IIT Bombay VCAS

In 1915, Emmy Noether proved that in characteristic 0, the invariant subring of a finite group acting on a polynomial ring is Noetherian. This talk will consider the long journey which this basic result has taken through commutative algebra, and introduce three important themes: reduction to positive characteristic and the use of the Frobenius endomorphism, the topic of uniformity, and the question of where coefficients lie in containment relations. The talk is intended for graduate students who have taken at least one year of commutative algebra, with knowledge of the Cohen Structure theorem, regular rings, and Cohen-Macaulay rings.

In this talk we will discuss a theory for affine threefolds of the form $x^my = F(x, z, t)$ which will yield several necessary and sufficient conditions for the coordinate ring of such a threefold to be a polynomial ring. For instance, we will see that this problem of four variables reduces to the equivalent but simpler two-variable question as to whether F(0, z, t) defines an embedded line in the affine plane. As one immediate consequence, one readily sees the non-triviality of the famous Russell-Koras threefold x^2y+x+z^2+t^3=0 (which was an exciting open problem till the mid 1990s) from the obvious fact that z^2+t^3 is not a coordinate. The theory on the above threefolds connects several central problems on Affine Algebraic Geometry. It links the study of these threefolds with the famous Abhyankar-Moh “Epimorphism Theorem” in characteristic zero and the Segre-Nagata lines in positive characteristic. We will also see a simplified proof of the triviality of most of the Asanuma threefolds (to be defined in the talk) and an affirmative solution to a special case of the Abhyankar-Sathaye Conjecture. Using the theory, we will also give a recipe for constructing infinitely many counterexample to the Zariski Cancellation Problem (ZCP) in positive characteristic. This will give a simplified proof of the speaker's earlier result on the negative solution for the ZCP.

In these two lectures we shall be concerned with some results and questions on the set theoretic complete intersections of ideals in polynomial rings over a field $K$. How many equations are necessary to define a given algebraic set in affine $n$-space over a field set-theoretically? Despite its simple formulation this question is highly non-trivial. The history of this problem and its variants are very interesting. We begin with a historical survey of known results and questions. It seems worthwhile to mention some problems which are still open. Further, we present a proof (assuming the following Theorem A.1) of the following theorem of Cowsik-Nori.

Theorem (Cowsik-Nori, 1978) Every algebraic curve $C$ in the affine $n$-space $A _K^n$, where $K$ is a field of characteristic $p>0$, is a set-theoretic complete intersection.

The following two results were used in the proof of the above theorem:

A.1 Theorem (P. Szpiro, 1979, Mohan Kumar, 1978) Affine algebraic curves which are locally complete intersections are set-theoretic complete intersections.

A.2 Theorem: Let $K$ be a perfect field and that $n\in\mathbb{N}$, $n\geq 2$. Let $a \subseteq R := K[X_{1},\ldots , X_{n}]$ be an ideal such that $B := R/a$ is reduced and $\dim B =1$. Then after a change of variables $R\rightarrow R$, $B$ is integral and birational over $K[X_1, X_2] /(a \cap K[X_1, X_2])$.

In this second lecture, we present a proof of Theorem A.2.

We will discuss a question by M. P. Murthy, which asks whether the minimal number of generators for $I/I^2$, where $I$ is an ideal in a polynomial ring in several variables over a field, is equal to the number of generators for $I$. We will give a proof of this in certain cases and deduce that any local complete intersection curve is a set-theoretic complete intersection. We will give an alternate proof of the latter fact by M. Boratynski. The main purpose of the talk will be to highlight several interesting problems and techniques in the area, accessible to graduate students with some commutative algebra background.

We will give a proof that Hankel determinantal rings are $F$-rational, at least if the characteristic of the residue field is suitably large. This is joint work with Aldo Conca, Maral Mostafazadehfard, and Matteo Varbaro.

