Related papers: Virtual criterion for generalized Eagon-Northcott …
What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this paper, we invoke the Bernstein--Gelfand--Gelfand…
The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon-Northcott and the Buchsbaum-Rim complexes, which are constructed from a…
Virtual resolutions are homological representations of finitely generated $\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a…
In the appendix of the famous book "Commutative Algebra with a View Towards Algebraic Geometry" one can find an infinite family of complexes indexed by integers. This family includes Eagon-Northcott and Buschsbaum-Rim complexes. The…
Previous examples of self-duality for generalized Eagon-Northcott complexes were given by computing the divisor class group for Hankel determinantal rings. We prove a new case of self-duality of generalized Eagon-Northcott complexes with…
Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a…
We construct two families of free resolutions that resolve the ideals of certain opposite Schubert varieties restricted to the big open cell. We conjecture that these examples have genericity properties translating to structure theorems for…
We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces. In particular, we provide sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length…
Let $Q=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with the standard $N^n$-grading. Let $\phi$ be a morphism of finite free $N^n$-graded $Q$-modules. We translate to this setting several notions and constructions that appear…
For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results…
Let $M$ be a perfect module of projective dimension 3 in a Gorenstein, local or graded ring $R$. We denote by $\FF$ the minimal free resolution of $M$. Using the generic ring associated to the format of $\FF$ we define higher structure…
While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising…
We provide a new characterization of enriched accessible categories by introducing the two new notions of virtual reflectivity and virtual orthogonality as a generalization of the usual reflectivity and orthogonality conditions for locally…
We prove a noetherian criterion for a sequence of modules with linear maps between them. This generalizes a noetherian criterion of Gan and Li for infinite EI categories. We apply our criterion to the linear categories associated to certain…
The main idea of [4] was that structures built from periodic prime ideals have better properties from the usual ones built from invariant ideals; but unable to work with periodic ideals alone, we had to generalise further to a somewhat…
In this paper, we give a sufficient condition which makes the total complex of a cube exact. This can be regarded as a variant of the Buchsbaum-Eisenbud theorem which gives a characterization of what makes a complex of finitely generated…
We give lower and upper bounds on the Buchsbaum-Rim multiplicity of finitely generated torsion-free modules over two-dimensional regular local rings, and conditions for them to attain the bounds. As consequences, we have formulae on the…
The classical "generalized principal ideal theorems" of Macaulay, Eagon-Northcott, and others give sharp bounds on the heights of determinantal ideals in arbitrary rings. But in regular local rings (or graded polynomial rings) these are far…
We describe a software package for constructing minimal free resolutions of GL_n(Q)-equivariant graded modules M over Q[x_1, ..., x_n] such that for all i, the ith syzygy module of M is generated in a single degree. We do so by describing…
Let d1,...,dn be a strictly increasing sequence of integers. Boij and S\"oderberg [arXiv:math/0611081] have conjectured the existence of a graded module M of finite length over any polynomial ring K[x_1,..., x_n], whose minimal free…