Matthew Mastroeni

Research

I am interested in computational and homological aspects of commutative algebra, especially the structure of free resolutions. Most recently, I have been studying the minimal free resolutions of commutative Koszul algebras.

1. Betti numbers of Koszul algebras defined by four quadrics (with P. Mantero) Submitted.
   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.
2. Quadratic Gorenstein rings and the Koszul property II (with H. Schenck and M. Stillman) Submitted.
   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.
3. Quadratic Gorenstein rings and the Koszul property I (with H. Schenck and M. Stillman) Submitted.
   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.
4. 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.
5. 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.