arXiv subjects

Srikanth B. Iyengar

Publications and source records attributed to Srikanth B. Iyengar.

At least 19 recordsLinked to original sources

arXiv2026

Some non-principal rigid ideals in Gorenstein domains of dimension one

We discuss an example of a rigid non-principal ideal in a one dimensional (commutative) Gorenstein domain, which contradicts a conjecture of C. Huneke and R. Wiegand. The construction and its analysis were discovered by Codex, when prompted by one of the authors to verify the conjecture or find a counterexample. A proof of the conjecture, also discovered by Codex, is presented when the ring is equicharacteristic and its embedding dimension is at most three.

math.AC↗

arXiv2026

The commutative algebra of congruence ideals and applications to number theory

In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to ``higher codimension''. A critical ingredient is a notion of congruence module in higher codimension: this has turned out to be a key definition whose utility extends beyond the role it plays in the numerical criterion. In this paper we trace the origin of some of the ideas that led to our work, both in number theory and commutative algebra, and new directions that emerge from it. We introduce a related notion of a congruence ideal. When applied to deformation theory of Galois representations and Hecke algebras, which is the setting of Wiles's work on Fermat's Last Theorem, our work leads to the notion of congruence ideals for local deformation rings. This sheds light on the classically studied congruence ideals for global deformation rings and Hecke algebras. We outline applications of the commutative algebra we have developed to: (i) integral modularity lifting theorems in the context of weight one forms, and (ii) factorization formulas for congruence ideals of global deformation rings at augmentations induced by newforms in which local congruence ideals enter as the local terms. The latter leads to surprising relations between these local congruence ideals and local Tamagawa ideals of Bloch-Kato associated to the rank 3 adjoint motive of $f$.

math.NT↗

arXiv2026

Commutative algebra inspired by modularity lifting

This article gives an overview of some recent results in commutative algebra that are inspired by the work of Wiles, Taylor and Wiles, Diamond, Lenstra and others on the modularity of elliptic curves.

math.AC↗

arXiv2026

High Frobenius pushforwards generate the bounded derived category

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme $X$ of prime characteristic. The main result is that when the Frobenius map on $X$ is finite, for any compact generator $G$ of $\mathsf{D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of $X$, an invariant that is a measure of the singularity of $X$. The conclusion holds for all positive integers $e$ when $X$ is locally complete intersection. The question of when one can take $G=\mathcal{O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

math.AG↗

arXiv2025

Ulrich modules over local rings of dimension two

It is proved that Ulrich modules exist for a large class of local rings of dimension two. This complements earlier work of the authors and Ziquan Zhuang that described complete intersection domains of dimension two that admit no Ulrich modules. As an application, it is proved that, for this class of rings, the length of a nonzero module of finite projective dimension is at least the multiplicity of the local ring.

math.AC↗

arXiv2025

Unstable elements in cohomology and a question of Lescot

In his work on the Bass series of syzygy modules of modules over a commutative noetherian local ring $R$, Lescot introduces a numerical invariant, denoted $σ(R)$, and asks whether it is finite for any $R$. He proves that this is so when $R$ is Gorenstein or Golod. In the present work many new classes of rings $R$ for which $σ(R)$ is finite are identified. The new insight is that $σ(R)$ is related to the natural map from the usual cohomology of the module to its stable cohomology, which permits the use of multiplicative structures to study the question of finiteness of $σ(R)$.

math.AC↗

arXiv2025

The spectrum of local dualisable modular representations

For a point $\mathfrak{p}$ in the spectrum of the cohomology ring of a finite group $G$ over a field $k$, we calculate the spectrum for the subcategory of dualisable objects inside the tensor triangulated category of $\mathfrak{p}$-local and $\mathfrak{p}$-torsion objects in the (big) stable module category of the group algebra $kG$.

math.RT↗

arXiv2025

Non-existence of Ulrich modules over Cohen-Macaulay local rings

Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal.

math.AC↗

arXiv2025

Homological properties of the module of differentials

These notes were produced by Jürgen Herzog to accompany his lectures in Recife, Brazil, in 1980, on the homological algebra of noetherian local rings. They are are concerned with two conjectures made by Wolmer Vasconcelos: if the conormal module of a local ring has finite projective dimension, or if the module of differentials, taken over an appropriate field, has finite projective dimension, then the ring must be complete intersection. The notes present an accessible and self-contained account of the strongest results known at the time in connection with these problems; this includes a number of ideas that have not appeared elsewhere. In the last section, Herzog turns his attention to the cotangent complex, and conjectures himself that if the cotangent complex of a local ring has bounded homology groups, then the ring must be complete intersection. Among other results, he proves that the conjecture holds for local rings of characteristic zero over which all modules have rational Poincaré series. Sadly Jürgen Herzog passed away in April of 2024. The notes in this form have been prepared in his memory, newly typeset and lightly edited. A short appendix has been added to survey some of the results of the intervening decades.

