92 citations · 473 across the 45 of their papers we have counts for
28 papers · 1 filter
Positivity and nonstandard graded Betti numbers
Michael K. Brown, Daniel Erman
A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of th…
Linear syzygies of curves in weighted projective space
Michael K. Brown, Daniel Erman
We develop analogues of Green's -conditions for subvarieties of weighted projective space, and we prove that such -conditions are satisfied for high degree embeddings of…
Linear strands of multigraded free resolutions
Michael K. Brown, Daniel Erman
We develop a notion of linear strands for multigraded free resolutions, and we prove a multigraded generalization of Green's Linear Syzygy Theorem.
Matrix factorizations of generic polynomials
Daniel Erman
We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduc…
Syzygies of : data and conjectures
Juliette Bruce, Daniel Corey, Daniel Erman +3
We provide a number of new conjectures and questions concerning the syzygies of . The conjectures are based on computing the graded Betti tables an…
Minimal free resolutions of differential modules
Michael K. Brown, Daniel Erman
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward st…
Characteristic dependence of syzygies of random monomial ideals
Caitlyn Booms, Daniel Erman, Jay Yang
When do syzygies depend on the characteristic of the field? Even for well-studied families of examples, very little is known. For a family of random monomial ideals, namely the Sta…
Small projective spaces and Stillman uniformity for sheaves
Daniel Erman, Steven V Sam, Andrew Snowden
We prove an analogue of Ananyan--Hochster's small subalgebra theorem in the context of sheaves on projective space, and deduce from this a version of Stillman's Conjecture for coho…
The Schur-Veronese package in Macaulay2
Juliette Bruce, Daniel Erman, Steve Goldstein +1
This note introduces the Macaulay2 package SchurVeronese, which gathers together data about Veronese syzygies and makes it readily accessible in Macaulay2. In addition to standard…
Interpolation over ZZ and torsion in class groups
John Berman, Daniel Erman
We prove an interpolation result for homogeneous polynomials over the integers, or more generally for PIDs with finite residue fields. Previous proofs of this result use the well-k…
Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
Daniel Erman, Steven V Sam, Andrew Snowden
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a mo…
Big polynomial rings with imperfect coefficient fields
Daniel Erman, Steven V Sam, Andrew Snowden
We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this…
Generalizations of Stillman's conjecture via twisted commutative algebras
Daniel Erman, Steven V Sam, Andrew Snowden
Combining recent results on noetherianity of twisted commutative algebras by Draisma and the resolution of Stillman's conjecture by Ananyan-Hochster, we prove a broad generalizatio…
Big polynomial rings and Stillman's conjecture
Daniel Erman, Steven V Sam, Andrew Snowden
The purpose of this paper is to prove that certain limits of polynomial rings are themselves polynomial rings, and show how this observation can be used to deduce some interesting…
Random Flag Complexes and Asymptotic Syzygies
Daniel Erman, Jay Yang
We use the probabilistic method to construct examples of conjectured phenomenon about asymptotic syzygies. In particular, we use the Stanley-Reisner ideals of random flag complexes…
Virtual Resolutions for a Product of Projective Spaces
Christine Berkesch, Daniel Erman, Gregory G. Smith
Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth proj…
Questions about Boij-Söderberg theory
Daniel Erman, Steven V Sam
Boij-Söderberg theory focuses on the properties and duality relationship between two types of numerical invariants. One side involves the Betti table of a graded free resolution ov…
A probabilistic approach to systems of parameters and Noether normalization
Juliette Bruce, Daniel Erman
We study systems of parameters over finite fields from a probabilistic perspective, and use this to give the first effective Noether normalization result over a finite field. Our c…
Divergent Series and Serre's intersection formula for graded rings
Daniel Erman
On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corres…
Three flavors of extremal Betti tables
Christine Berkesch, Daniel Erman, Manoj Kummini
We discuss extremal Betti tables of resolutions in three different contexts. We begin over the graded polynomial ring, where extremal Betti tables correspond to pure resolutions. W…
Categorified duality in Boij-Söderberg Theory and invariants of free complexes
David Eisenbud, Daniel Erman
We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Söderberg conjectures describing numerical invariants of syz…
The cone of Betti diagrams over a hypersurface ring of low embedding dimension
Christine Berkesch, Jesse Burke, Daniel Erman +1
We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a fi…
Shapes of free resolutions over a local ring
Christine Berkesch, Daniel Erman, Manoj Kummini +1
We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly…
Tensor complexes: Multilinear free resolutions constructed from higher tensors
Christine Berkesch Zamaere, Daniel Erman, Manoj Kummini +1
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 th…