Papers

Listed in (reverse chronological) order of first upload to arXiv (recentmost first).
My arXiv author page

9. Abelian extensions of equicharacteristic regular rings need not be Cohen-Macaulay
(arXiv) (PDF)
with Anurag K. Singh and Prashanth Sridhar

Abstract

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.

8. Homological properties of invariant rings of permutation groups
(arXiv) (PDF)

Abstract

Consider the action of a subgroup $G$ of the permutation group on the polynomial ring $S := k[x_{1}, \ldots, x_{n}]$ via permutations. We show that if $k$ does not have characteristic two, then the following are independent of $k$: the $a$-invariant of $S^{G}$, the property of $S^{G}$ being quasi-Gorenstein, and the Hilbert functions of $H_{\mathfrak{m}}^{n}(S)^{G}$ as well as $H_{\mathfrak{n}}^{n}(S^{G})$; moreover, these Hilbert functions coincide. In particular, being independent of characteristic, they may be computed using characteristic zero techniques, such as Molien's formula. In characteristic two, we show that the ring of invariants is always quasi-Gorenstein and compute the $a$-invariant explicitly, and show that the Hilbert functions of $H_{\mathfrak{m}}^{n}(S)^{G}$ and $H_{\mathfrak{n}}^{n}(S^{G})$ agree up to a shift, given by the number of transpositions. Lastly, we determine when the inclusion $S^{G} \hookrightarrow S$ splits, thereby proving the Shank--Wehlau conjecture for permutation subgroups.

7. The Scarf complex of squarefree powers, symbolic powers of edge ideals, and cover ideals of graphs
(arXiv) (PDF) (Journal)
with Trung Chau and Nursel Erey
Communications in Algebra

Abstract

Every monomial ideal $I$ has a Scarf complex, which is a subcomplex of its minimal free resolution. We say that $I$ is Scarf if its Scarf complex is also its minimal free resolution. In this paper, we fully characterize all pairs $(G,n)$ of a graph $G$ and an integer $n$ such that the squarefree power $I(G)^{[n]}$ or the symbolic power $I(G)^{(n)}$ of the edge ideal $I(G)$ is Scarf. We also determine all graphs $G$ such that its cover ideal $J(G)$ is Scarf, with an explicit description when $G$ is either chordal or bipartite.

6. Edge ideals with linear quotients and without homological linear quotients
(arXiv) (PDF)
with Trung Chau and Kanoy Kumar Das

Abstract

A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear quotients, then $G$ is co-chordal. We construct a family of co-chordal graphs $\{\mathrm{H}_n^c\}_{n\geq 6}$ and propose a conjecture that an edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and $\mathrm{H}_n^c$-free for any $n\geq 6$. In this paper, we prove one direction of the conjecture. Moreover, we study possible patterns of pairs $(G,k)$ of a co-chordal graph $G$ and integer $k$ such that $\mathrm{HS}_k(I(G))$ has linear quotients.

5. Polynomial invariants of classical subgroups of \(\operatorname{GL}_{2}\): Conjugation over finite fields
(arXiv) (PDF)

Abstract

Consider the conjugation action of the general linear group $\operatorname{GL}_{2}(K)$ on the polynomial ring $K[X_{2 \times 2}]$. When $K$ is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when $K$ is a finite field, and show that it is a hypersurface. We also consider the other classical subgroups, and the polynomial rings corresponding to other subspaces of matrices such as the traceless and symmetric matrices. In each case, we show that the invariant ring is either a polynomial ring or a hypersurface.

4. Splitting the difference: Computations of the Reynolds operator in classical invariant theory
(arXiv) (PDF) (Journal)
Bulletin of the London Mathematical Society

Abstract

If $G$ is a linearly reductive group acting rationally on a polynomial ring $S$, then the inclusion $S^{G} \hookrightarrow S$ possesses a unique $G$-equivariant splitting, called the Reynolds operator. We describe algorithms for computing the Reynolds operator for the classical actions as in Weyl's book. The groups are the general linear group, the special linear group, the orthogonal group, and the symplectic group, with their classical representations: direct sums of copies of the standard representation and copies of the dual representation.

3. Monomial ideals with minimal generalized Barile-Macchia resolutions
(arXiv) (PDF) (Journal)
with Trung Chau and Tài Huy Hà
Vietnam Journal of Mathematics

Abstract

We identify several classes of monomial ideals that possess minimal generalized Barile-Macchia resolutions. These classes of ideals include generic monomial ideals, monomial ideals with linear quotients, and edge ideals of hypertrees. We also characterize connected unicyclic graphs whose edge ideals are bridge-friendly and, in particular, have minimal Barile-Macchia resolutions. Barile-Macchia and generalized Barile-Macchia resolutions are cellular resolutions and special types of Morse resolutions.

2. Minimal cellular resolutions of powers of graphs
(arXiv) (PDF)
with Trung Chau and Tài Huy Hà
Electronic Journal of Combinatorics (To appear)

Abstract

Let $G$ be a connected graph and let $I(G)$ denote its edge ideal. We classify when $I(G)^n$, for $n \ge 1$, admits a minimal Lyubeznik resolution. We also give a characterization for when $I(G)^n$ is bridge-friendly, which, in turn, implies that $I(G)^n$ has a minimal Barile-Macchia cellular resolution.

1. Linear quotients of connected ideals of graphs
(arXiv) (PDF) (Journal) (MathSciNet)
with H. Ananthnarayan and Omkar Javadekar
Journal of Algebraic Combinatorics, Volume 61, Issue 34 (2025)

Abstract

As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the connected subsets of size $t$. For chordal graphs, we show that $\operatorname{J}_{t}$ has a linear resolution iff the tree is $t$-gap-free, and that this is equivalent to having linear quotients. We then show that if $G$ is any gap-free and $t$-claw-free graph, then $\operatorname{J}_{t}(G)$ has linear quotients and, hence, linear resolution.