Commutative Algebra Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Feb. 22, 2016
Gennady Lyubeznik :
2 p.m. in SEO 1227
Abstract
Holonomic D-modules play an important role in commutative algebra. Their definition and proofs of their elementary properties are quite simple over polynomial rings but considerably more complicated over formal power series. In this talk I will present the very recent work of my student Peyman Gharemani which considerably simplifies the proofs of some important aspects of the theory in the formal power series case.
April 4, 2016
Nicholas Switala :
2 p.m. in SEO 1227
Abstract
Let $R$ be a formal power series ring over a field $k$ of characteristic zero, and let $D$ be the ring of $k$-linear differential operators on $R$. By interpreting the Matlis dual of an $R$-module $M$ in terms of certain $k$-linear maps from $M$ to $k$ (following SGA2), we show that given any left $D$-module, there is a natural structure of left $D$-module on its Matlis dual, whose de Rham cohomology is $k$-dual to that of the original module in the holonomic case. We give an application to Hartshorne's theory of de Rham homology and cohomology for complete local rings, showing that the associated Hodge-de Rham spectral sequences (beginning with their $E_2$ terms) are independent of the embedding used in their definition and consist of finite-dimensional $k$-spaces.
April 11, 2016
Mordechai Katzman :
2 p.m. in SEO 1227
Abstract
In this talk I will briefly review the connection between matrix factorizations and maximal Cohen-Macaulay modules and apply these to the study of hypersurfaces in prime characteristic twisted via Frobenius. We will then apply these to specific hypersurfaces. This is joint work with Khaled Alhazmy.
Aug. 29, 2016
- :
2 p.m. in SEO 427
Sept. 12, 2016
No Seminar :
2 p.m. in SEO 427
Sept. 19, 2016
No Seminar :
2 p.m. in SEO 427
Sept. 26, 2016
Wenliang Zhang :
2 p.m. in SEO 427
Abstract
In 1960, Lech conjectured that, if R-->S is a flat local map between noetherian local rings, then the Hilbert-Samuel multiplicity of R is less than or equal to the one of S. This conjecture has been open in dimension 3 or higher since then. The purpose of this talk is to give an overview of this conjecture, especially the recent breakthrough made by Linquan Ma in dimension 3.
Oct. 3, 2016
Kevin Tucker :
1 p.m. in SEO 427
Abstract
Following last week's overview, we will continue the discussion of Ma's proof of Lech's conjecture in dimension 3, focusing on the key technical construction in his proof.
Oct. 10, 2016
Linquan Ma :
1 p.m. in SEO 427
Abstract
In this talk, we will sketch the proof of Lech's conjecture in dimension three in equal characteristic. We will also discuss some partial results and related open questions in higher dimensions.
Oct. 17, 2016
Eric Canton :
1 p.m. in SEO 427
Abstract
We have long known, or implicitly used, that to a potential splitting \phi of the Frobenius homomorphism on a ring R of positive characteristic one can assign a divisor on Spec(R), and these divisors describe the Frobenius splitting behavior of \phi at height one primes. We generalize this process and assign coefficients called log discrepancies to valuations on Spec(R/P) for primes P of R. Using these log discrepancies, we can characterize the Frobenius splitting behavior of \phi at any point of Spec(R).
Oct. 24, 2016
No Seminar :
1 p.m. in SEO 427
Oct. 28, 2016
Thomas Polstra :
1 p.m. in SEO 512
Abstract
The F-signature of a local ring, a numerical invariant shown to exist by Tucker, is the asymptotic measurement of the number of Frobenius splittings for which a local ring of prime characteristic admits. The F-signature serves as a measurement of singularities. Most notably, the F-signature of a local ring is 1 if and only if the ring is regular, by work of Huneke and Leuschke, and the F-signature of a local ring is positive if and only if the ring is strongly F-regular, by work of Aberbach and Leuschke. We will discuss how the notion and existence of F-signature extends to all rings which are F-finite but not necessarily local. Our methods our made meaningful by extending Huneke's and Leuschke's and Aberbach's and Leuschke's theorems to the non-local case. This is based on joint work with Alessandro De Stefani and Yongwei Yao.
Oct. 31, 2016
Nicholas Switala :
1 p.m. in SEO 427
Abstract
The goal of this talk is to introduce the categories of Cartier modules and Cartier crystals introduced by M. Blickle and G. Boeckle in their 2011 Crelle paper, as well as the basic finiteness results proved there.
