Research

Code that I have written in Macaulay2 for my research is available on my Github (see here for some example computations).

Papers/Preprints

We describe a collection of open questions that arise from the study of cohomology of line bundles on flag varieties in positive characteristic. This note was prepared for the proceedings of the Open Problems in Algebraic Combinatorics (OPAC) conference.

We classify (up to quasi-isomorphism) the free differential modules whose homology is equal to a given module M by developing a theory for deforming an arbitrary free complex into a differential module and explore applications to related rank conjectures.

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. We provide explicit computations of certain stable sheaf cohomology groups and use these computations to prove an effective characteristic-free vanishing statement for Koszul modules.

We prove a generalized version of the Total Rank Conjecture over non Cohen-Macaulay rings for any algebra over a field (independent of characteristic).

We investigate a connection between Koszul algebras and a generalization of the classical notion of Schur functors.

We give GL-equivariant descriptions of Tor and Ext between a natural set of modules over Veronese subalgebras of a polynomial ring. 

Differential modules are natural generalizations of complexes. In this paper, we study differential modules with complete intersection homology, comparing and contrasting the theory of these differential modules with that of the Koszul complex. We construct a Koszul differential module that directly generalizes the classical Koszul complex and investigate which properties of the Koszul complex can be generalized to this setting.

In this paper we construct a GL-equivariant complex of Schur modules over a ring of positive characteristic that can be used to deduce classical alternating sum identities for Schur polynomials. This complex globalizes to a complex of vector bundles and can also be used to give an explicit construction of an exact sequence predicted by work of Grayson involving Adams operations identities on the algebraic K-theory of a given scheme X. The more general complex gives an explicit construction that reproves the aforementioned Adams operations identities in full generality. 

In this paper we study the extent to which Golodness may be transferred along morphisms of DG-algebras. In particular, we show that if I is a so-called fiber invariant ideal, then Golodness of I is equivalent to Golodness of the initial ideal of I. We use this to transfer Golodness results for monomial ideals to more general classes of ideals. We also prove that any so-called rainbow monomial ideal with linear resolution defines a Golod ring; this result encompasses and generalizes many known Golodness results for classes of monomial ideals. We then combine the techniques developed to give a concise proof that maximal minors of (sparse) generic matrices define Golod rings, independent of characteristic.

In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies--Welker to show that the L-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the ``spanning tree condition'' of Almousa--Floystad--Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called ``restricted powers'' of the graded maximal ideal.

In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a $3$-dimensional regular local ring (or $3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a Golod ring for any proper ideal $I \subset R$. We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade $\geq 4$, there exists an ideal $J \subseteq I$ such that $IJ$ is not Golod. We conclude by showing that if $I$ is any proper ideal in a $3$-dimensional regular local ring and $\mfa \subseteq I$ a complete intersection, then $\mfa I$ is Golod. 

A restricted $d$th power of an ideal $I$ is obtained by restricting the exponent vectors allowed to appear on the ``natural" generating set of $I^d$, for some integer $d$. In this paper, we study homological properties of restricted powers of complete intersections. We construct an explicit minimal free resolution for any restricted power of a complete intersection which generalizes the $L$-complex construction of Buchsbaum and Eisenbud. We use this resolution to compute an explicit basis for the Koszul homology which allows us to deduce that the quotient defined by any restricted $d$th power of a complete intersection is a Golod ring for $d \geq 2$. Finally, using techniques of Miller and Rahmati, we show that the minimal free resolution of the quotient defined by any restricted power of a complete intersection admits the structure of an associative DG-algebra. 

In this paper, we extend constructions and results for the Taylor complex to the generalized Taylor complex constructed by Herzog. We construct an explicit DG-algebra structure on the generalized Taylor complex and extend a result of Katth\"an on quotients of the Taylor complex by DG-ideals. We introduce a generalization of the Scarf complex for families of monomial ideals, and show that this complex is always a direct summand of the minimal free resolution of the sum of these ideals. We also give an example of an ideal where the generalized Scarf complex strictly contains the standard Scarf complex. Moreover, we introduce the notion of quasitransverse monomial ideals, and prove a list of results relating to Golodness, Koszul homology, and other homological properties for such ideals.

In this paper we extend the well-known iterated mapping cone procedure to monomial ideals in strongly Koszul algebras. We study properties of ideals generated by monomials in commutative Koszul algebras and show that the linear strand of ideals generated by linear forms is obtained as a subcomplex of the Priddy complex. In the case of strongly Koszul algebras, this shows that the minimal free resolution of a monomial ideal admitting linear quotients is obtained as an iterated mapping cone, immediately extending results for such ideals in polynomial rings to strongly Koszul algebras. We then consider monomial ideals admitting a so-called regular ordering, a generalization of regular decomposition functions, and show that the comparison maps in the iterated mapping cone construction can be computed explicitly. In particular, this gives a closed form for the minimal free resolution of monomial ideals admitting a regular ordering over strongly Koszul algebras.

