arXiv subjects

Dipankar Ghosh

Publications and source records attributed to Dipankar Ghosh.

At least 19 recordsLinked to original sources

arXiv2026

Complexity, curvature and homological dimension of modules under linkage

In this article, we analyze how (projective and injective) complexity, curvature, and complete intersection dimension behave under linkage of modules and ideals. Let $R$ be a Gorenstein local ring. Consider a Gorenstein perfect ideal $\mathfrak{a}$ (e.g., $\mathfrak{a}$ is generated by an $R$-regular sequence). Let $M$ and $N$ be two Cohen-Macaulay $R$-modules linked by $\mathfrak{a}$. We prove that $\mathrm{cx}_R(M)= \mathrm{inj\,cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{inj\,curv}_R(N)$. In particular, when $R$ is complete intersection, $\mathrm{cx}_R(M)= \mathrm{cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{curv}_R(N)$. Furthermore, we show that $\mathrm{pd}_R(M)= \mathrm{pd}_R(N)$ and $\operatorname{CI-dim}_R(M)= \operatorname{CI-dim}_R(N)$. If any of these dimensions is finite, it is equal to $\mathrm{ht}(\mathfrak{a})$. Similar results are obtained for linkage of ideals. All these results highly extend a classical result of Peskine and Szpiro in many directions. We construct several examples that complement our results. These also show how properties like `integrally closed', `$\mathfrak{m}$-full' and `Burch' behave under linkage of ideals.

math.AC↗

arXiv2026

Asymptotic prime divisors and Vasconcelos invariant

Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. In this article, we prove that $$\mathrm{Ass}_R(M/I^{n} M) = \mathrm{Ass}_R(0:_{M} I) \cup \mathrm{Ass}_R(I^{n-1} M/I^{n} M) \text{ for all } n \gg 0.$$ We then investigate the asymptotic behaviour of the (local) Vasconcelos invariant of $M/I^{n} M$ as a function of $n$, when $R$ is $\mathbb{N}$-graded, $I$ is homogeneous, and $M$ is $\mathbb{Z}$-graded. When $I$ is generated by elements of positive degree, we show that, for sufficiently large n, the (local) Vasconcelos invariant of $M/I^{n} M$ either coincides with that of the colon submodule $(0 :_{M} I)$, or is a polynomial in $n$ of degree one whose leading coefficient is one of the degrees of the generators of $I$. This dichotomy depends exclusively on two cases determined by $(0:_{M} I)$. Thus, we recover and considerably strengthen the main results of Fiorindo-Ghosh [Nagoya Math. J. 258 (2025), 296-310.], where asymptotic linearity was shown under the additional assumption that $(0:_{M} I)=0$.

math.AC↗

arXiv2025

Complexity and curvature of pairs of Burch modules and ideals

The complexity and curvature of a module were first introduced by Avramov to distinguish modules of infinite homological dimension. Later, Avramov-Buchweitz extended the notion of complexity from a single module to pairs of modules, measuring the polynomial growth rate of the minimal number of generators of their Ext-modules. By taking one of the modules in the pair to be the residue field, one recovers the standard projective and injective complexity of modules, whereas the vanishing of the complexity of a pair is equivalent to the eventual vanishing of Ext-modules, giving rise to the study of what are popularly known as Ext-pd and Ext-id test modules. Dao studied a similar notion of Tor-complexity. In the same vein, the vanishing of the Tor-complexity of pairs gives rise to Tor-pd test modules. On the other hand, the concept of Burch ideals was introduced by Dao-Kobayashi-Takahashi, motivated by the classical work of Burch, and subsequently extended to modules by Dey-Kobayashi. It follows from a result of Avramov that Burch modules exhibit extremal complexity and curvature. Moreover, Dey-Kobayashi and Ghosh-Saha showed, respectively, that Burch modules are Tor-pd and Ext-pd test, and that Burch ideals are Ext-id test. In this paper, we unify and significantly extend these two themes of extremal complexity and curvature, and Ext/Tor vanishing results of Burch modules. A key new ingredient in our proofs, particularly in dealing with Burch modules of depth zero, is the independence of the Burch property under embedding.

math.AC↗

arXiv2025

Coherent functors, powers of ideals, and asymptotic stability

Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded ring with $S_{\underline{0}} = R$, and $\mathcal{M} $ be a finitely generated $\mathbb{Z}^r$-graded $S$-module. For $ \underline{n} = (n_1,\dots,n_r) \in \mathbb{N}^r$, set $G_{\underline{n}} := \mathcal{M}_{\underline{n}}$ or $G_{\underline{n}} := M/{\bf I}^{\underline{n}} N$, where ${\bf I}^{\underline{n}} = I_1^{n_1} \cdots I_r^{n_r}$. Suppose $F$ is a coherent functor on the category of finitely generated $R$-modules. We prove that the set $\rm{Ass}_R \big(F(G_{\underline{n}}) \big)$ of associate primes and $\rm{grade}\big(J, F(G_{\underline{n}})\big)$ stabilize for all $\underline{n} \gg 0$, where $J$ is a non-zero ideal of $R$. Furthermore, if the length $\lambda_R(F(G_{\underline{n}}))$ is finite for all $\underline{n} \gg 0$, then there exists a polynomial $P$ in $r$ variables over $\mathbb{Q}$ such that $\lambda_R(F(G_{\underline{n}})) = P(\underline{n})$ for all $\underline{n}\gg 0$. When $R$ is a local ring, and $G_{\underline{n}} = M/{\bf I}^{\underline{n}} N$, we give a sharp upper bound of the total degree of $P$. As applications, when $R$ is a local ring, we show that for each fixed $i \geq 0$, the $i$th Betti number $\beta_i^R(F(G_{\underline{n}}))$ and Bass number $\mu^i_R(F(G_{\underline{n}}))$ are given by polynomials in $\underline{n}$ for all $\underline{n} \gg 0$. Thus, in particular, the projective dimension $\rm{pd}_R(F(G_{\underline{n}}))$ (resp., injective dimension $\rm{id}_R(F(G_{\underline{n}}))$) is constant for all $\underline{n}\gg 0$.

math.AC↗

arXiv2024

Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor

In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these test behaviour, along with other results, to characterize various kinds of local rings, including hypersurface rings of multiplicity at most two, surprisingly requiring only finitely many vanishing of Ext or Tor involving such CM modules. As further applications, we verify the long-standing (Generalized) Auslander-Reiten Conjecture for every CM module of minimal multiplicity over a Noetherian local ring, thus vastly extending a result of Huneke-\c{S}ega-Vraciu.

math.AC↗

arXiv2024

Complexity and curvature of (pairs of) Cohen-Macaulay modules, and their applications

The complexity and curvature of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by these notions, we define Ext and Tor curvature of pairs of modules. The aim of this article is to study (Ext and Tor) complexity and curvature of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity and curvature of pairs of modules in terms of that of a single module. It is known that among all modules, the residue field has maximal complexity and curvature, moreover they characterize complete intersection local rings. As applications of our results, we provide some upper bounds of the curvature of the residue field in terms of curvature and multiplicity of any nonzero CM module. As a final upshot, these allow us to characterize complete intersection local rings (including hypersurfaces and regular rings) in terms of complexity and curvature of pairs of certain CM modules. In particular, under some additional hypotheses, we characterize complete intersection and regular local rings via injective curvature of the ring and that of the module of K\"{a}hler differentials respectively. Thus, we make partial progress towards a question of Christensen-Striuli-Veliche, as well as another by Vasconcelos.

math.AC↗

arXiv2024

Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal

Let $R$ be a Noetherian $\mathbb{N}$-graded ring. Let $L$, $M$ and $N$ be finitely generated graded $R$-modules with $N \subseteq M$. For a homogeneous ideal $I$, and for each fixed $k \in \mathbb{N}$, we show the asymptotic linearity of v-numbers of the graded modules $ {\rm Ext}_R^{k}(L,{I^{n}M}/{I^{n}N})$ and ${\rm Tor}_k^{R}(L,{I^{n}M}/{I^{n}N})$ as functions of $n$. Moreover, under some conditions on ${\rm Ext}_R^k(L,M)$ and ${\rm Tor}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\rm Ext}_R^{k}(L,{M}/{I^{n}N})$ and $ {\rm Tor}_k^{R}(L,{M}/{I^{n}N})$. The last result is obtained by proving the asymptotic linearity of v-number of $(U+I^{n}V)/I^{n}W$, where $U$, $V$ and $W$ are graded submodules of a finitely generated graded $R$-module such that $W \subseteq V$ and $(0:_{U}I) = 0$.

math.AC↗

arXiv2024

Auslander-Reiten conjecture for modules whose (self) dual has finite complete intersection dimension