Nov. 7, 2016
N/A :
1 p.m. in SEO 427
Abstract
Bhargav Bhatt will speak from 4:30-7:00 on Yves André's solution to the Direct Summand Conjecture at UChicago (see http://math.uchicago.edu/research/calendar/).
Nov. 14, 2016
Emily Witt :
1 p.m. in SEO 427
Abstract
The F-pure threshold is an invariant in characteristic p > 0 that measures the severity of a singularity; it is analogous to the log canonical threshold in characteristic zero. In this talk, we consider a consequence of the Ascending Chain Condition conjecture for F-pure thresholds, focusing on hypersurfaces with isolated singularities.
Jan. 23, 2017
Axel Stabler :
1 p.m. in SEO 427
Abstract
In my talk I will report on joint work with Manuel Blickle. I will explain how one can generalize the definition of test ideals \tau to so-called Cartier modules in a functorial way. We obtain several transformation rules with respect to f^! and f_* for various classes of morphisms f: X \to Y, e.g. for f smooth one has an isomorphism f^! \tau = \tau f^!. Part of the reason for working in this generality is that one has an equivalence with constructible etale p-torsion sheaves up to nilpotence of Cartier modules and these results further support the idea that the test module construction relates to etale nearby cycles similarly to the complex situation where multiplier ideals relate to complex nearby cycles.
Feb. 16, 2017
Claudia Miller :
2 p.m. in SEO 427
Abstract
Using an idea of Atiyah from 1966, we develop Adams operations on the Grothendieck groups of perfect complexes with support and of matrix factorizations using cyclic group actions on tensors powers. In the former setting, Gillet and Soule developed these using the Dold-Kan correspondence and used them to solve Serre's Vanishing Conjecture in mixed characteristic (also proved independently by P. Roberts using localized Chern characters). Their approach cannot be used in the setting of matrix factorizations, so we use Atiyah's approach, avoiding simplicial theory altogether.
As an application, we prove a conjecture of Dao and Kurano on the vanishing of Hochster's theta pairing for pairs of modules over an isolated hypersurface singularity in the remaining open case of mixed characteristic. Our proof is analogous to that of Gillet and Soule for the vanishing of Serre's intersection multiplicity.
This is joint work with Michael Brown, Peder Thompson, and Mark Walker.
March 3, 2017
Janet Page :
11 a.m. in SEO 1227
Abstract
Cartier algebras and their duals, rings of Frobenius operators, have come up in the study of Frobenius splittings, which have been useful in many topics ranging from singularity theory in algebraic geometry to representation theory. When $R$ is a local ring of characteristic $p >0$, the Cartier algebra $\mathcal{C}(R)$, which is the ring of all potential Frobenius splittings of $R$, is dual to the ring of Frobenius operators ($p^{e}$-linear maps) on the injective hull of the residue field. This ring of Frobenius operators need not be finitely generated over $R$, which led Enescu and Yao to define Frobenius complexity as a measure of its non-finite generation. In their examples Frobenius complexity is not always even rational, but its limit as $p \rightarrow \infty$ is an integer. Few other examples have been computed. In this talk, I will discuss a method to compute limit Frobenius complexity for Hibi rings, which are a class of toric rings defined from finite posets. I will show that this computation can be read directly from the defining poset in nice cases
March 13, 2017
Claudia Miller :
1 p.m. in SEO 427
Abstract
We will describe a construction of Adams operations on the Grothendieck group of matrix factorizations with support via an idea of Atiyah from the 60s using the cyclic group action on tensor powers and as an application prove a vanishing theorem for Hochster's theta invariant. This is joint work with Michael Brown, Peder Thompson, and Mark Walker.
Although this talk is a continuation of my last talk, it is completely independent as the focus will be on Adams operations this time.
March 28, 2017
Gennady Lyubeznik :
2 p.m. in SEO 427
Abstract
N. Switala has developed a Matlis duality theory for D-modules. In this talk I am going to show how Switala's theory leads to a generalization of a recent result of R. Hartshorne and C. Polini on the structure of some local cohomology modules.
Dec. 4, 2017
Alex Constantinescu :
1 p.m. in SEO 427
Abstract
We show that the virtual cohomological dimension of a Coxeter group is essentially the same as the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of its nerve. Using this connection, we modify a construction of Osajda in group theory to find for every positive integer r a quadratic monomial ideal, with linear syzygies, and regularity of the quotient equal to r. This answers a question of Dao, Huneke and Schweig, and shows that Gromov asked essentially the same question about the virtual cohomological dimension of hyperbolic Coxeter groups. For monomial quadratic Gorenstein ideals with linear syzygies we prove that the regularity of their quotients can not exceed four, which implies that for d > 4 every triangulation of a d-manifold has an induced square or a hollow simplex. All results are in collaboration with Thomas Kahle and Matteo Varbaro.
