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Prashanth Sridhar

Publications and source records attributed to Prashanth Sridhar.

15 recordsLinked to original sources
arXiv2025

Abelian extensions of equicharacteristic regular rings need not be Cohen-Macaulay

By a theorem of Roberts, the integral closure of a regular local ring in a finite abelian extension of its fraction field is Cohen-Macaulay, provided that the degree of the extension is coprime to the characteristic of the residue field. We show that the result need not hold in the absence of this requirement on the characteristic: for each positive prime integer $p$, we construct polynomial rings over fields of characteristic $p$, whose integral closure in an elementary abelian extension of order $p^2$ is not Cohen-Macaulay. Localizing at the homogeneous maximal ideal preserves the essential features of the construction.

math.AC↗
arXiv2025

Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture.

math.AC↗
arXiv2025

A note concerning the vanishing of local cohomology for roots in mixed characteristic

The goal of this note is to record the following curious fact: let $(S,\n)$ be an unramified regular local ring of mixed characteristic $p>0$ and dimension $d$. Let $L$ denote the quotient field of $S$ and $K=L(\omega)$ with $\omega^p\in L$. Let $R$ denote the integral closure of $S$ in $K$. Then $R$ is Cohen-Macaulay if and only if $\mathrm{H}^{d-1}_{\n}(R)=0$, i.e., the obstruction to the Cohen-Macaulayness of $R$ lies in a single local cohomology module. Furthermore, this is equivalent to the dual module $\Hom_S(R,S)$ satisfying Serre's condition $(S_3)$.

math.AC↗
arXiv2025

Existence of balanced dualizing dg-modules

We describe cohomological conditions that are necessary and sufficient for the existence of balanced dualizing dg-modules, generalizing a theorem of Van den Bergh for balanced dualizing complexes over graded algebras. As a consequence, we show that a dg-algebra satisfying certain finiteness conditions admits a balanced dualizing dg-module if and only if its zeroth cohomology algebra admits a balanced dualizing complex. Additionally, we obtain a host of new examples of dg-algebras whose associated noncommutative spaces satisfy Serre duality.

math.RA↗
arXiv2025

Module-Theoretic Characterizations of Gorenstein Morphisms

The Gorenstein property in local algebra admits several characterizations via its module category. The goal of this paper is to collect and generalize such characterizations to the relative setting, i.e., to Gorenstein morphisms as defined by [AF92]. We achieve this by proving these characterizations more generally for graded-commutative Gorenstein dg-algebras.

math.AC↗
arXiv2024

Serre duality for dg-algebras

We generalize Yekutieli-Zhang's noncommutative Serre Duality Theorem to the setting of noncommutative spaces associated to dg-algebras. As an application, we establish some finiteness properties of derived global sections over such noncommutative spaces. Along the way, we generalize Yekutieli's notion of a balanced dualizing complex to the setting of dg-algebras and establish some cases in which they exist.

math.RA↗
arXiv2024

On Abelian extensions in mixed characteristic and ramification in codimension one

A theorem of Paul Roberts states that the integral closure of a regular local ring in a generically abelian extension is Cohen-Macaulay, provided the characteristic of the residue field does not divide the order of the Galois group. An example of Koh shows the conclusion is false in the modular case. After a modification to the statement concerning ramification over $p$ in codimension one, we give an extension of Roberts's theorem to the modular case for unramified regular local rings in mixed characteristic when the $p$-torsion of the Galois group is annihilated by $p$.

math.AC↗
arXiv2024

The multigraded BGG correspondence in Macaulay2

We give an overview of a Macaulay2 package for computing with the multigraded BGG correspondence. This software builds on the package BGG due to Abo-Decker-Eisenbud-Schreyer-Smith-Stillman, which concerns the standard graded BGG correspondence. In addition to implementing the multigraded BGG functors, this package includes an implementation of differential modules and their minimal free resolutions, and it contains a method for computing strongly linear strands of multigraded free resolutions.

math.AC↗
arXiv2023

Orlov's Theorem for dg-algebras

A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. We generalize this theorem to the setting of differential graded algebras. As an application, we obtain new cases of the Lattice Conjecture in noncommutative Hodge theory.

math.AG↗
arXiv2023

Periodicity of ideals of minors in free resolutions

We study the asymptotic behavior of the ideals of minors in minimal free resolutions over local rings. In particular, we prove that such ideals are eventually 2-periodic over complete intersections and Golod rings. We also establish general results on the stable behavior of ideals of minors in any infinite minimal free resolution. These ideals have intimate connections to trace ideals and cohomology annihilators. Constraints on the stable values attained by the ideals of minors in many situations are obtained, and they can be explicitly computed in certain cases.

math.AC↗
arXiv2022

Finitistic dimensions over commutative DG-rings

In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude. We prove that any DG-module $M$ of finite flat dimension over such a DG-ring satisfies $\mathrm{projdim}_A(M) \leq \mathrm{dim}(\mathrm{H}^0 (A)) - \inf(M)$. We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds $\mathrm{dim}(\mathrm{H}^0 (A)) - \mathrm{amp}(A) \leq \mathsf{FPD}(A) \leq \mathrm{dim}(\mathrm{H}^0(A))$. Moreover, we prove that DG-rings exist which achieve either bound. As a direct application, we prove new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras.

math.AC↗
arXiv2021

Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$.

math.AC↗
arXiv2021

On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.

math.AC↗