Related papers: Linearity defects of modules over commutative ring…

We discover a new connection between Koszul theory and representation theory. Let $\La$ be a quadratic algebra defined by a locally finite quiver with relations. Firstly, we give a combinatorial description of the local Koszul complexes and…

Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^{\phi}\!M$ be the abelian group $M$ with…

This article is concerned with nonzero modules of finite length and finite projective dimension over a local ring. We show the Loewy length of such a module is larger than the regularity of the ring whenever the ring is strict…

Let R be a Cohen-Macaulay local ring. In this paper we study the structure of Ulrich $R$-modules mainly in the case where R has minimal multiplicity. We explore generation of Ulrich R-modules, and clarify when the Ulrich R-modules are…

Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. Old results are reviewed and a number of new results are…

We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of…

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring $(R,m)$ with $m^3=0$. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes,…

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective…

For any ideal $I$ of finite projective dimension in a commutative noetherian local ring $R$, we prove that if the conormal module $I/I^2$ has finite projective dimension over $R/I$, then $I$ must be generated by a regular sequence. This…

We construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure…

Let $\mathfrak{q}$ denote an ideal in a Noetherian local ring $(A,\mathfrak{m})$. Let $\underline{a}=a_1,\ldots,a_d \subset \mathfrak{q}$ denote a system of parameters in a finitely generated $A$-module $M$. This note investigate an…

Over Cohen--Macaulay rings admitting a pointwise dualizing module, we show that the class of modules of restricted projective dimension bounded by any integer is finitely deconstructible and that the class of modules of restricted flat…

Using linear functional-based duality of modules, we generalize the syndrome decoding algorithm of linear codes over finite fields to those over finite commutative rings. Moreover, If the ring is local the algorithm is simplified by…

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over…

In this work, we prove that if a graded, commutative algebra $R$ over a field $k$ is not Koszul then, denoting by $\mathfrak{m}$ the maximal homogeneous ideal of $R$ and by $M$ a finitely generated graded $R$-module, the nonzero modules of…

In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general…

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for…

Repetitiveness in projective and injective resolutions and its influence on homological dimensions are studied. Some variations on the theme of repetitiveness are introduced, and it is shown that the corresponding invariants lead to very…

We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example…

Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective…