arXiv subjects

Srikanth B. Iyengar

Publications and source records attributed to Srikanth B. Iyengar.

At least 91 records · Page 5Linked to original sources

arXiv2011

Stratifying triangulated categories

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in T; and results concerning the support of the R-module of homomorphisms Hom_T^*(C,D) leading to an analogue of the tensor product theorem for support varieties of modular representation of groups.

math.CT↗

arXiv2011

Stratifying modular representations of finite groups

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

math.RT↗

arXiv2011

Module categories for finite group algebras

This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.

math.RT↗

arXiv2011

Colocalizing subcategories and cosupport

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.

math.RT↗

arXiv2011

Detecting flatness over smooth bases

Given an essentially finite type morphism of schemes f: X --> Y and a positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the d-fold fiber product, X^{d}, of X over Y and π_i: X^{d} --> X the i'th canonical projection. When Y smooth over a field and F is a coherent sheaf on X, it is proved that F is flat over Y if (and only if) f^{d} maps the associated points of the tensor product sheaf \otimes_{i=1}^d π_i^*(F) to generic points of Y, for some d greater than or equal to dim Y. The equivalent statement in commutative algebra is an analog---but not a consequence---of a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings.

math.AG↗

arXiv2010

Short Koszul modules

This article is concerned with graded modules M with linear resolutions over a standard graded algebra R. It is proved that if such an M has Hilbert series $H_M(s)$ of the form $ps^d+qs^{d+1}$, then the algebra R is Koszul; if, in addition, M has constant Betti numbers, then $H_R(s)=1+es+(e-1)s^{2}$. When $H_R(s)=1+es+rs^{2}$ with $r\leq e-1$, and R is Gorenstein or $e=r+1\le 3$, it is proved that generic R-modules with $q\leq(e-1)p$ are linear.

math.AC↗

arXiv2010

Cohomology over complete intersections via exterior algebras

A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.

math.AC↗

arXiv2010

Reflexivity and rigidity for complexes, II: Schemes

We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I, which dealt with commutative rings.

math.AG↗

arXiv2010

Support and injective resolutions of complexes over commutative rings

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 semi-injective complex need not be semi-injective.

math.AC↗

arXiv2009

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite type flat maps f: X->Y of noetherian schemes, with f^!(O_Y) in place of D.

math.AC↗

arXiv2009

Reflexivity and rigidity for complexes. I. Commutative rings

A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the way, a number of new results concerning derived reflexivity with respect to C are established. Noteworthy is the statement that derived C-reflexivity is a local property; it implies that a finite R-module M has finite G-dimension over R if it is locally of finite G-dimension.

math.AC↗

arXiv2009

Homological dimensions and regular rings

A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings. A key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules. Analogous results for flat dimension and projective dimension are also established.

math.AC↗

arXiv2009

Linearity defects of modules over commutative rings

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module.

math.AC↗

arXiv2009

Free resolutions over commutative Koszul algebras

For R=Q/J with Q a commutative graded algebra over a field and J non-zero, we relate the slopes of the minimal resolutions of R over Q and of k=R/R_{+} over R. When Q and R are Koszul and J_1=0 we prove Tor^Q_i(R,k)_j=0 for j>2i, for each non-negative integer i, and also for j=2i when i>dim Q-dim R and pd_QR is finite.

math.AC↗

arXiv2009

Dimensions of triangulated categories via Koszul objects

Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.

math.CT↗

arXiv2008

On the existence of star products on quotient spaces of linear Hamiltonian torus actions

We discuss BFV deformation quantization of singular symplectic quotient spaces in the special case of linear Hamiltonian torus actions. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms, Gotay and Jennings for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.

math.QA↗

arXiv2008

Free resolutions over short local rings

The structure of minimal free resolutions of finite modules M over commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families of Koszul modules are identified. When R is Gorenstein the non-Koszul modules are classified. Structure theorems are established for the graded k-algebra Ext_R(k,k) and its graded module Ext_R(M,k).

math.AC↗

arXiv2008

Local cohomology and support for triangulated categories

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the case of group representations, this allows us to correct and establish a conjecture of Benson.

math.KT↗