IIT Bombay VCAS

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\R(I)$ of a graded ideal or the symmetric algebra $\Sym(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs. This is a report on joint work with Takayuki Hibi and Somayeh Moradi.

In these expository talks, we will discuss the Rees valuations and Rees valuation rings associated with an ideal in a Noetherian ring, as well as their applications to asymptotic prime divisors and various multiplicities. If time permits, we will describe the Rees valuations associated with a finitely generated torsion-free module.

In this talk we recall the notion of Hilbert-Kunz density function for graded rings. This function was introduced to understand the Hilbert-Kunz multiplicity which is a difficult characteristic $p$ invariant to compute and to make speculation about its properties. It turns out the HK density function is also related to another characteristic $p$-invariant namely $F$-threshold. Here we describe its properties and give its applications to HK multiplicity, $F$-thershold and to a conjecture of Watanabe-Yoshida. The talk is partly based on a joint work with K.I. Watanabe.

The talks will give a survey about multiplicity based criteria for the integral dependence of ideals. This subject has close connections with equisingularity theory and intersection theory, which will be discussed as well. The first numerical criterion for integral dependence was proved in the 1960s by Rees who treated the case of zero-dimensional ideals using the Hilbert-Samuel multiplicity. Criteria for arbitrary ideals require generalized notions of multiplicities. We will discuss various such notions and talk about how they are used to detect integral dependence. The most recent results are from joint work with Claudia Polini, Ngo Viet Trung, and Javid Validashti.

The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this talk we explore the structure of the associated graded ring of stretched $\m$-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring $(A,\m)$. As an application, we present complete descriptions of the associated graded ring of stretched $\m$-primary ideals with small reduction number.

Let $P$ be a property of local rings (such as regular, Gorenstein, or complete). We say that $P$ deforms if a local ring $R$ enjoys property $P$ provided there exists a nonzerodivisor $x$ such that $R/xR$ is $P$. (For example, the properties of being regular or Gorenstein deform, but the property of being complete does not deform). The deformation problem, as it pertains to the prime characteristic singularity classes of $F$-regular, $F$-rational, $F$-pure, and $F$-injective singularities, has a rich history that dates to work of Fedder in the 1980's and remains an active research area. We will survey the history of the deformation problem of these four prime characteristic singularity classes and discuss a recent solution to the deformation of $F$-purity problem in rings which are $\mathbb{Q}$-Gorenstein. This talk is based on a collaboration with Austyn Simpson.

Given their importance in constructing counterexamples to the Eisenbud-Goto Conjecture, it is reasonable to study the algebra and geometry of Rees-like algebras further. Given a graded ideal I of a polynomial ring S, its Rees-like algebra is S[It, t^2], where t is a new variable. Unlike the Rees algebra, whose defining equations are difficult to compute in general, the Rees-like algebra has a concrete minimal generating set in terms of the generators and first syzygies of I. Moreover, the free resolution of this ideal is well understood. While it is clear that the Rees-like algebra of an ideal is never normal and only Cohen-Macaulay if the ideal is principle, I will explain that it is often seminormal, weakly normal, or F-pure. I will also discuss the computation of the singular locus, how the singular locus is affected by homogenization, and the structure of the canonical module, class group, and Picard group. This talk is joint work with Paolo Mantero and Lance E. Miller.

In 1985, Stanley submitted a paper titled "Two Poset Polytopes", which was published in 1986, in which he defined the order and chain polytopes of a finite partially ordered set (poset for short). On the other hand, Hibi presented a notion of an algebra with straightening law (ASL for short) on a finite distributive lattice, which nowadays is called a Hibi ring, in a conference held in Kyoto 1985. This result was published in 1987. It turned out that the Hibi ring on a distributive lattice D is the Ehrhart ring of the order polytope of the poset consisting of join-irreducible elements of D. In the first talk, we recall the definition of Ehrhart rings, order and chain polytopes, and Hibi rings. We recall some basic properties of Ehrhart rings and describe the canonical module of them. Using these facts, we state some basic facts of Hibi rings, i.e., the Ehrhart rings of the order polytopes of posets. We also state some basic facts of the Ehrhart rings of chain polytopes of posets. In the second talk, we focus on the structure of the canonical modules of the Ehrhart rings of order and chain polytopes of a poset. We describe the generators of the canonical modules in terms of the combinatorial structure of the poset and characterize the level property. If time permits, we describe the radical of the trace of the canonical module of these rings and describe the non-Gorenstein locus. This final part is a joint-work with Janet Page.

In the talk we explain a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on $n$. An integer $N_*$ is determined such that the optima of these integer programs are a quasi-linear function of $n$ for all $n\ge N_*$. Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo-Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.