Over a commutative Noetherian ring, we show that the Auslander-Reiten conjecture holds true for the class of (finitely generated) modules whose dual has finite complete intersection dimension. We provide another result that validates the conjecture for the class of modules whose self dual has finite complete intersection dimension and either the module or its dual has finite Gorenstein dimension. Thus we combine and strengthen a number of results in the literature, due to Auslander-Ding-Solberg, Dey-Ghosh and Rubio-P\'{e}rez.

math.AC↗

arXiv2024

Asymptotic behaviour of Vasconcelos invariants for products and powers of graded ideals

Let $R$ be a commutative Noetherian $\mathbb{N}$-graded ring. Let $N\subseteq M$ be finitely generated $\mathbb{Z}$-graded $R$-modules. Let $I_1,\ldots,I_r$ be nonzero proper homogeneous ideals of $R$. Denote ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$ for $\underline{n}=(n_1,\dots,n_r)\in\mathbb{N}^r$. In this paper, we prove that the (local) Vasconcelos invariant of ${\bf I}^{\underline{n}}M/{\bf I}^{\underline{n}}N$ is eventually the minimum of finitely many linear functions in $\underline{n}$. The same holds for $M/{\bf I}^{\underline{n}}N$ under certain conditions. Some specific examples are provided, where these functions are not eventually linear in $\underline{n}$. However, when $R$ is a polynomial ring over a field, we show that the global Vasconcelos invariants of $R/{\bf I}^{\underline{n}}$ and ${\bf I}^{\underline{n}}/{\bf I}^{\underline{n}+\underline{1}}$ are, in fact, asymptotically linear in $\underline{n}$ with the leading coefficients given by the initial degrees of $I_1,\ldots,I_r$. The last result is surprising: It differs from the Castelnuovo-Mumford regularity, which is not always linear even over polynomial rings, as shown by Bruns-Conca.

math.AC↗

arXiv2024

On the asymptotic behaviour of the Vasconcelos invariant for graded modules

The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behaviour of the minimum distance of projective Reed-Muller type codes. We initiate the study of this invariant for graded modules. Let $R$ be a Noetherian $\mathbb{N}$-graded ring, and $M$ be a finitely generated graded $R$-module. The v-number $v(M)$ can be defined as the least possible degree of a homogeneous element $x$ of $M$ for which $(0:_Rx)$ is a prime ideal of $R$. For a homogeneous ideal $I$ of $R$, we mainly prove that $v(I^nM)$ and $v(I^nM/I^{n+1}M)$ are eventually linear functions of $n$. In addition, if $(0:_M I)=0$, then $v(M/I^{n}M)$ is also eventually linear with the same leading coefficient as that of $v(I^nM/I^{n+1}M)$. These leading coefficients are described explicitly. The result on the linearity of $v(M/I^{n}M)$ considerably strengthens a recent result of Conca which was shown when $R$ is a domain and $M=R$, and Ficarra-Sgroi where the polynomial case is treated.

math.AC↗

arXiv2023

Finite homological dimension of Hom, vanishing of Ext, and applications to divisor class group

For finitely generated modules $M$ and $N $ over a commutative Noetherian local ring $R$, we give various sufficient criteria for detecting freeness of $M$ or $N$ via vanishing of some finitely many Ext modules $\textrm{Ext}^i_R(M,N)$ and finiteness of certain homological dimension of $\textrm{Hom}_R(M,N)$. Some of our results provide partial progress towards answering a question of Ghosh-Takahashi and also generalize their main results in many ways, for instance, by reducing the number of vanishing. Certain special cases of our results allow us to address the Auslander-Reiten conjecture for modules whose (self-) dual has finite projective dimension. Along the way, we establish a new characterization of $I$-Ulrich modules of Dao-Maitra-Sridhar which we then apply to provide a negative answer to a question of Gheibi-Takahashi concerning characteristic modules. Among other techniques, we introduce and study certain generalizations of the notion of residually faithful modules of Brennan-Vasconcelos and Goto-Kumashiro-Loan, which play a crucial role in our study. As some applications of our results, we provide affirmative answers to two questions raised by Tony Se on $n$-semidualizing modules. Namely, we show that over a local ring of depth $t$, every $(t-1)$-semidualizing module of finite G-dimension is free. Moreover, we establish that for normal domains which satisfy Serre's condition $(S_3)$ and are locally Gorenstein in codimension two, the class of $1$-semidualizing modules forms a subgroup of the divisor class group. These two groups coincide when, in addition, the ring is locally regular in codimension two.

