Anurag K. Singh
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Anurag K. Singh has a total of 37 co-authors. Here are the top co-authors by number of shared publications:
- Uli Walther
Purdue Univ, Dept Math, 150 N Univ St, W Lafayette, IN 47907 USA
9 shared publications - Bhargav Bhatt
University of Michigan
8 shared publications - Gennady Lyubeznik
Department of Mathematics, University of Minnesota, Minneapolis, USA
7 shared publications - Manuel Blickle
Institut für Mathematik, Fachbereich 08, Johannes Gutenberg-Universität Mainz, Mainz, Germany
7 shared publications - Kei-Ichi Watanabe
Nihon Univ, Coll Humanities & Sci, Dept Math, Setagaya Ku, Tokyo 1568550, Japan
5 shared publications - Jack Jeffries
University of Michigan
4 shared publications
Latest Publications
- Open Access
Ideals generated by power sums
We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key case of a conjecture of Conca, Krattenthaler, and Watanabe, and prove other results in that direction.
Open AccessIdeals generated by power sums
We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key case of a conjecture of Conca, Krattenthaler, and Watanabe, and prove other results in that direction.
- 6 August 2024
- Published by World Scientific Pub Co Pte Ltd in Journal of Algebra and Its Applications
- Vol. 24 (13n14)
Invariant rings of the special orthogonal group have nonunimodal h-vectors
For [ ] an infinite field of characteristic other than two, consider the action of the special orthogonal group [ ] on a polynomial ring via copies of the regular representation. When [ ] has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when [ ] has positive characteristic, the invariant ring is [ ]-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for [ ] as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the [ ]-invariant and information on the Hilbert series. Indeed, we use this to show that the [ ]-vector of the invariant ring for [ ] need not be unimodal.
- 6 August 2024
- Published by World Scientific Pub Co Pte Ltd in Journal of Algebra and Its Applications
- Vol. 24 (13n14)
Invariant rings of the special orthogonal group have nonunimodal h-vectors
For [ ] an infinite field of characteristic other than two, consider the action of the special orthogonal group [ ] on a polynomial ring via copies of the regular representation. When [ ] has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when [ ] has positive characteristic, the invariant ring is [ ]-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for [ ] as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the [ ]-invariant and information on the Hilbert series. Indeed, we use this to show that the [ ]-vector of the invariant ring for [ ] need not be unimodal.
- Open Access
Local cohomology of modular invariant rings
For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We study how the local cohomology module $H^n_{\mathfrak{n}}(R^G)$ compares with $H^n_{\mathfrak{n}}(R)^G$. Various results on the $a$-invariant and on the Hilbert series of $H^n_\mathfrak{n}(R^G)$ are obtained as a consequence.Open AccessLocal cohomology of modular invariant rings
For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We study how the local cohomology module $H^n_{\mathfrak{n}}(R^G)$ compares with $H^n_{\mathfrak{n}}(R)^G$. Various results on the $a$-invariant and on the Hilbert series of $H^n_\mathfrak{n}(R^G)$ are obtained as a consequence. - 13 March 2024
- Published by Springer Nature in Transformation Groups
- Vol. 31 (1) , 647-663
Local Cohomology of Modular Invariant Rings
- 13 March 2024
- Published by Springer Nature in Transformation Groups
- Vol. 31 (1) , 647-663
Local Cohomology of Modular Invariant Rings
- Open Access
On an example of Nagarajan
K. R. Nagarajan constructed an example of a formal power series ring of dimension two, over a field of characteristic two, with the action of a cyclic group of order two, such that the ring of invariants is not noetherian. We point out how Nagarajan's example readily extends to each positive prime characteristic, and also to a characteristic zero example: There exists a formal power series ring of dimension two, over a field of characteristic zero, with an action of the infinite cyclic group, such that the ring of invariants is not noetherian. Both the positive characteristic and the characteristic zero examples are sharp in multiple ways.
Open AccessOn an example of Nagarajan
K. R. Nagarajan constructed an example of a formal power series ring of dimension two, over a field of characteristic two, with the action of a cyclic group of order two, such that the ring of invariants is not noetherian. We point out how Nagarajan's example readily extends to each positive prime characteristic, and also to a characteristic zero example: There exists a formal power series ring of dimension two, over a field of characteristic zero, with an action of the infinite cyclic group, such that the ring of invariants is not noetherian. Both the positive characteristic and the characteristic zero examples are sharp in multiple ways.
- Open Access
On Segre products, $F$-regularity, and finite Frobenius representation type
We study the behavior of various properties of commutative Noetherian rings under Segre products, with a special focus on properties in positive prime characteristic defined using the Frobenius endomorphism. Specifically, we construct normal graded rings of finite Frobenius representation type that are not Cohen-Macaulay.
Open AccessOn Segre products, $F$-regularity, and finite Frobenius representation type
We study the behavior of various properties of commutative Noetherian rings under Segre products, with a special focus on properties in positive prime characteristic defined using the Frobenius endomorphism. Specifically, we construct normal graded rings of finite Frobenius representation type that are not Cohen-Macaulay.
- 11 July 2023
- Published by Springer Nature in Acta Mathematica Vietnamica
- Vol. 49 (1) , 129-138
On Segre Products, F-regularity, and Finite Frobenius Representation Type
- 11 July 2023
- Published by Springer Nature in Acta Mathematica Vietnamica
- Vol. 49 (1) , 129-138
On Segre Products, F-regularity, and Finite Frobenius Representation Type
- Open Access
When are the natural embeddings of classical invariant rings pure?
Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Pl\"ucker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of $S$, equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is linearly reductive.Open AccessWhen are the natural embeddings of classical invariant rings pure?
Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Pl\"ucker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of $S$, equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is linearly reductive. - Open Access
Frobenius on the cohomology of thickenings
We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.
Open AccessFrobenius on the cohomology of thickenings
We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.
- Open AccessDifferential operators on classical invariant rings do not lift modulo $p$Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with very few exceptions, these hypersurfaces -- and, more generally, classical invariant rings -- do not admit a mod $p^2$ lift of the Frobenius endomorphism.Open AccessDifferential operators on classical invariant rings do not lift modulo $p$Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with very few exceptions, these hypersurfaces -- and, more generally, classical invariant rings -- do not admit a mod $p^2$ lift of the Frobenius endomorphism.