arXiv subjects

Anjan Gupta

Publications and source records attributed to Anjan Gupta.

11 recordsLinked to original sources

arXiv2026

A criterion for modules over Gorenstein local rings to have rational Poincar\'e series

We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincar\'e series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over $R$ have rational Poincar\'e series sharing a common denominator. By a result of \c Sega, it follows that $R$ satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincar\'e series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincar\'e series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.

math.AC↗

arXiv2025

On Tor-vanishing of local rings

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the Tor-vanishing property if the converse holds, that is, $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $pd_R(M) < \infty$ or $pd_R(N) < \infty$. Interest in the Tor-vanishing property stems from the fact that Cohen-Macaulay local rings satisfying this property also satisfy the Auslander-Reiten conjecture. In this article, we study a variant of this property. If $R$ is a generalized Golod ring, we prove that $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $\{curv_R M, curv_R N \} \cap \{0, 1\} \neq \emptyset$. A key intermediate step in our proof is to show that $curv_R M \in \{0, 1, curv_R k\}$ for any module $M$ over a generalized Golod ring $R$. As an application, we prove that generic Gorenstein local rings, non-trivial connected sums of generalized Golod-Gorenstein rings satisfy the Tor-vanishing property and consequently the Auslander-Reiten conjecture. Our method suggests a uniform approach and recovers many old results on the Tor-vanishing property.

math.AC↗

arXiv2025

On Projective modules over graded $R$-subalgebras of $R[X,1/X]$

Let $R$ be a Noetherian ring of dimension $d$ and $A$ be a graded $R$-subalgebra of $R[X,1/X]$. Let $P$ be a projective module over $A$ of rank $r \geq \max\{d+1,2\}$ and $\v=(a,p)$ be a unimodular element of $A \oplus P$. We find an elementary automorphism $\tau$ such that $\tau (\v) = (1, 0)$. Consequently, we obtain the cancellative property of $P$. We show that $P$ splits off a free summand of rank one. When $A = R[X]$ or $R[X, 1/ X]$, the results are well-known due to the contributions by various authors.

math.AC↗

arXiv2025

A generalization of Rao's theorem to graded $R$-subalgebras of $R[t]$

Let $R$ be a Noetherian local ring of Krull dimension $d$ such that $(d!)R = R$, and let $A$ be a graded $R$-subalgebra of the polynomial algebra $R[t]$. We prove that every unimodular row of length $d + 1$ over $A$ can be completed to an invertible matrix. This is a generalization of a classical result by Rao, who proved that in the same setting, every unimodular row of length $d + 1$ over $R[t]$ admits a completion to an invertible matrix.

math.AC↗

arXiv2017

Detecting Koszulness and related homological properties from the algebra structure of Koszul homology

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

math.AC↗

arXiv2017

Ascent and descent of the Golod property along algebra retracts

We study ascent and descent of the Golod property along an algebra retract. We characterise trivial extensions of modules, fibre products of rings to be Golod rings. We present a criterion for a graded module over a graded affine algebra of characteristic zero to be a Golod module.

math.AC↗

arXiv2017

A Study of Good and Bad Artinian Gorenstein local Rings

We say that a local ring $R$ is good, in the sense of Roos, if all finitely generated $R$-modules have rational Poincar\'e series that share a common denominator; otherwise, $R$ is said to be bad. An important class of good rings is the class of generalized Golod rings. In this paper, we show that connected sums of Artinian Gorenstein generalized Golod rings are good. We provide a criterion for decomposing Artinian Gorenstein local rings as connected sums. As a key application, we prove that a Gorenstein local ring $R$ with maximal ideal $\mathfrak{m}$ is good under either of the following conditions: (1) the multiplicity of $R$ is at most $12$ and its $h$-vector is different from $(1, 5, 5, 1)$, (2) $\mathfrak{m}^4$ = 0 and $\mathfrak{m}^2$ is generated by at most four elements. The above result records partial progress towards resolving a question posed by L.~Avramov. We also present examples of bad Artinian Gorenstein local rings of any multiplicity greater than or equal to $18$. In all these cases, the results establishing that the rings are good are obtained by showing that the rings are generalized Golod rings.

math.AC↗

arXiv2015

Optimal injective stability for the symplectic $K_1Sp$ group

If $R$ is a commutative ring, $I$ an ideal of $R$ and $v, w \in Um_{2n}(R, I)$ then we show that $v, w$ are in the same orbit of elementary action if and only if they are in the same orbit of elementary symplectic action. We also show that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $d! A = A$, $d \equiv 2 \pmod 4$ and $I$ an ideal of $A$, then $Um_d(A, I) = e_1{Sp}_d(A, I)$. As a consequence it is proved that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $(d + 1)!A = A$, $d \equiv 1 \pmod 4$ and $I$ a principal ideal then $Sp_{d-1}(A, I) \cap {ESp}_{d+1}(A, I) = {ESp}_{d -1}(A, I)$. We give an example to show that the above result does not hold true for an affine algebra over a $C_2$ field and also show by an example that the above stability estimate is optimal.

math.KT↗

arXiv2015

Characterizations of regular local rings via syzygy modules of the residue field

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 dimension', then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.

math.AC↗

arXiv2014

On the existence of unimodular elements and cancellation of projective modules over noetherian and non-noetherian rings

Let $R$ be a commutative ring of dimension $d$, $S = R[X]$ or $R[X, 1/X]$ and $P$ a finitely generated projective $S$ module of rank $r$. Then $P$ is cancellative if $P$ has a unimodular element and $r \geq d + 1$. Moreover if $r \geq \dim (S)$ then $P$ has a unimodular element and therefore $P$ is cancellative. As an application we have proved that if $R$ is a ring of dimension $d$ of finite type over a Prüfer domain and $P$ is a projective $R[X]$ or $R[X, 1/X]$ module of rank at least $d + 1$, then $P$ has a unimodular element and is cancellative.

math.KT↗

arXiv2013

A nice group structure on the orbit space of unimodular rows-II

We establish an Excision type theorem for niceness of group structure on the orbit space of unimodular rows of length $n$ modulo elementary action. This permits us to establish niceness for relative versions of results for the cases when $n = d+1$, $d$ being the dimension of the base algebra. We then study and establish niceness for the case when $n = d$, and also establish a relative version, when the base ring is a smooth affine algebra over an algebraically closed field.

math.KT↗