In the first lecture of this series Huneke explained how the work on invariants of groups, due to Hilbert and Noether, lead to some of the modern developments in commutative algebra. In my talks I will discuss a different connection between representation theory of groups and commutative algebra. A starting point for this is the work of Jon Carlson, from the 1980s, on 'rank varieties' for modular representations of abelian groups of the form $(\ZZ/p\ZZ)^c,$ where p is some prime number. The group algebra of such an elementary abelian group is a complete intersection ring and Carlson's theory of rank varieties has been extended to apply to all complete intersections. This development was initiated by Avramov and Buchweitz, and is still an area of active research. The aim of my talks is to give an introduction to these ideas, starting with the work of Carlson.

Bezout's theorem states that projective curves of degrees a and b meet in ab points if ''counted properly''. The correct number to count at a point of intersection is the intersection-multiplicity defined in Serre's book ''Local Algebra and Intersection-Multiplicity''. The talk, meant for graduate students, will be an introduction to the subject. The definitions will be looked at from various angles. This will be followed by a report on the progress towards Serre's conjectures.

In this talk we shall introduce the notion of absolute integral closure of a domain and mention some of its basic properties. Along with some other results we shall prove the Newton-Puiseux theorem and the fact that for a powers series ring A of finitely many variables over a field of positive characteristic, the absolute integral closure of A is flat over A.

In the first talk, we will look at some applications of big Cohen-Macaulay algebras. In the second we will give an outline of the proof by Huneke and Lyubeznik that the absolute integral closure of a noetherian local domain that is a homomorphic image of a Gorenstein local ring is a (big) Cohen-Macaulay algebra over it.

In 1965, Alexander Grothendieck introduced the notion of excellent rings, a class of well-behaved rings, which is a frequently used hypothesis in the literature. A well-known result of E. Kunz says that in prime characteristic, any F-finite rings are excellent. He showed the backward implication for Noetherian local rings of characteristic p with F-finite residue fields. Recently, R. Datta and K. E. Smith came up with another criterion when the converse holds. They proved that a reduced, Noetherian ring of characteristic p with F-finite total quotient ring is excellent if and only if it is F-finite. In this talk, we will give a sketch of the proof of this theorem. Datta and Smith also gave examples of non-excellent rings in the framework of discrete valuation rings of a function field of characteristic p. We will see that such examples are abundant and easy to construct.

We discuss the theory of multiplicities and mixed multiplicities of filtrations of m-primary ideals. We show that many classical formulas are true in this setting. We also consider the case of equality in Minkowski's inequality. We give some general theorems characterizing when this condition hold, giving generalizations of classical theorems of Rees, Sharp, Teissier, Katz and others.

A finitely generated module $M$ over a commutative ring $R$ is called reflexive if the natural map from $M$ to $M^{**} = Hom(Hom(M,R), R)$ is an isomorphism. In understanding reflexive modules, the case of dimension one is crucial. If $R$ is Gorenstein, then any maximal Cohen-Macaulay module is reflexive, but in general it is quite hard to understand reflexive modules even over well-studied one-dimensional singularities. In this work, joint with Sarasij Maitra and Prashanth Sridhar, we will address this problem and give some partial answers.

In this talk we will discuss a theory for affine threefolds of the form $x^my = F(x, z, t)$ which will yield several necessary and sufficient conditions for the coordinate ring of such a threefold to be a polynomial ring. For instance, we will see that this problem of four variables reduces to the equivalent but simpler two-variable question as to whether F(0, z, t) defines an embedded line in the affine plane. As one immediate consequence, one readily sees the non-triviality of the famous Russell-Koras threefold x^2y+x+z^2+t^3=0 (which was an exciting open problem till the mid 1990s) from the obvious fact that z^2+t^3 is not a coordinate. The theory on the above threefolds connects several central problems on Affine Algebraic Geometry. It links the study of these threefolds with the famous Abhyankar-Moh “Epimorphism Theorem” in characteristic zero and the Segre-Nagata lines in positive characteristic. We will also see a simplified proof of the triviality of most of the Asanuma threefolds (to be defined in the talk) and an affirmative solution to a special case of the Abhyankar-Sathaye Conjecture. Using the theory, we will also give a recipe for constructing infinitely many counterexample to the Zariski Cancellation Problem (ZCP) in positive characteristic. This will give a simplified proof of the speaker's earlier result on the negative solution for the ZCP.

