Related papers: A note on cancellation of reflexive modules
We prove that certain modules are faithful. This enables us to draw consequences about the reduction number and the integral closure of some classes of ideals.
The Module Cancellation Problem solicits hypotheses that, when imposed on modules $K$, $L$, and $M$ over a ring $S$, afford the implication $K\oplus L\cong K\oplus M\Longrightarrow L\cong M$. In a well-known paper on basic element theory…
In this paper we prove that small cancellation rings under some natural restrictions are non-amenable and contain non-commutative free associative algebra.
The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root…
Let R be a ring of dimension d and A be one of R[Y] or R[Y,Y^{-1}]. If P is a projective A-module of rank \geq d+1 satisfying some condition, then we show that E(A\oplus P) acts transitively on Um(A\oplus P). When P is free, this result is…
We give a criterion for rings with $\m^3=0$ which are obtained as connected sums of two other rings to have non-trivial totally acyclic modules.
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
Given a finitely generated module over a commutative noetherian ring that satisfies certain reflexivity conditions, we show how failure of the semidualizing property for the module manifests in a disconnection of the prime spectrum of the…
We define, via Gorenstein homomorphisms, a class of local rings over which there exist non-trivial totally reflexive modules. We also provide a general construction of such rings, which indicates their abundance.
We prove complete reducibility for an integrable module for an affine Lie algebra where the canonical central element acts non-trivially. We further prove that integrable modules does not exists for most of the super affine Lie algebras…
Motivated by a recent result of Yoshino, and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over commutative Noetherian local rings. Our main result considers modules…
For a standard graded Cohen-Macaulay ring $S$, if the quotient $S/(\underline{x})$ admits non-free totally reflexive modules, where $\underline{x}$ is a system of parameters consisting of elements of degree one, then so does the ring $S$.…
In this paper, sufficient conditions for finitely generated modules over a commutative noetherian ring to be projective are given in terms of vanishing of Ext modules. One of the main results of this paper asserts that the Auslander--Reiten…
Reflexive functors of modules naturally appear in Algebraic Geometry, mainly in the theory of linear representations of group schemes, and in "duality theories". In this paper we study and determine reflexive functors and we give many…
The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen,…
Since 2020, finite weight modules have been studied over twisted affine Lie superalgebras. To complete the characterization of modules over affine Lie superalgebras, we need some information regarding modules over untwisted affine Lie…
Examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read off the minimal semi-injective resolution of the complex. The same examples also show that a localization of a…
We set up some basic module theory over semirings, with particular attention to what is needed in scheme theory over semirings. We show that while not all the usual definitions of vector bundle agree over semirings, all the usual…
The Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a finitely generated module is projective when certain Ext-modules vanish. But what if those Ext-modules do not vanish? We study the annihilators of these…
For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge…