Using divisibility relations between the generators of a square-free monomial ideal I, we describe divisibility relations between the generators of the second power I^2. We then employ discrete Morse theory to produce a cellular free resolution of I^2 which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of I^2 when the generators of I satisfy at least one divisibility relation.

A divisibility relation between the generators of a square-free monomial ideal formally encodes the situation when one generator divides the least common multiple of some other generators. The divisibility relations contribute to the deletion of some parts of the Taylor resolution of the ideal, and therefore lead to finding a resolution closer to the minimal one. Motivated by this observation, for a given set 𝒟 of divisibility relations, we study all square-free monomials satisfying the relations in 𝒟. We define a class of square-free monomial ideals called 𝒟-extremal ideals ℰ_𝒟 , and show it is optimal in the sense that it is an ideal satisfying exactly those divisibility relations coming from 𝒟, and no others. We then show that ℰ_𝒟 is extremal in the sense that the resolution and betti numbers of the powers of any square-free monomial ideal satisfying the relations in 𝒟 are bounded by those of the same powers of ℰ_𝒟.

This paper is concerned with finding bounds on betti numbers and describing combinatorially and topologically (minimal) free resolutions of powers of ideals generated by a fixed number q of square-free monomials. Among such ideals, we focus on a specific ideal Eq, which we call extremal, and which has the property that for each r≥1 the betti numbers of Eqr are an upper bound for the betti numbers of Ir for any ideal I generated by q square-free monomials (in any number of variables). We study the Scarf complex of the ideals Eqr and use this simplicial complex to extract information on minimal free resolutions. In particular, we show that Eqr has a minimal free resolution supported on its Scarf complex when q≤4 or when r≤2, and we describe explicitly this complex. For any q and r, we also show that β1(Eqr) is the smallest possible, or in other words equal to the number of edges in the Scarf complex. These results lead to effective bounds on the betti numbers of Ir, with I as above. For example, we obtain that pd(Ir)≤5 for all ideals I generated by 4 square-free monomials and any r≥1.

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.

This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one, by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal.

Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power of any square-free monomial ideal.

Let I be a square-free monomial ideal of projective dimension one. Starting with the Taylor complex on the generators of $$I^r$$ , we use discrete Morse theory to describe a CW complex that supports a minimal free resolution of $$I^r$$ . To do so, we concretely describe the acyclic matching on the faces of the Taylor complex.

We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.

We investigate some general machinery for describing semidualizing modules over generic constructions like ladder determinantal rings with coefficients in a normal domain. We also pose and investigate natural localization questions that arise in the process.

(1) Combinatorics and differential operators, leaders Christine Berkesch, University of Minnesota, and Laura Matusevich, Texas A&M Univesity. (2) Methods in prime characteristic, leaders Karen Smith, University of Michigan, and Emily Witt, University of Kansas. (3) Combinatorial commutative algebra, leaders Sara Faridi, Dalhousie University, and Susan Morey, Texas State University. (4) Rees algebra, leaders Elisa Gorla, University of Neuchatel, Switzerland, and Claudia Polini, Notre Dame University. (5) Finite resolutions and complexes, leaders Claudia Miller, Syracuse University, and Alexandra Seceleanu, University of Nebraska. (6) Tropical commutative algebra, leaders Diane Maclagan, University of Warwick, United Kingdom, and Josephine Yu, Georgia Tech University.

We synthesize the recent work done on conditionally defined Lucas and Fibonacci numbers, tying together various definitions and results generalizing the linear recurrence relation. Allowing for any initial conditions, we determine the generating function and a Binet-like formula for the general sequence, in both the positive and negative directions, as well as relations among various sequence pairs. We also determine conditions for periodicity of these sequences and graph some recurrent figures in Python.

The Hilbert–Kunz multiplicity and F-signature are important invariants for researchers in commutative algebra and algebraic geometry. We provide software, and describe the automation, for the calculations of the two invariants in the case of intersection algebras over polynomial rings.

We continue our study of ladder determinantal rings over a field k from the perspective of semidualizing modules. In particular, given a ladder of variables Y, we show that the associated ladder determinantal ring k[Y]/I2(Y) admits exactly 2n non-isomorphic semidualizing modules where n is determined from the combinatorics of the ladder Y: the number n is essentially the number of non-Gorenstein factors in a certain decomposition of Y. From this, for each n, we show explicitly how to find ladders Y such that k[Y]/I2(Y) admits exactly 2n non-isomorphic semidualizing modules. This is in contrast to our previous work, which demonstrates that large classes of ladders have exactly 2 non-isomorphic semidualizing modules.

We identify all semidualizing modules over certain classes of ladder determinantal rings over a field 𝗄. Specifically, given a ladder of variables Y, we show that the ring 𝗄[Y]/I_t(Y) has only trivial semidualizing modules up to isomorphism in the following cases: (1) Y is a one-sided ladder, and (2) Y is a two-sided ladder with t=2 and no coincidental inside corners.

We investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals one. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number with indices (1,2) of the local ring at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.

We continue the study of intersection algebras [Formula: see text] of two ideals [Formula: see text] in a commutative Noetherian ring [Formula: see text]. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when [Formula: see text] is a polynomial ring over a field and [Formula: see text] are principal monomial ideals. Specifically, we calculate the [Formula: see text]-signature, divisor class group, and Hilbert–Samuel and Hilbert–Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formulæ for the [Formula: see text]-signature and Hilbert–Kunz multiplicity, dependent on families of parameters, are provided.

We discuss the generalization, in higher dimensions, of the tropical semiring, whose two binary operations on the set of real numbers together with infinity are defined to be the minimum and the sum of a pair, respectively. In particular, our objects are closed convex sets, and for any pair, we take the convex hull of their union and their Minkowski sum, respectively, as the binary operations. We consider the semiring in several different cases, determined by a recession cone.

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $\operatorname{Spec}(R)-V(J)$, and the (finite simple) graph $\Gamma_R$ with vertex set $\operatorname{Min}(R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of maximal ideals of $T$ is equal to the number of connected components of $\Gamma_R$. We further investigate this graph by exhibiting a technique for showing that a given graph $G$ can be realized as one of the form $\Gamma_R$.

We investigate torsion elements in the kernel of the map on divisor class groups of excellent local normal domains A and A/I, for an ideal I of finite projective dimension. The motivation for this work is a result of Griffith-Weston which applies when I is principal.

We study H. Dao’s invariant \({\eta_c^R}\) of pairs of modules defined over a complete intersection ring R of codimension c having an isolated singularity. Our main result is that \({\eta_c^R}\) vanishes for all pairs of modules when R is a graded complete intersection ring of codimension c > 1 having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are c-Tor-rigid.