The normal Hilbert coefficients are important numerical invariants associated with an ideal in an analytically unramified local ring. They play an important role in determining the homological properties of the blow-up algebras. This will be an expository talk on the normal Hilbert coefficients, and its relation with blow-up algebras. We will also discuss recent developments on this topic.

Splittings of Frobenius have been employed to study the singularities and cohomology of rings. In this talk we will employ ideas inspired by this technique to obtain results of symbolic powers of monomial and determinantal ideals.

Lyubeznik numbers are certain Bass numbers of local cohomology modules associated to local rings containing a field. These numerical invariants are known to have many interesting homological, geometric and topological properties and have been an active area of research. In this talk we plan to give a brief overview of these.

This is an expository talk introducing the methods of reducing to characteristic $p$. The main tools and general notions necessary to reduce a problem to characteristic $p$ will be discussed in this talk. It is based on chapter 2 of the excellent resource "Tight Closures in Characterisitic zero" by Hochster and Huneke. We will be restricting ourselves to the case of affine algebras since it is more accessible.

We shall discuss Hilbert-Kunz density function of a Noetherian standard graded ring over a perfect field of characteristic $p \geq 0$. We will also talk about the second coefficient of the Hilbert-Kunz function and the possibility of existence of a $\beta$-density function for this coefficient.

Watanabe and Eto have shown that Hilbert-Kunz multiplicity of affine monoid rings with respect to a monomial ideal of finite colength can be expressed as relative volume of certain nice set arising from the convex geometry associated to the ring. In this talk, we shall discuss similar expression for the density functions of polytopal monoid algebra with respect to the homogeneous maximal ideal in terms of the associated convex geometric structure. This is a joint work with Prof. V. Trivedi. We shall also discuss the existence of $\beta$-density function for monomial prime ideals of height one of these rings in this context.

We discuss Eisenbud operators over a complete intersection. As an application, we prove that if $A$ is a strict complete intersection of positive dimension and if $M$ is a maximal CM $A$-module with bounded betti numbers, then the Hilbert function of $M$ is non-decreasing.

These two talks are about the following theorem: If $I$ is an ideal of finite projective dimension in a ring $R$, and the conormal module $I/I^2$ has finite projective dimension over $R/I$, then $I$ is locally generated by a regular sequence. This was conjectured by Vasconcelos, after he and (separately) Ferrand established the case that the conormal module is projective.

The key tool is the homotopy Lie algebra, an object sitting at the centre of a bridge between commutative algebra and rational homotopy theory. In the first part I will explain what the homotopy Lie algebra is, and how it can be constructed by differential graded algebra techniques, following the work of Avramov. In the second part I will bring all of the ingredients together and, hopefully, present the proof of Vasconcelos' conjecture.

F-singularities are singularities in positive characteristic defined using the Frobenius map and there are four basic classes of F-singularities: F-regular, F-pure, F-rational and F-injective singularities. They conjecturally correspond via reduction modulo $p$ to singularities appearing in complex birational geometry. In the first talk, I will survey basic properties of F-singularities. In the second talk, I will explain what is known and what is not known about the correspondence of F-singularities and singularities in birational geometry. If the time permits, I will also discuss its geometric applications.