Oct. 1, 2018
Takumi Murayama :
3 p.m. in 612 SEO
Abstract
A common trend in algebraic geometry and commutative algebra is to associate asymptotic invariants to various objects, such as line bundles or families of ideals. In particular, Seshadri constants measure how close a line bundle is to being very ample, and can be defined by combining the volume function from geometry with the Hilbert-Samuel multiplicity function from algebra, thereby combining both geometric and local algebraic information into one invariant. We will describe how this point of view together with Frobenius techniques in commutative algebra can be used instead of vanishing theorems to prove effective positivity results over the complex numbers in the direction of Fujita's conjecture.
Oct. 8, 2018
Karl Schwede :
3 p.m. in 612 SEO
Abstract
I will discuss applications of the
existence and functoriality of big Cohen-Macaulay algebras (due to Andre,
Gabber and others) to study singularities in mixed characteristic. In
particular, we introduce analogs of rational/F-rational and log terminal/F-regular
singularities in mixed characteristic and study their properties. As an
application, we will obtain new results on F-rationality and F-regularity
as the characteristic varies.
Oct. 22, 2018
Daniel Smolkin :
3 p.m. in 612 SEO
Abstract
An important question in commutative algebra is the relationship between ordinary and symbolic powers of ideals. Indeed, Huneke--Katz--Validashti ask: for a domain R, does there always exist a number h so that the hn-th symbolic power of every prime ideal P is contained in its n-th ordinary power? I will present recent progress on answering this question, joint with Janet Page and Kevin Tucker, that was done by studying p-inverse linear maps compatible with higher diagonal embeddings. Time permitting, we will investigate how the set of such maps reflects the singularities of R.
Nov. 19, 2018
Uli Walther :
11 a.m. in 427 SEO
Abstract
I will define GKZ-systems, and talk a little about their properties from the
algebraic, analytic, and combinatorial point of view. Then I will
discuss a theorem of Gelfand et al, and a sharpening by Mathias Schulze and
myself, on the question which GKZ-systems arise as (D-module-)direct
image of a natural D-module on a torus. In such cases, the GKZ-system can
inherit a mixed Hodge module structure. I will then explain work with
Thomas Reichelt that computes the weight filtration of this MHM structure on a
class of GKZ-systems that comes up naturally in mirror symmetry. This
complements work of Reichelt and Christian Sevenheck who computed the Hodge
filtration, and supersedes computations of Batyrev who determined the
weight filtration in a generic point. Very few of such explicitly
computed structures seem to be known.
April 29, 2019
Eric Canton :
4 p.m. in 427 SEO
Abstract
K-stability is concerned with one parameter families of a fixed projective variety that are generated by a one-dimensional torus action; stated more algebraically, Grobner degenerations of the homogenous ideals defining the variety under various embeddings. This has been re-interpreted in recent years in terms of singularities of pairs arising in the minimal model program, having implications for certain moduli spaces. These connections are surprising and intriguing, given that Tian (and Donaldson) defined K-stability complex-analytically (and algebraically, respectively) to study the question of existence of Kahler-Einstein metrics on smooth Fano varieties. Many results require techniques from complex differential geometry, the minimal model program, or non-Archimedean geometry. In this talk, I'll give some background on this topic and a flavor of ongoing work studying K-stability via free resolutions.
Sept. 11, 2019
Anurag Singh :
4 p.m. in 1227 SEO
Abstract
Work of Levasseur and Stafford describes the rings of differential operators on various classical invariant rings of characteristic zero; in each of these cases, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of reductive groups over the complex numbers, Smith and Van den Bergh asked if reduction modulo p works for differential operators in this context. In joint work with Jack Jeffries, we establish that this is not the case for various classical groups.
Oct. 30, 2019
Monica A. Lewis :
4 p.m. in 1227 SEO
Abstract
Let S be a complete intersection presented as R/J for R a regular ring and J a parameter ideal. Let I be an ideal containing J. It is well known that the set of associated primes of H^i_I(S) can be infinite, but far less is known about the set of minimal primes. In 2017, Hochster and Núñez-Betancourt showed that if R has prime characteristic p > 0, then the finiteness of Ass H^i_I(J) implies the finiteness of Min H^{i-1}_I(S), raising the following question: is Ass H^i_I(J) always finite? We give a positive answer when i=2 but provide a counterexample when i=3. The counterexample crucially requires Ass H^2_I(S) to be infinite. The following question, to the best of our knowledge, is open: (under suitable hypotheses on R) does the finiteness of Ass H^{i-1}_I(S) imply the finiteness of Ass H^i_I(J)? When S is a domain, we give a positive answer when i=3. When S is locally factorial, we extend this to i=4. Finally, if R has prime characteristic p > 0 and S is regular, we give a complete answer by showing that Ass H^i_I(J) is finite for all values of i.
Nov. 1, 2019
Eloísa Grifo :
2 p.m. in 427 SEO
Abstract
The powers of an ideal are easy to compute, though difficult to
describe geometrically; in contrast, symbolic powers are difficult to
compute while having a natural geometric description. In trying to
compare symbolic and ordinary powers, Harbourne conjectured that a
famous containment by Ein--Lazersfeld--Smith, Hochster--Huneke, and
Ma--Schwede could be tightened. Harbourne's Conjecture is a statement
depending on n that unfortunately has been disproved for particular
values of n. However, recent evidence points towards a stable version
of Harbourne's conjecture, where we ask only for n to be large enough.
Some of that evidence is joint work with Craig Huneke and Vivek
Mukundan.
Nov. 20, 2019
Austin Simpson :
4 p.m. in 1227 SEO
Abstract
I discuss joint work with Rankeya Datta which shows that the Hilbert-Kunz multiplicity, a prime characteristic numerical invariant that detects regularity, does not increase upon taking general hyperplane sections of an equidimensional quasi-projective variety.
Jan. 29, 2020
Ben Antieau :
4 p.m. in 1227 SEO
Feb. 5, 2020
Kevin Tucker :
4 p.m. in 1227 SEO
Feb. 12, 2020
Rankeya Datta :
4 p.m. in 1227 SEO
Feb. 19, 2020
Wenliang Zhang :
4 p.m. in 1227 SEO
March 18, 2020
Florian Enescu :
4 p.m. in 1227 SEO
April 8, 2020
Jason McCullough :
4 p.m. in 1227 SEO
Jan. 13, 2021
Benjamin Briggs :
4 p.m. in Zoom
Abstract
For any ideal I of finite projective dimension in a local ring R, Vasconcelos conjectured that I is complete intersection if and only if the conormal module I/I^2 has finite projective dimension over R/I. Quillen made a similar conjecture earlier: I is complete intersection if and only if the cotangent complex of R/I over R has finite projective dimension (this was established by Avramov in 1999).
I'll try to explain why the similarity between these conjectures is not just superficial, and how you can in fact prove a result generalising the two about "higher conormal modules" (I'll explain what these are, and also give some background on the cotangent complex). This is joint work with Srikanth Iyengar.
Feb. 24, 2021
Eamon Quinlan-Gallego :
4 p.m. in Zoom
Abstract
The Bernstein-Sato polynomial of a holomorphic function is an invariant that originated in complex analysis, and with now strong applications to birational geometry and singularity theory over the complex numbers. For example, it detects the log-canonical threshold as well as the eigenvalues of the monodromy action on the cohomology of the Milnor fibre. In this talk I will present an analogue of this invariant for polynomials with Z/p^m coefficients and explain some connections to the characteristic-0 theory. This is joint work with T. Bitoun.
March 3, 2021
Zhan Jiang :
4 p.m. in Zoom
March 10, 2021
Linquan Ma :
4 p.m. in Zoom
Abstract
We generalize the theory of globally F-regular pairs to mixed characteristic, which we call +-regularity, and introduce certain stable sections of adjoint line bundles. This is inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure. We will discuss some applications of these results to birational geometry in mixed characteristic. Joint work with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.
March 31, 2021
Michael Perlman :
4 p.m. in Zoom
Abstract
Given a subvariety Z in affine space, the local cohomology modules with support in Z are mixed Hodge modules. This implies that each local cohomology module is endowed with two filtrations subtly encoding the singularities of Z: the Hodge filtration and the weight filtration. We will discuss new calculations of these filtrations in the case when Z is a generic determinantal variety. Joint work with Claudiu Raicu.
April 7, 2021
Austyn Simpson :
4 p.m. in Zoom
Abstract
Given a local ring R of prime characteristic p>0 and a non-zero-divisor f such that R/(f) is F-pure, is it necessarily the case that R is F-pure? That is, does F-purity deform? Fedder answered this question affirmatively if R is Gorenstein, but non-Q-Gorenstein counterexamples exist due to both Fedder and Singh. In this talk, I'll present a recent solution to this deformation question when R is assumed to be Q-Gorenstein. Joint work with Thomas Polstra.
