Matthew Mastroeni

Research

I am interested in computational, combinatorial, and homological aspects of commutative algebra, especially the structure of free resolutions, Koszul algebras, linkage, and Rees algebras.

1. Depth and singular varieties of exterior edge ideals
   (with J. McCullough, A. Osborne, J. Rice, and C. Willis) Submitted.
   The homological properties of edge ideals of graphs are well-studied over polynomial rings. In this paper, we study analogously defined edge ideals over exterior algebras, focusing on the depth and singular varieties of such ideals. We prove upper bounds on the depth of edge ideals and explicitly compute the depths of various families of graphs including cycles, complete multipartite graphs, spider graphs, and whiskered graphs.
2. Chow rings of matroids are Koszul
   (with J. McCullough) To appear in Math. Ann.
   Chow rings of matroids were instrumental in the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow ring of a matroid is a commutative, graded, Artinian, Gorenstein algebra with linear and quadratic relations defined by the matroid. We establish a conjecture of Dotsenko that the Chow ring of any matroid is also Koszul.
3. Computing generalized Frobenius powers of monomial ideals
   (with C. Francisco, J. Mermin, and J. Schweig) Submitted.
   Generalized Frobenius powers of an ideal were introduced in work of Hernández, Teixeira, and Witt as characteristic-dependent analogs of test ideals. However, little is known about the Frobenius powers and critical exponents of specific ideals, even in the monomial case. We describe an algorithm to compute the critical exponents of monomial ideals and use this algorithm to prove some results about their Frobenius powers and critical exponents.
4. The InvariantRing package for Macaulay2
   (with L. Ferraro, F. Galetto, F. Gandini, H. Huang, and X. Ni) Submitted.
   A significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups as well as invariants of arbitrary linearly reductive group actions.
5. The structure of Koszul algebras defined by four quadrics
   (with P. Mantero) J. Algebra 601 (2022), 280–311.
   Building on previous work, we prove structure theorems for the defining ideals of all Koszul algebras defined by height two ideals of four quadrics over an algebraically closed field. As a result, the structure of all Koszul algebras defined by at most four quadrics is well-understood, and we give applications to the Backelin-Roos property.
6. Quadratic Gorenstein rings and the Koszul property II
   (with H. Schenck and M. Stillman) Int. Math. Res. Not. IMRN 2023, no. 2, 1461–1482.
   With the aim of answering the question for which codimensions \(c\) and regularities \(r\) is every quadratic Gorenstein ring \(R\) with \(\codim R = c\) and \(\reg R = r\) Koszul, we prove that quadratic Gorenstein rings with \(c = r + 1\) are always Koszul, and for all \(c \geq r + 2 \geq 6\), we construct quadratic Gorenstein rings that are not Koszul.
7. Betti numbers of Koszul algebras defined by four quadrics
   (with P. Mantero) J. Pure Appl. Algebra 225 (2021), no. 2, Paper No. 106504.
   We give an affirmative answer to a question of Avramov, Conca, and Iyengar about the Betti numbers of Koszul algebras for all Koszul algebras defined by 4 quadrics. In the process, we completely describe the Betti tables of Koszul algebras of codimension 2 defined by 4 quadrics and prove a structure theorem for those of multiplicity 2.
8. Quadratic Gorenstein rings and the Koszul property I
   (with H. Schenck and M. Stillman) Trans. Amer. Math. Soc. 374 (2021), no. 2, 1077-1093.
   We negatively answer a question of Conca, Rossi, and Valla about whether every quadratic Gorenstein ring of regularity 3 is Koszul by using idealization to construct non-Koszul quadratic Gorenstein rings of regularity 3 for every codimension greater than or equal to 9.
9. Koszul almost complete intersections
   J. Algebra 501 (2018), 285-302.
   We prove a structure theorem for the defining ideals of Koszul almost complete intersections and, in the process, give an affirmative answer for all such rings to a question of Avramov, Conca, and Iyengar about the Betti numbers of Koszul algebras.
10. Matrix factorizations and singularity categories in codimension two
    Proc. Amer. Math. Soc. 146 (2018), no. 11, 4605–4617.
    We show how to functorially connect the Eisenbud-Peeva matrix factorizations of a complete intersection of codimension two to its singularity category by way of the graded matrix factorizations of Burke and Walker.

Software

1. LatticeChowRings
   A Macaulay2 package for working with the Feichtner-Yuzvinsky Chow rings of atomic lattices with respect to a given building set, including the augmented Chow rings and graded Möbius algebras of matroids.
2. InvariantRing
   (with L. Ferraro, F. Galetto, F. Gandini, T. Hawes, H. Huang, and X. Ni)
   A Macaulay2 package for computing invariants of group actions on polynomial rings. The package includes a variety of methods for computing invariants of finite groups, diagonal actions of tori and finite abelian groups, and actions of linearly reductive groups.