Given an arbitrary point on a Schubert (sub)variety in a Grassmannian, how to compute the Hilbert function (and, in particular, the multiplicity) of the local ring at that point? A solution to this problem based on "standard monomial theory" was conjectured by Kreiman-Lakshmibai circa 2000 and the conjecture was proved about a year or two later by them and independently also by Kodiyalam and the speaker. The two talks will be an exposition of this material aimed at non-experts in the sense that we will not presume familiarity with Grassmannians (let alone flag varieties) or Schubert varieties.

There are two steps to the solution. The first translates the problem from geometry to algebra and in turn to combinatorics. The second is a solution of the resulting combinatorial problem, which involves establishing a bijection between two combinatorially defined sets. The two talks will roughly deal with these two steps respectively.

Three aspects of the combinatorial formulation of the problem (and its solution) are noteworthy: (A) it shows that the natural determinantal generators of the tangent cone (at the given point) form a Groebner basis (in any "anti-diagonal" term order); (B) it leads to an interpretation of the multiplicity as counting certain non-intersecting lattice paths; and (C) as was observed by Kreiman some years later, the combinatorial bijection is a kind of Robinson-Schensted-Knuth correspondence, which he calls the "bounded RSK".

Determinantal varieties arise as tangent cones of Schubert varieties (in the Grassmannian), and thus one recovers multiplicity formulas for these obtained earlier by Abhyankar and Herzog-Trung. (The multiplicity part of the Kreiman-Lakshmibai conjecture was also proved by Krattenthaler, but by very different methods.)

What about Schubert varieties in other (full or partial) flag varieties (G/Q with Q being a parabolic subgroup of a reductive algebraic group G)? The problem remains open in general, even for the case of the full flag variety GL(n)/B, although there are several papers over the last two decades by various authors using various methods that solve the problem in various special cases. Time permitting, we will give some indication of these results, without however any attempt at comprehensiveness.

We begin by discussing $K_0$ and defining $K_1$ for a ring $R$ and the exact sequence connecting them on localization with respect to a multiplicative set $S$. More generally, there is a similar localization exact sequence for an open set $V(I)^c$ of Spec(R) connecting $K_0$ and $K_1$, and we relate the properties of the ideal $I$ with the intermediate term in the sequence.

The theory of local cohomology has been developed significantly during the six decades of research after its introduction by Grothendieck. The study of the finiteness properties of local cohomology modules initiated, in 1962, with a question asked by Grothendieck in his algebraic geometry seminar.

In this survey talk, first, we will recall the basic definitions and properties in the theory of local cohomology. Then, we shall list some major problems on the finiteness properties of local cohomology modules. Next, we will focus on the notion of cofiniteness. We shall continue by examining some generalizations of local cohomology modules, and the reformulation of finiteness properties for them.

To study finiteness properties for the widest generalization of local cohomology modules, if time allows, at the end of the second session, we will review the derived category approach to local cohomology.

In these two talks I take a pedagogic approach to Quillen $K$-theory. What it takes to teach (and learn) Quillen $K$-theory? I am at the tail end of completing a book on this, which would eventually be available through some outlet. This is based on a course I taught. Current version has nearly 400 pages, in eleven chapters. I finish with Swan’s paper on quadrics. I tried to do it in a reader friendly way, and tried to avoid expressions like “left to the readers”. I would give an overview and a road map.

To justify the title, let me remind you that $K$-theory used to be part of Commutative algebra. In this endeavor, I consolidate the background needed, in about 100 pages, for a commutative algebraist to pick up the book and give a course, or learn. There is a huge research potential in this direction. This is because, with it, topologists have done what they are good at. However, these higher $K$-groups have not been descried in a tangible manner. That would be the job of commutative algebraist, and would require such expertise.

In these two talks I take a pedagogic approach to Quillen $K$-theory. What it takes to teach (and learn) Quillen $K$-theory? I am at the tail end of completing a book on this, which would eventually be available through some outlet. This is based on a course I taught. Current version has nearly 400 pages, in eleven chapters. I finish with Swan’s paper on quadrics. I tried to do it in a reader friendly way, and tried to avoid expressions like “left to the readers”. I would give an overview and a road map.