April 14, 2021
Patricia Klein :
4 p.m. in Zoom
Abstract
Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, we will describe an explicit connection between these approaches. In particular, we describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras.
This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel’s strategy of using liaison to establish Gr\"obner bases. We describe briefly, as an application of this work, a proof of a recent conjecture of Hamaker, Pechenik, and Weigandt on diagonal Gr\"obner bases of Schubert determinantal ideals.
This talk is based on joint work with Jenna Rajchgot.
April 21, 2021
Swaraj Pande :
4 p.m. in Zoom
Abstract
Multiplier ideals are very refined invariants that measure the singularities of algebraic varieties. They give rise to many other interesting invariants, for example, the log canonical threshold and more generally, jumping numbers. This talk is about another related invariant, namely multiplicities of jumping numbers that measure the difference between successive multiplier ideals. The main theorem here is that these multiplicities fit into a quasi-polynomial. We'll also discuss when the various components of the quasi-polynomial have the highest possible degree relating it to Rees valuations. Finally, we'll consider the special case of monomial ideals where these invariants have a combinatorial description in terms of the Newton polyhedron.
April 27, 2021
Neil Epstein :
4 p.m. in Zoom
Abstract
Broadening existing results in the literature to much wider classes of rings, we prove among other things:
1. Reduced quotients of excellent regular rings of characteristic p admit big test elements,
2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of ℚ, and
3. If R is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of R.
To do so, we develop the parallel theories of Ohm-Rush and intersection flat algebras. We show that both properties can be checked locally in flat maps of Noetherian rings. We show that intersection-flatness admits a content theory parallel to that of Ohm-Rush content for Ohm-Rush algebras. We develop descent results for these properties. Using the descent result for intersection flatness, we obtain a local condition under which a faithfully flat map of Noetherian rings must be intersection-flat. The local condition for intersection-flatness allows us to conclude that finitely generated faithfully flat algebras over a Noetherian ring are intersection-flat. Combining the local condition for intersection flatness with results of Kunz and Radu yields the conclusion that the Frobenius endomorphism associated to a locally excellent regular ring of prime characteristic is intersection-flat, thus answering a question of Sharp. Applications of the latter result include the three enumerated results above. We also get applications to tight closure and parameter test ideals.
Sept. 15, 2021
Thomas Polstra :
3 p.m. in Zoom
Abstract
Let $(R,\mathfrak{m},k)$ be local normal Cohen-Macaulay domain and $I\subseteq R$ an ideal of pure height $1$. For each natural number $N$ let $I^{(N)}$ denote the $N$th symbolic power of $I$. We consider annihilators of the local cohomology modules $H^i_{\mathfrak{m}}(R/I^{(N)})$. When $R$ is of prime characteristic $p>0$ and $I$ is a multiple of an anticanonical ideal of $R$ then understanding the annihilators $H^{i}_{\mathfrak{m}}(R/I^{(p^e)})$ as $e$ varies through the natural numbers sheds light on the weak implies strong conjecture from tight closure theory. This talk is based on joint work with Ian Aberbach.
Nov. 4, 2021
Jonathan Montano :
3 p.m. in Zoom
Abstract
We study F-purity and strong F-regularity of blowup algebras. Our main focus is on algebras given by symbolic and ordinary powers of different types of determinantal ideals. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their projective dimension stabilizes. To obtain these results we develop the notion of F-pure filtrations and symbolic F-purity. This is joint work with Alessandro De Stefani and Luis Núñez-Betancourt.
Nov. 10, 2021
Justin Chen :
3 p.m. in Zoom
Abstract
The difference between nonnegative polynomials and sums of
squares is an important topic in real algebraic geometry, and the
Hankel index of a variety is a natural (but subtle) invariant that
quantifies this difference. Surprising connections were found between
the Hankel index and the commutative algebra of the variety, namely the
N_{2,p} property of linear syzygies for the free resolution - although
this only provides half the story. For curves of almost minimal degree,
we complete the picture, by determining the Hankel index in terms of a
new rank called almost real rank, which interpolates between real
(Waring) rank and complex border rank. This is joint work with Greg
Blekherman and Jaewoo Jung.
Nov. 17, 2021
Claudiu Raicu :
3 p.m. in Zoom
Abstract
Let X denote the incidence correspondence (or partial flag variety) parametrizing pairs consisting of a point in projective space and a hyperplane containing it. I will explain how to characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X over an arbitrary field. For the projective plane, the results are contained in the thesis of Griffith from the 70s, while in characteristic zero the cohomology groups are described in any dimension by the Borel-Weil-Bott theorem. Joint work with Zhao Gao.
Dec. 2, 2021
Cancelled :
3 p.m. in Zoom
Jan. 26, 2022
Sarasij Maitra :
3 p.m. in Zoom
Abstract
We shall define an invariant that helps in obtaining information regarding the torsion submodule of a finitely generated rank one module over a one-dimensional Noetherian local domain. We shall briefly go through some properties of this and see how this can be applied to a long-standing conjecture of R. Berger concerning the module of differentials over an analytic $k$-algebra. If time permits, then we shall also state and prove an explicit computation of the invariant in this case.
Feb. 9, 2022
Vaibhav Pandey :
3 p.m. in Zoom
Abstract
Over an infinite field, a generic determinantal ring is the
fixed subring of an action of the general linear group on a polynomial
ring; this is the natural embedding of the title. If the field has
characteristic zero, the general linear group is linearly reductive, and
it follows that the invariant ring is a split subring of the polynomial
ring. We determine if the natural embedding is split in the case of a
field of positive characteristic. Time permitting, we will address the
corresponding question for Pfaffian and symmetric determinantal rings.
This is ongoing work with Mel Hochster, Jack Jeffries, and Anurag Singh.
March 2, 2022
Jack Jeffries :
2 p.m. in Zoom
Abstract
Bernstein's inequality is a fundamental result in the theory of D-modules on smooth varieties, with many important consequences. However, rings of differential operators are much more poorly behaved on singular varieties in general. In this talk we will discuss a version of Bernstein's inequality that holds on certain singular singularities. This is based on joint work with Josep Àlvarez Montaner, Daniel J. Hernández, Luis Núñez-Betancourt, Pedro Teixeira, and Emily E. Witt.
March 16, 2022
Jiamin LI :
3 p.m. in Zoom
Abstract
In 2005, Cutkosky, Ha, Srivinivasan and Theodorescu studied the generalized multiplicity defined by the 0-th local cohomology, followed by the work by Katz-Validashti in 2010, Ulrich-Validashti in 2011, and Cutkosky in 2014, and was considered as a generalization of the classical Hilbert-Samuel multiplicity. In 2019 Dao and Montano generalized this notion to the j-th local cohomology (here we call the j-multiplicity) of R/I^t. However its existence and rationality is unknown in general. Using the representation theoretical tools developed by Raicu et al, in this talk we will show the closed formula of the j-multiplicity of determinantal thickenings of maximal minors in the m x n matrix for a suitable choice of j, which extended the work by Jeffries, Montano and Varbaro in 2015, and the work by Kenkel in 2019.
March 30, 2022
Jason McCullough :
3 p.m. in Zoom
Abstract
Chow rings of matroids are a generalization of the cohomology rings of wonderful compactifications of complements of complex hyperplane arrangements introduced by de Concini and Procesi. They have been employed in the proof of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz on the log-concavity of the coefficients of the characteristic polynomial of a graph (or more generally a matroids); they were also employed in the recent proof of the Top Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. Among other properties, Chow rings of matroids are commutative, graded, Artinian, quadratic, Gorenstein K-algebras. Dotsenko conjectured that they are additionally Koszul. While previous work of Mastroeni, Schenck, and Stillman showed that not all quadratic, Artinian, Gorenstein K-algebras are Koszul, we show that all Chow rings of matroids are Koszul. In particular, this places restrictions on the possible Hilbert Series of Chow rings and implies that their Poincare series are rational. This is joint work with Matt Mastroeni and can be found in our preprint: https://arxiv.org/abs/2111.00393.
April 13, 2022
Alapan Mukhopadhyay :
3 p.m. in Zoom
Abstract
We shall discuss a natural generalization of the classical Hilbert-Kunz multiplicity theory when the underlying objects are graded. More precisely, given a graded ring $R$ and a finite co-length homogeneous ideal $I$ in a positive characteristic $p$ and for any complex number $y$, we shall show that the limit
$$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(R)}\sum \limits_{j= -\infty}^{\infty}\lambda \left( (\frac{R}{I^{[p^n]}R})_j\right)e^{-iyj/p^n}$$
exists. This limit as a function in the complex variable $y$ is a natural refinement of the Hilbert-Kunz multiplicity of the pair $(R,I)$: the value of the limiting function at the origin is the Hilbert-Kunz multiplicity of the pair $(R,I)$. We name this limiting function the <i>Frobenius-Poincare function</i> of $(R,I)$. We shall establish that Frobenius-Poincare functions are holomorphic everywhere in the complex plane. We shall discuss properties of Frobenius-Poincare functions, give examples and describe these functions in terms of the sequence of graded Betti numbers of $\frac{R}{I^{[p^n]}R}$. On the way, we shall mention some questions on the structure and properties of Frobenius-Poincare functions.