J. Herzog proved in 1969 that the possible values of the first Betti number (minimal number of generators of the defining ideal) of numerical semigroup rings in embedding dimension 3 are 2 (complete intersection and Gorenstein) and 3 (the almost complete intersection). In a conversation about this work, O.Zariski indicated a possible relation between Gorenstein rings and symmetric value semigroups. In response to that, E.Kunz proved (in 1970) that a one-dimensional, local, Noetherian, the reduced ring is Gorenstein if and only if its value semigroup is symmetric. A question that remains open to date is whether the Betti numbers (or at least the first Betti number) of every numerical semigroup ring in embedding dimension e, are bounded above by a function of e.

In the years 1974 and 1975, two interesting classes of examples were given by T. Moh and H. Bresinsky. Moh’s example was that of a family of algebroid space curves and Bresinsky’s examples was about a family of numerical semigroups in embedding dimension 4, with the common feature that there is no upper bound on the Betti numbers. Therefore, for embedding dimension 4 and above, the Betti numbers (or at least the first Betti number) are not bounded above by some “good” function of the embedding dimension e. A question that emerges is the following: Is there a natural way to generate such numerical semigroups in arbitrary embedding dimension? In this talk we will discuss some recent observations in this direction, which is a joint work of the author with his collaborators Joydip Saha and Ranjana Mehta.

Chairperson: Dilip P. Patil, IISc Bangalore, India

Similar to Castelnouvo-Mumford regularity, reduction number is a measure for the complexity of the structure of an algebra. For a projective subscheme $X$, there is a degree upper bound, $\deg(X) \leq \binom{e+r}{r},$ where $e$ is the codimension and $r$ is the reduction number of the homogeneous coordinate ring. In this talk, I discuss the maximal case $\deg(X)=\binom{e+r}{r}$ and the almost maximal case $\deg(X)=\binom{e+r}{r}-1$. In these cases, it is possible to describe explicitly certain initial ideal of the defining ideal of $X$ and consequently one obtains an explicit description of the Betti table. I also discuss how componentwise linearity is helpful for computing the Betti tables of projective varieties with almost maximal degree. This is a joint work with Sijong Kwak.

Chairperson: Le Tuan Hoa, Institute of Mathematics, Hanoi

We will describe an algebraic criterion for two morphisms of the spectrum of a henselian regular local ring into a smooth projective surface to be naively A^1-homotopic. If time permits, we will explain the significance of this criterion to purely algebro-geometric questions related to A^1-connected components. The talk is based on joint work with Chetan Balwe.

Chairperson: Ravi Rao, Narsee Monjee Institute of Management Studies, Mumbai

This talk will report on joint work with Laurent Busé and Navid Nemati. First, the classical situation of the theory of resultants and Sylvester forms in a standard graded algebra will be presented, as it was developed by Jouanolou in a series of monographs. Then we will explain the extension of this theory to the multi-graded case (which corresponds to a product of projective spaces, in place of a single one). This builds on the previous work of two Ph.D. students of Jouanolou (Chaichaa and Chkiriba) and an extension of a duality result from the classical case to this more general setting. We will illustrate these in a very simple case, by providing a family of formulas that extends the work of Dixon.

Chairperson: Juergen Herzog, University of Duisburg-Essen, Germany

In the first part of the talk, we present some elementary new facts about Koszul and \v{C}ech complexes with respect to a single element. We construct free resolutions of the \v{C}ech complex for a system of elements in a commutative ring. This is used in order to construct quasi-isomorphisms between the \v{C}ech complexes and certain Koszul complexes. The free resolution of the \v{Cech} complex is applied in order to find relations to the left derived functors of the completion as a certain Koszul homology. This material provides an elementary introduction to some of the results of the speakers joint work with A.-M. Simon (see "Completion, \v{C}ech and local homology and cohomology. Interactions between them. Springer Monograph, 2018") as well as some further developments. One focus is the study of weakly proregular sequences and of proregular sequences which provides a new insight for the local cohomology as well as the left derived functors of the completion. Finally we shall present an application to prisms in the sense of Bhatt and Scholze.

Chairperson: Le Tuan Hoa, Institute of Mathematics, Hanoi

We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relationship between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra. This is joint work with Jonathan Montaño. Time permitting, we will also speak about some joint work with Fatemeh Mohammadi and Leonid Monin regarding "multi-graded algebras and multi-graded linear series".

In the second talk, I start talking about rough ideas of Andre's proof of the direct summand conjecture. Then I move to basics of almost rings and formulate the almost purity theorem. I also show some guidelines for learning perfectoids by showing some fundamental results on perfectoid geometry. Finally, I talk about my recent contributions (joint with K. Nakazato and S. Ishiro).

This is a joint work with Kazufumi Eto (Nippon Institute of Technology). This work was inspired by the talk of M.E. Rossi (Univ. Genova) at VCAS on Dec. 1, 2020. Let H be a numerical semigroup and K[H] be its semigroup ring over any field K. If $H = (n_1,...,n_e)$, we express $K[H]$ as $K[x_1,...,x_e]/I_H$ and we want to express $K[H]/(t^h)$ by "Inverse polynomials" of Macaulay. We study the defining ideal of a numerical semigroup ring K[H] using the inverse polynomial attached to the Artinian ring $K[H]/(t^h)$ for $h \in H_+.$ I believe this method to express by inverse polynomials is very powerful and can be used for many purposes. At present, we apply this method for the following cases. (1) To give a criterion for H to be symmetric or almost symmetric. (2) Characterization of symmetric numerical semigroups of small multiplicity. (3) A new proof of Bresinsky’s Theorem for symmetric semigroups generated by 4 elements.