To justify the title, let me remind you that $K$-theory used to be part of Commutative algebra. In this endeavor, I consolidate the background needed, in about 100 pages, for a commutative algebraist to pick up the book and give a course, or learn. There is a huge research potential in this direction. This is because, with it, topologists have done what they are good at. However, these higher $K$-groups have not been descried in a tangible manner. That would be the job of commutative algebraist, and would require such expertise.

Let $I$ be a monomial ideal. Even though there may not exist any proper reduction of $I$ which is monomial (or even homogeneous), the intersection of all reductions, the core, is again a monomial ideal. The integral closure and the adjoint of a monomial ideal are again monomial ideals and can be described in terms of the Newton polyhedron of $I$. Such a description cannot exist for the core, since the Newton polyhedron only recovers the integral closure of the ideal, whereas the core may change when passing from $I$ to its integral closure. When attempting to derive any kind of combinatorial description for the core of a monomial ideal from the known colon formulas, one faces the problem that the colon formula involves non-monomial ideals, unless $I$ has a reduction $J$ generated by a monomial regular sequence. Instead, in joint work with Ulrich and Vitulli, we exploit the existence of such non-monomial reductions to devise an interpretation of the core in terms of monomial operations. This algorithm provides a new interpretation of the core as the largest monomial ideal contained in a general locally minimal reduction of $I$. In recent joint work with Fouli, Montano, and Ulrich, we extend this formula to a large class of monomial ideals and we study the core of lex-segment monomial ideals generated in one-degree.

One of the hardest problems that come up in affine algebraic geometry is to decide whether a certain d-dimensional factorial affine domain is ``trivial'', i.e., isomorphic to the polynomial ring in d variables. There are instances when the ring of invariants of a suitably chosen G_a-action has been able to distinguish between two rings (i.e., to prove they are non-isomorphic), when all other known invariants failed to make the distinction. It was using one such invariant that Makar-Limanov proved the non-triviality of the Russell-Koras threefold, leading to the solution of the Linearization Problem; and again, it was using an invariant of G_a-actions that Neena Gupta proved the nontriviality of a large class of Asanuma threefolds leading to her solution of the Zariski Cancellation Problem in positive characteristic.

G_a actions are also involved in the algebraic characterisation of the affine plane by M. Miyanishi and the algebraic characterisation of the affine 3-space.by Nikhilesh Dasgupta and Neena Gupta. Miyanishi's characterisation had led to the solution of Zariski's Cancellation Problem for the affine plane. Using G_a-actions, a simple algebraic proof for this cancellation theorem was obtained three decades later by Makar-Limanov.

In this talk (in two parts), we will discuss the concept of G_a-actions along with the above applications, and the closely related theme of Invariant Theory. The concept of G_a-action can be reformulated in the convenient ring-theoretic language of ``locally nilpotent derivation'' (in characteristic zero) and ``exponential map'' (in arbitrary characteristic). The ring of invariants of a G_a- action corresponds to the kernel of the corresponding locally nilpotent derivation (in characteristic zero) and the ring of invariants of an exponential map. We will recall these concepts. We will also mention a theorem on G_a actions on affine spaces (or polynomial rings) due to C.S. Seshadri.

We will also discuss the close alignment of the kernel of a locally nilpotent derivation on a polynomial ring over a field of characteristic zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem had its genesis in a problem on Invariant Theory, Hilbert's fourteenth problem seeks a further generalisation: Zariski generalises it still further. The connection with locally nilpotent derivations has helped construct some low-dimensional counterexamples to Hilbert's problem. We will also mention an open problem about the kernel of a locally nilpotent derivation on the polynomial ring in four variables; and some partial results on it due to Daigle-Freudenburg, Bhatwadekar-Daigle, Bhatwadekar-Gupta-Lokhande and Dasgupta-Gupta. Finally, we will state a few technical results on the ring of invariants of a G_a action on the polynomial ring over a Noetherian normal domain, obtained by Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dutta-Gupta.