In this paper, we study ideals I whose linear strand can be supported on a regular CW complex. We provide a sufficient condition for the linear strand of an arbitrary subideal of I to remain supported on an easily described subcomplex. In particular, we prove that a certain class of rainbow monomial ideals always have linear strand supported on a regular CW complex, including any initial ideal of the ideal of maximal minors of a generic matrix. We also provide a sufficient condition for these ideals to have linear resolution, which is also an equivalence under mild assumptions. We then employ a result of Almousa, Fløystad, and Lohne to apply these results to polarizations of Artinian monomial ideals. We conclude with further questions relating to cellularity of certain classes of squarefree monomial ideals and the relationship between initial ideals of maximal minors and algebra structures on certain resolutions.

In this paper, we construct an explicit linear strand for the initial ideal with respect to any diagonal term order of an arbitrary DFI. In particular, we show that if Δ has no 1-nonfaces, then the Betti numbers of the linear strand of J_Δ and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has at most 2 maximal cliques. More generally, we show that the linear strand of the initial ideal (with respect to <)  of any DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the complex of boxes, introduced by Nagel and Reiner.

In this paper, we consider the iterated trimming complex associated to data yielding a complex of length 3. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the  problem of realizability for Tor-algebras of grade 3 perfect ideals, and show that under mild hypotheses, the process of “trimming” an ideal preserves Tor-algebra class. In particular,  we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes G(r) and H(p, q) for a prescribed set of homological data.

Two ideals I and J are called transverse if I ∩ J = IJ. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always 0. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative multiplicative structures on the minimal free resolution of the quotient defined by products of transverse ideals.

Consider the ideal (x1,…,xn)^d ⊆ k[x1,…,xn], where k is any field. This ideal can be resolved by both the L-complexes of Buchsbaum and Eisenbud, and the Eliahou-Kervaire resolution. Both of these complexes admit the structure of an associative DG algebra, and it is a question of Peeva as to whether these DG structures coincide in general. In this paper, we construct an isomorphism of complexes between the aforementioned complexes that is also an isomorphism of algebras with their respective products, thus giving an affirmative answer to Peeva's question.

Let R = k[x1, . . . , xn] denote the standard graded polynomial ring over a field k. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the generators. We then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using a result of splitting mapping cones by Miller and Rahmati, we construct the minimal free resolutions for all ideals under consideration explicitly and conclude with questions about extra structure on these complexes.

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic nxm matrix indexed by the facets of a simplicial complex. We consider the more general notion of an r-DFI, which is generated by a subset of r-minors of a generic matrix indexed by the facets of a simplicial complex. We define and study so-called lcm-closed and interval DFIs, and show that the minors parametrized by the facets of Δ form a reduced Grobner basis with respect to any diagonal term order in both of these cases. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that lcm-closedness is necessary for being a Grobner basis in some cases. We also give conditions on the maximal cliques of Δ ensuring that lcm-closed and interval DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of r-DFIs, and provide a proof of this conjecture for Cohen-Macaulay interval DFIs.

We build a DG structure on the length 4 big from small construction through characteristic free techniques. More precisely,  we construct a morphism from a Tate-like complex to an acyclic DG-algebra exhibiting Poincare duality. The induced chain homotopies may be suitably modified in order to produce morphisms satisfying a list of properties perfectly encapsulating the needed associativity and Leibniz rule axioms.

We study ideals of arbitrary type with socle minimally generated in degree s and 2s-1 for s>2. We prove that all such ideals arise as iterated trimmings of grade 3 Gorenstein ideals. We use the machinery of trimming complexes to produce a resolution of all such ideals that is minimal in generic cases. We use this to show that there exist rings of arbitrarily large type with Tor algebra class G(r), for any r >1.

We produce a family of complexes called trimming complexes and explore applications. We study how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by subsets of a generating set for an arbitrary ideal I. In particular, explicit Betti tables are computed for an infinite class of determinantal facet ideals.

We study ideals defining type 2 compressed rings with socle minimally generated in degrees s and 2s-1 for s>2. We prove that all such ideals arise as trimmings of grade 3 Gorenstein ideals and construct an explicit resolution. In particular, we give bounds on parameters arising in the Tor-algebra classification and construct explicit ideals attaining all intermediate values for every s. This partially answers a question of realizability of Tor-algebra structures.

We study the behavior of the (Uniform) Auslander Condition under localization. In particular, we construct a ring R satisfying UAC with prime ideal p such that the localization at p does not satisfy AC. Similarly, we study the behavior of isomorphism classes of semidualizing modules under localization. All such examples are constructed as fiber products of rings over their common residue field. We conclude by characterizing all nontrivial CM fiber products of finite CM type.