April 20, 2022
Cheng Meng :
3 p.m. in Zoom
Abstract
We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R, \mathfrak{m}) \to (S, \mathfrak{n})$ is a flat local extension of Noetherian local rings with $\dim R=\dim S$, the completion of $(S, \mathfrak{n})$ is the completion of a standard graded ring $(S_g, \mathfrak{n}_g)$ over a field $k$ and the completion of $I=\mathfrak{m}S$ is extended from a homogeneous ideal $I_g$, then $e(R) \leq e(S)$.
Sept. 6, 2022
Hunter Simper :
11 a.m. in 636 SEO
Abstract
Let $R$ be the ring of polynomial functions in $mn$ variables with coefficents in $\mathbb{C}$, where $m>n$. Set $X$ to be the matrix in these variables and $I$ the ideal of maximal minors of this matrix. I will discuss the R-module structure of certain Ext and local cohomology modules arising from the rings $R/I^t$. In particular, for $i$ equal to the cohomological dimension of $I$, I will discuss the embedding of $Ext^i_R(R/I^t,R)$ into $H_\frak{m}^{mn}(R)$, explicitly describing this embedding when $X$ is size $n \times (n-1)$. More generally for $X$ of arbitrary size I will describe the annihilator of $Ext^i_R(R/I^t,R)$ and thereby completely determine the $R$-module structure of $H_\frak{m}^{mn-i}(R)$.
Sept. 20, 2022
Michael Perlman :
11 a.m. in 636 SEO
Abstract
Given a closed subvariety Z in a smooth complex variety X, the local cohomology sheaves with support in Z are holonomic D-modules, and thus have finite filtration with simple composition factors. We will discuss work in progress concerning the D-module structure on local cohomology in the case when X is a Grassmannian and Z is a Schubert variety, including combinatorial formulas describing the composition factors and levels of the weight filtration. Upon restriction to the "opposite big cell", these calculations recover several previously known results concerning local cohomology with support in determinantal varieties.
Oct. 11, 2022
Shiji Lyu :
11 a.m. in 636 SEO
Abstract
The splinter property is a classical subject in positive- and mixed-characteristic commutative algebra. The long-standing conjecture that a regular Noetherian ring is a splinter was solved only recently. In this talk, we will discuss a relative version of this problem, and how ultrapower shows up in the attack.
Oct. 18, 2022
Anna Brosowsky :
11 a.m. in 636 SEO
Abstract
For a local $F$-finite ring, the splitting prime is an "obstruction" to strong $F$-regularity. In this talk, we will define the self-map on the Frobenius split locus of a ring $R$ which sends a point $P$ to the splitting prime of $R_P$, and discuss the generalization of this map to the setting of Cartier algebras. We will go over some properties of this map and, as an application, will show how it behaves for Stanley-Reisner rings.
Nov. 1, 2022
Luis Nuñez Betancourt :
11 a.m. in Zoom
Abstract
The functional equation gives a systematic way to decrease the power of a polynomial via the ring of differential operators. This tool has been used to study singularities, multiplier ideals, local cohomology and zeta functions. In this talk we will discuss a functional equation for meromorphic equations and several consequences of its existence. This is joint work with Josep Àlvarez Montaner, Manuel González Villa, and Edwin León-Cardenal.
Nov. 8, 2022
No Seminar :
11 a.m. in 636 SEO
Nov. 29, 2022
Matteo Varbaro :
11 a.m. in Zoom
Abstract
Quasi-Gorenstein rings are, roughly speaking,"Gorenstein rings which possibly fail the Cohen-Macaulay property”. They are much more than Gorenstein rings, and in some situation they are more natural: For example, the Stanley-Reisner ring of an orientable manifold is quasi-Gorenstein, while the only orientable manifolds whose Stanley-Reisner ring is Gorenstein are (homology) spheres. In this talk I will discuss some features of quasi-Gorenstein rings, a liaison theory by quasi-Gorenstein ideals generalizing the classical one by Gorenstein ideals, and two applications of the latter: one on the combinatorics of the minimal prime ideals of a quasi-Gorenstein ring, and the other one explaining a connection with the topological Lefshetz duality. All this is based on a joint work with Hongmiao Yu.