Let $(R, m)$ be a Noetherian local ring of dimension $d > 0$ and $M$ be a finitely generated $R$-module of finite length. Suppose char R = $p > 0$ and $d = 1.$ De Stefani, Huneke and Núñez-Betancourt explored the question: what vanishing conditions on the Frobenius Betti numbers force projective dimension of $M$ to be finite. In this talk we will discuss the question for $d ≥ 1.$ This is joint work with Ian Aberbach.

Let $R$ be a commutative noetherian ring. Let $M$ and $N$ be finitely generated $R$-modules. When can we get $M$ from $N$ by taking direct summands, extensions and syzygies? This question is closely related to classification of resolving subcategories of finitely generated $R$-modules. In this talk, I will explain what I have got so far on this topic.

Let $(R, m, k)$ be a (Noetherian) local ring of positive prime characteristic $p.$ Assume also, for simplicity, that $R$ is complete (or, more generally, excellent). In such rings we have the notion of tight closure of an ideal, defined by Hochster and Huneke, using the Frobenius endomorphism. The tight closure of an ideal sits between the ideal itself and its integral closure. When the tight closure of an ideal $I$ is $I$ itself we call $I$ tightly closed. For particularly nice rings, e.g., regular rings, every ideal is tightly closed. We call such rings weakly $F$-regular. Unfortunately, tight closure is known not to commute with localization, and hence this property of being weakly $F$-regular is not known to localize. To deal with this problem, Hochster and Huneke defined the notion of strongly $F$-regular (assuming $R$ is $F$-finite), which does localize, and implies that $R$ is weakly $F$-regular. It is still an open question whether or not the two notions are equivalent, although it has been shown in some classes of rings. Not much progress has been made in the last 15-20 years. I will discuss the problem itself, the cases that are known, and also outline recent progress made by myself and Thomas Polstra.

I will present some results, old and new, about the algebraic objects that are naturally associated with a finite set of subspaces of a given vector space.

Zariski conjectured that if the module of derivations of a local ring $R$ at a point on an algebraic variety defined over a field of chararacteristic $0$ is a free $R$-module then $R$ is regular. In these two talks we will survey most of the interesting results proved affirming the conjecture.

Results of Lipman, Scheja-Storch, Becker, Hochster, Steenbrink-van Straten, Flenner, Kallstrom, Biswas-Gurjar-Kolte, and some general results which can be deduced by combining some of these results will be discussed. An interesting proposed counterexample due to Hochster will be introduced. Some unsolved cases in the paper of Biswas-Gurjar-Kolte will be mentioned.

The Jordan type of a pair (A,x), where x is in the maximum ideal of a standard graded Artinian algebra A, is the partition P giving the Jordan block decomposition of the multiplication map by x on A. When A is Artinian Gorenstein, we say that (A,x) is weak Lefschetz if the number of parts in the Jordan type P_x is the Sperner number of A – the highest value of the Hilbert function H(A). We say that (A,x) is strong Lefschetz if P_x is the conjugate of the Hilbert function.

Weak and strong Lefschetz properties of A for a generic choice of x have been studied, due to the connection with topology and geometry, where A is the cohomology ring of a topological space or a variety X. We discuss some of the properties of Jordan type, and its use as an invariant of A, its behavior for tensor products and free extensions (defined by T. Harima and J. Watanabe).

If there is time, we will discuss an application to the study of local Artinian Gorenstein algebras of fixed Hilbert function H; in recent work with Pedro Macias Marques we show that in codimension three the properties of Jordan type and of symmetric decompositions show that certain families Gor(H) in codimension three or greater have several irreducible components. The first part of the talk is based on work with Chris McDaniel and Pedro Marques (arXiv:math.AC/1802.07383, to appear JCA).