math.AC↗

arXiv2023

Integrally closed $\mathfrak{m}$-primary ideals have extremal resolutions

We show that every integrally closed $\mathfrak{m}$-primary ideal $I$ in a commutative Noetherian local ring $(R,\mathfrak{m},k)$ has maximal complexity and curvature, i.e., $ {\rm cx}_R(I) = {\rm cx}_R(k) $ and $ {\rm curv}_R(I) = {\rm curv}_R(k) $. As a consequence, we characterize complete intersection local rings in terms of complexity, curvature and complete intersection dimension of such ideals. The analogous results on projective, injective and Gorenstein dimensions are known. However, we provide short proofs of these results as well.

math.AC↗

arXiv2023

The (ir)regularity of Tor and Ext

We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.

math.AC↗

arXiv2023

Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology

The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module $M$ of finite Gorenstein dimension $g$ such that ${\rm type}(M) \le \mu( {\rm Ext}_R^g(M,R) )$ (e.g., ${\rm type}(M)=1$). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero $R$-module $M$ of depth $\ge d - 1$ such that the injective dimensions of $M$, ${\rm Hom}_R(M,M)$ and ${\rm Ext}_R^1(M,M)$ are finite, then $M$ has finite projective dimension and $R$ is Gorenstein. In the 2nd part, we assume that $R$ is CM with a canonical module $\omega$. For CM $R$-modules $M$ and $N$, we show that the vanishing of one of the following implies the same for others: ${\rm Ext}_R^{\gg 0}(M,N^{+})$, ${\rm Ext}_R^{\gg 0}(N,M^{+})$ and ${\rm Tor}_{\gg 0}^R(M,N)$, where $M^{+}$ denotes ${\rm Ext}_R^{d-\dim(M)}(M,\omega)$. This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that $R$ is Gorenstein.

math.AC↗

arXiv2023

Complexity and rigidity of Ulrich modules, and some applications

We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$-Tor-rigid in general (where $s\geq 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.

math.AC↗

arXiv2022

Homological dimensions of Burch ideals, submodules and quotients

The notion of Burch ideals and Burch submodules were introduced (and studied) by Dao-Kobayashi-Takahashi in 2020 and Dey-Kobayashi in 2022 respectively. The aim of this article is to characterize various local rings in terms of homological invariants of Burch ideals, Burch submodules, or that of the corresponding quotients. Specific applications of our results include the following: Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $M=I$ be an integrally closed ideal of $R$ such that ${\rm depth}(R/I)=0$, or $M = \mathfrak{m} N \neq 0$ for some submodule $N$ of a finitely generated $R$-module $L$ such that either ${\rm depth}(N)\ge 1$ or $L$ is free. It is shown that: (1) $I$ has maximal projective $($resp., injective$)$ complexity and curvature. (2) $R$ is Gorenstein if and only if ${\rm Ext}_R^n(M,R)=0$ for any three consecutive values of $n \ge \max\{{\rm depth}(R)-1,0\}$. (3) $R$ is CM (Cohen-Macaulay) if and only if CM-$\dim_R(M)$ is finite.

math.AC↗

arXiv2021

Auslander-Reiten conjecture and finite injective dimension of Hom

For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${\rm Hom}_R(M,R)$ and ${\rm Hom}_R(M,M)$ has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.

math.AC↗

arXiv2020

Gorensteinness of short local rings in terms of the vanishing of Ext and Tor

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring which contains a regular sequence $ \underline{x} = x_1,\ldots,x_d \in \mathfrak{m} \smallsetminus \mathfrak{m}^2 $ such that $ \mathfrak{m}^3 \subseteq (\underline{x}) $. Let $ M $ be a finite $ R $-module with maximal complexity or curvature, e.g., $ M $ can be a nonzero direct summand of some syzygy module of the residue field $ R/\mathfrak{m} $. It is shown that the following are equivalent: (1) $R$ is Gorenstein, (2) $\mathrm{Ext}_R^{\gg 0}(M,R)=0$, and (3) $\mathrm{Tor}_{\gg 0}^R(M,ω) = 0$, where $ω$ denotes a canonical module of $R$. It gives a partial answer to a question raised by Takahashi. Moreover, the vanishing of $\mathrm{Ext}_R^{\gg 0}(ω,N)$ for certain $ R $-module $ N $ is also analyzed. Finally, it is studied why Gorensteinness of such local rings is important.

math.AC↗