Jan. 30, 2023
Mahrud Sayrafi :
3 p.m. in 636 SEO
Abstract
Building on a result of Swanson, Cutkosky--Herzog--Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for large enough n the regularity of I^n is exactly dn+e.
Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) as a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules. This is joint work with Juliette Bruce and Lauren Cranton Heller.
March 6, 2023
Andras Lorincz :
3 p.m. in 636 SEO
Abstract
I present results on the geometry of equivariant, proper maps from homogeneous bundles over flag varieties, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image of a collapsing has rational singularities in characteristic zero. We extend this to positive characteristics showing that such an image is strongly F-regular if its coordinate ring has a good filtration, and give criteria for the existence of the latter. We further show that the restrictions of such collapsing maps to Schubert varieties are F-rational in positive characteristic and have rational singularities in characteristic zero.
These results give a uniform, characteristic-free approach for the study of the geometry of some remarkable varieties, such as: multicones over Schubert varieties, various determinantal varieties in spaces of matrices, varieties of complexes, subspace varieties, higher rank varieties.
March 14, 2023
Florian Enescu :
2 p.m. in 427 SEO
Abstract
Prompted by the definition of the Frobenius complexity of a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of toric rings for which these generating functions are rational. We will discuss this class of rings and aspects of the rationality of the complexity generating function. This work is joint with Yongwei Yao.
April 24, 2023
No Seminar :
3 p.m. in No Seminar
April 10, 2024
Jonah Tarasova :
3 p.m. in 1227 SEO
Abstract
Consider a group acting on a polynomial ring $S$ over a field $K$ by degree-preserving $K$-algebra automorphisms. The invariant ring $R$ is a graded subring of $S$; let $\mathfrak{m}_R$ denote the homogeneous maximal ideal of $R$. Several key properties of the invariant ring and its embedding in $S$ can be deduced by studying the nullcone $S/\mathfrak{m}_R S$ of the group action. This includes, for example, the finite generation of the invariant ring and the purity of the embedding. In this talk, we study the nullcone arising from the natural action of the symplectic group.
For the natural representation of the symplectic group (via copies of the standard representation), the invariant ring is the ring defined by the principal Pfaffians of a fixed even size of a generic alternating matrix. We show that the nullcone of this embedding is a strongly $F$-regular ring in positive characteristic, and hence in characteristic zero, a ring of strongly $F$-regular type. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein.
April 24, 2024
Josh Pollitz :
3 p.m. in 1227 SEO
Abstract
By now it is quite classical that one can understand singularities in prime characteristic local algebra/algebraic geometry, through properties of the Frobenius endomorphism. A foundational result illustrating this is the celebrated theorem of Kunz characterizing the regularity of a noetherian scheme (in prime characteristic) in terms of whether a Frobenius pushforward on that scheme is flat. In this talk, I'll discuss a structural explanation of, that also recovers, the theorem of Kunz and other theorems of this ilk. Namely, I’ll discuss recent joint work with Ballard, Iyengar, Lank, and Mukhopadhyay where we show that over an F-finite noetherian scheme of prime characteristic high enough Frobenius pushforwards generate the bounded derived category.
Jan. 15, 2025
Adam LaClair :
11 a.m. in 1227 SEO
Abstract
Binomial edge ideals provide a way to associate to any graph a binomial ideal. Many researchers have investigated the algebraic properties of binomial edge ideals in terms of the combinatorics of the graph. One such question is: When are binomial edge ideals F-pure? F-purity describes a class of rings in positive characteristic that exhibit a "mild" singularity, such rings have assumed a prominent position amongst the study of F-singularities. In 2012, K. Matsuda introduced the class of weakly closed graphs (i.e., incomparability graphs), and he proved that the binomial edge ideal associated to such graphs is F-pure in any positive characteristic. He conjectured that the converse should hold in characteristic 2. In this talk, we will review binomial edge ideals, F-purity, and other relevant background, and then we will present the main ideas going into the proof of Matsuda's conjecture.
April 9, 2025
Jack J. Garzella :
3 p.m. in 1227 SEO
Abstract
We first describe a recent generalization of F-split singularities known as quasi-F-split singularities, with particular emphasis on a connection to an object which shows up in number theory called the *newton polygon*. We then describe how techniques from computational number theory can be used to compute these invariants much faster than the previous state of the art. This covers joint work with Batubara and Pan.