arXiv subjects

Keller VandeBogert

Publications and source records attributed to Keller VandeBogert.

35 records · Page 2Linked to original sources

arXiv2022

DG Structure on Length 3 Trimming Complexes and Applications to Tor Algebras

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.

math.AC↗

arXiv2022

Iterated Mapping Cones for Strongly Koszul Algebras

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.

math.AC↗

arXiv2022

Products of Ideals and Golod Rings

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.

math.AC↗

arXiv2022

Polarizations and Hook Partitions

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--Fløystad--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.

math.AC↗

arXiv2022

Minimal Free Resolutions of Certain Equigenerated Monomial Ideals

Let $R = k[x_1, \dotsc , x_n]$ 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 on 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.

math.AC↗

arXiv2022

Linear Strands of Initial Ideals of Determinantal Facet Ideals

A determinantal facet ideal (DFI) is an ideal $J_Δ$ generated by maximal minors of a generic matrix parametrized by an associated simplicial complex $Δ$. 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 \emph{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 \emph{any} DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the \emph{complex of boxes}, introduced by Nagel and Reiner.

math.AC↗

arXiv2022

DG Structure on the Length 4 Big From Small Construction

The big from small construction was introduced by Kustin and Miller and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced Kustin for building matrix factorizations and construct a DG-algebra structure on the length $4$ big from small construction. The techniques employed involve the construction of a morphism from a Tate-like complex to an acyclic DG-algebra exhibiting Poincaré duality. This induces homomorphisms which, after suitable modifications, satisfy a list of identities that end up perfectly encapsulating the required associativity and DG axioms of the desired product structure for the big from small construction.

math.AC↗

arXiv2021

On Restricted Powers of Complete Intersections

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.

math.AC↗

arXiv2021

On Constructions Related to the Generalized Taylor Complex

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 Katthän 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.

math.AC↗

arXiv2021

Vanishing of Avramov Obstructions for Products of Sequentially Transverse Ideals

Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. 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.

math.AC↗

arXiv2021

Linear Strands Supported on Regular CW Complexes

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.

math.AC↗

arXiv2021

Resolution and Tor Algebra Structures of Grade 3 Ideals Defining Compressed Rings

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and $\ell \geq 1$ are integers. The case for $\ell =1$ was studied in a previous paper by the author; a generically minimal resolution was constructed for all such ideals. More recently, this resolution is generalized in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals $I$ such that $R/I$ is a ring of Tor algebra class $G (r)$ for some fixed $r \geq2$, and $R/I$ may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman.

math.AC↗

arXiv2020

Trimming Complexes and Applications to Resolutions of Determinantal Facet Ideals

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, we compute the Betti table for removing an arbitrary generator from the ideal of submaximal pfaffians of a generic skew symmetric matrix $M$. We also explicitly compute the Betti table for the ideal generated by certain subsets of the generating set of the ideal of maximal minors of a generic $n \times m$ matrix. Such ideals are a subset of a class of ideals called determinantal facet ideals, whose higher degree Betti numbers had not previously been computed.

math.AC↗

arXiv2020

Structure Theory for a Class of Grade 3 Homogeneous Ideals Defining Type 2 Compressed Rings

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman. We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we can give bounds on the minimal number of generators $μ(I)$ of $I$ depending only on $s$; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all $s\geq 3$. Finally, we study the Tor-algebra structure of $R/I$. It is shown that these rings have Tor algebra class $G(r)$ for $s \leq r \leq 2s-1$. Furthermore, we produce ideals $I$ for all $s \geq 3$ and all $r$ with $s \leq r \leq 2s-1$ such that $\textrm{Soc} (R/I ) = k(-s) \oplus k(-2s+1)$ and $R/I$ has Tor-algebra class $G(r)$, partially answering a question of realizability posed by Avramov.

math.AC↗

arXiv2018

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen-Macaulay fiber products of finite Cohen-Macaulay type.

math.AC↗

arXiv2018

On Gorenstein fiber products and applications

We show that a non-trivial fiber product $S\times_k T$ of commutative noetherian local rings $S,T$ with a common residue field $k$ is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both $S$ and $T$ are regular rings of dimension 1. We also give some applications of this result.

math.AC↗

arXiv2016

Random Polygon to Ellipse: A Generalization

This paper generalizes the result of Elmachtoub et al to any weighted barycenter, where a transformation is considered which takes an arbitrary point of division $ξ\in (0,1)$ of the segments of a polygon with $n$ vertices. We then consider connecting these new points to form another polygon, and iterate this process. After considering properties of our generalized transformation matrix, a surprisingly elegant interplay of elementary complex analysis and linear algebra is used to find a closed form for our iterative process. We then specify the new limiting ellipse, $\mathcal{E}$, which has oscillating semi-axes. Along the way we find that the case for $ξ= 1/2$ enjoys some special optimality conditions, and periodicity of the ellipse $\mathcal{E}$ is analyzed as well. To conclude, an even more generalized case is considered: taking a different point of division for every segment of our polygon $\mathcal{P} (\vec{x}^{(0)}, \vec{y}^{(0)})$.

math.MG↗