math.AC↗

arXiv2025

A freeness criterion for complexes with derived actions

Inspired by the patching method of Calegari and Geraghty, and a conjecture of de Smit that has been proved by the first author, we present a conjectural freeness criterion without patching for complexes over commutative noetherian local rings with derived actions, and verify it in several cases.

math.AC↗

arXiv2024

Proxy-small objects present compactly generated categories

We develop a correspondence between presentations of compactly generated triangulated categories as localizations of derived categories of ring spectra and proxy-small objects, and explore some consequences. In addition, we give a characterization of proxy-smallness in terms of coproduct preservation of the associated corepresentable functor `up to base change'.

math.CT↗

arXiv2024

Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions

We define a congruence module $Ψ_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $λ\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra regular at $\mathfrak{p}$, the kernel of $λ$, and $M$ a finitely generated $A$-module. We establish a numerical criterion for $M$ to have a free direct summand over $A$ of positive rank. It is in terms of the lengths of $Ψ_A(M)$ and the torsion part of $\mathfrak{p}/\mathfrak{p}^2$. It generalizes results of Wiles, Lenstra, and Diamond, that deal with the case when the codimension of $\mathfrak{p}$ is zero. Number theoretic applications include integral (non-minimal) $R=\mathbb T$ theorems in situations of positive defect conditional on certain standard conjectures. Here $R$ is a deformation ring parametrizing certain Galois representations and $\mathbb T$ is a Hecke algebra. An example is a modularity lifting for 2-dimensional $\ell$-adic Galois representations over an imaginary quadratic field. The proofs combine our commutative algebra results with a generalization due to Calegari and Geraghty of the patching method of Wiles and Taylor--Wiles and level raising arguments that go back to Ribet. The results provide new evidence in favor of the intriguing, and as yet fledgling, torsion analog of the classical Langlands correspondence. We also prove unconditional integral $R=\mathbb T$ results for Hecke algebras $\mathbb T$ acting on weight one cohomology of Shimura curves over $\mathbb Q$. This leads to a torsion Jacquet--Langlands correspondence comparing integral Hecke algebras acting on weight one cohomology of Shimura curves and modular curves. In this case the cohomology has abundant torsion and so our correspondence cannot be deduced by means of the classical Jacquet--Langlands correspondence.

math.NT↗

arXiv2024

Lim Ulrich sequences and Boij-Söderberg cones

This paper extends the results of Boij, Eisenbud, Erman, Schreyer, and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.

math.AC↗

arXiv2024

Locally dualisable modular representations and local regularity

This work concerns the stable module category of a finite group over a field of characteristic dividing the group order. The minimal localising tensor ideals correspond to the non-maximal homogeneous prime ideals in the cohomology ring of the group. Given such a prime ideal, a number of characterisations of the dualisable objects in the corresponding tensor ideal are given. One characterisation of interest is that they are exactly the modules whose restriction along a corresponding $π$-point are finite dimensional plus projective. A key insight is the identification of a special property of the stable module category that controls the cohomological behaviour of local dualisable objects. This property, introduced in this work for general triangulated categories and called local regularity, is related to strong generation. A major part of the paper is devoted to developing this notion and investigating its ramifications for various special classes of objects in tensor triangulated categories.

math.RT↗

arXiv2024

Locally dualizable modules abound

It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the derived category of $A$ than those that are obtained as retracts of images of perfect $A$-complexes. An analogous result is established dealing with the stable module category of the group algebra, over a countable field of positive characteristic $p$, of an elementary abelian $p$-group of rank at least 3.

math.AC↗

arXiv2023

Congruence modules in higher codimension and zeta lines in Galois cohomology

This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation is proved. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms.

math.NT↗

arXiv2023

Lattices over finite group schemes and stratification

This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full subcategory of all $G$-modules projective over $R$ generated by the lattices. The stable category of such $G$-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of $G$. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

math.RT↗

arXiv2023

A class of Gorenstein algebras and their dualities

In the recent paper "The Nakayama functor and its completion for Gorenstein algebras", a class of Gorenstein algebras over commutative noetherian rings was introduced, and duality theorems for various categories of representations were established. The manuscript on hand provides more context to the results presented in the aforementioned work, identifies new classes of Gorenstein algebras, and explores their behaviour under standard operations like taking tensor products and tilting.

math.RT↗