve the previously open case for residual characteristic two.
We prove that if a local ring admits a (pd-)test module of finite complete intersection dimension, then it is a complete intersection ring. This answers, positively, a question proposed by Celikbas, Dao and Takahashi. To this aim, we first investigate another question raised by Celikbas and Sather- Wagstaff concerning ascent properties of (pd-)test complexes under weakly regular homomorphisms.
We show that any quasi-Gorenstein deformation of a $3$-dimensional quasi-Gorenstein Buchsbaum local ring with $I$-invariant $1$ admits a maximal Cohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such a class of rings includes two instances of unique factorization domains constructed by Marcel-Schenzel and by Imtiaz-Schenzel, respectively. Apart from this result, motivated by the small Cohen-Macaulay conjecture in prime characteristic, we examine a question about when the Frobenius pushforward $F^e_*(M)$ of an $R$-module $M$ comprises a maximal Cohen-Macaulay direct summand in both local and graded cases.
We reduce Hochster's Canonical Element Conjecture (theorem since 2016) to a localization problem in a characteristic free way. We prove the validity of a new variant of the Canonical Element Theorem (CET) and explain how a characteristic free deduction of the new variant from the original CET would provide us with a characteristic free proof of the CET. We also show that the Balanced Big Cohen-Macaulay Module Theorem can be settled by a characteristic free proof if the big Cohen-Macaulayness of Hochster's modification module can be deduced from the existence of a maximal Cohen-Macaulay complex in a characteristic-free way.
For any $d\ge 4$, by deformation theory of schemes, we present examples of (complete or excellent) $d$-dimensional mixed characteristic normal local domains admitting no small Cohen-Macaulay algebra, but admitting instances of small (maximal) Cohen-Macaulay modules. It is also shown that a graded normal domain over a field whose Proj is an Abelian variety admits a graded small (maximal) Cohen-Macaulay module.
Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen-Macaulay R-module B_M such that the socle of B_M\otimes_RM is zero. When R is a quasi-specialization of a G-regular local ring or when M has complete intersection dimension zero, we show the existence of such a big Cohen-Macaulay R-module. It is conjectured that if R admits a non-zero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We prove this conjecture if either R is a quasi-specialization of a G-regular local ring or a quasi-Buchsbaum local ring.
In this article, we propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection $R$. The question can be stated in terms of the acyclicity of certain (finite) residual approximation complexes whose $0$-th homologies are the residue field of $R$. We show that our question has an affirmative answer for the first Koszul homology of any almost complete intersection, as well as for all positive Koszul homologies of certain system of parameters which exist in some almost complete intersection rings with small multiplicities. The statement about the first Koszul homology is shown to be equivalent to the Monomial Conjecture and thus follows from its validity.
In the present article, we investigate the following deformation problem. Let $(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a (homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is quasi-Gorenstein. Then is $R$ quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition requires some delicate studies.
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.
In this article, applying the quasi-Gorenstein analogous of the Ulrich's deformation of certain Gorenstein rings we show that some homological conjectures, including the Monomial Conjecture, Big Cohen-Macaulay Algebra Conjecture as well as the Small Cohen-Macaulay Conjecture reduce to the excellent unique factorization domains. Some other reductions of the Monomial Conjecture are also proved. We, moreover, show that certain almost complete intersections adhere the Monomial Conjecture.
Let R be a commutative Noetherian ring of prime characteristic and M be an x-divisible right R[x,f]-module that is Noetherian as R-module. We give an affirmative answer to the question of Sharp and Yoshino in the case where R is semi-local and prove that the set of graded annihilators of R[x,f]-homomorphic images of M is finite. We also give a counterexample in the general case.