We explore spinor coordinates on free resolutions of codimension four Gorenstein ideals. Our analysis of spinor coordinate examples shows notable differences between Gorenstein ideals with 4, 6, 7, and 8 generators compared to those with more generators. Also, we discuss the codimension four Gorenstein ideals resulting from doubling codimension three perfect ideals. We see, in particular, the construction of generic doublings of almost complete intersection perfect ideals of codimension three.

Let R be a Cohen-Macaulay local ring (or positively graded K-algebra) with canonical module \omega_R. The trace of the latter, tr(\omega_R), is by definition, the ideal generated by the images of all R-module homomorphisms from \omega_R into R. Since this ideal describes the non-Gorenstein locus of R, it can be viewed as a way to measure how far is R from being Gorenstein. In terms of this ideal, new classes of rings have been introduced, and their properties are under scrutiny. We discuss some of these approaches, with a special focus on families of examples coming from combinatorics. This talk is based on joint works with J. Herzog, T. Hibi, R. Jafari and S. Kumashiro.

In this talk, we consider a problem that lies in the confluence of two topics.

On one hand, we have maximal Cohen-Macaulay (MCM) modules - these are classical objects that have been studied extensively from algebraic and geometric viewpoints. There is a rich theory of MCM modules over Cohen-Macaulay (CM) rings and many beautiful connections to the singularities of the ring have been discovered. However, in the absence of the CM property in the ring, not as much is known - even the object's existence is largely unclear.

On the other hand, we have a mixed characteristic phenomenon. In 1980, Paul Roberts showed that the integral closure of a regular local ring in an Abelian extension of its quotient field is CM, provided the characteristic of the residue field does not divide the degree of the extension. This fails in the "modular case" in mixed characteristic.

We will look at some past results in the literature before considering the question of existence of MCMs in the modular case of Roberts's theorem.

Seshadri constants of line bundles on projective varieties were defined by J-P. Demailly in 1990, motivated by an ampleness criterion of C. S. Seshadri. They are a measure of local positivity of line bundles and have interesting connections to different areas in mathematics. We will discuss some important questions that drive research on Seshadri constants. We will also discuss their relationship with Waldschmidt constants which are invariants attached to homogeneous ideals in polynomial rings.

In this talk, we shall present some results regarding the relation between the Chow groups and Euler class groups of affine varieties over algebraically closed fields. We shall show how these results allow one to deduce an old conjecture of Murthy. If time permits, we shall discuss some related questions on Chow-Witt groups of affine varieties over non-algebraically closed.

Chairperson: Mrinal Kanti Das, ISI, Kolkata

Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call "tautological bundles (classes)" of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

Using combinatorial structures to obtain resolutions of monomial ideals is an idea that traces back to

Diana Taylor’s thesis, where a simplex associated to the generators of a monomial ideal was used to

construct a free resolution of the ideal. This concept has been expanded over the years, with various

authors determining conditions under which simplicial or cellular complexes can be associated to

monomial ideals in ways that produce a free resolution.  

In a research project initiated at a BIRS workshop “Women in Commutative Algebra” in Fall 2019, the

authors studied simplicial and cellular structures that produced resolutions of powers of monomial

ideals. The optimal structure to use depends upon the structure of the monomial ideal. This talk will

focus on powers of square-free monomial ideals of projective dimension one. Faridi and Hersey proved

that a monomial ideal has projective dimension one if and only if there is an associated tree (one

dimensional acyclic simplicial complex) that supports a free resolution of the ideal. The talk will show

how, for each power $r >1$, to use the tree associated to a square-free monomial ideal $I$ of projective

dimension one to produce a cellular complex that supports a free resolution of $I^r$. Moreover, each of

these resolutions will be minimal resolutions. These cellular resolutions can also be viewed as strands of

the resolution of the Rees algebra of $I$. This talk will contain joint work with Susan Cooper, Sabine El

Khoury, Sara Faridi, Sarah Mayes-Tang, Liana Sega, and Sandra Spiroff.

Chairperson - Takayuki Hibi

In this talk I will describe old and new results on free resolutions of Gorenstein ideals of codimension 4.

In the first part I will discuss situation in codimension 3 and Kustin-Miller results. Then I will describe new ideas related to spinor structures and connection to the exceptional root systems.

Chairperson - Bernd Ulrich

We consider the edge ideals of edge-weighted very well-covered graphs and discuss their unmixed and Cohen-Macaulay properties.

This is based on a joint work with S.A. Seyed Fakhari, K. Shibata and S. Yassemi.

Chairperson - Siamak Yassemi