In the first half, we shall discuss the importance of Castelnuovo-Mumford regularity and the main motivation to define this invariant along with examples. This part is aimed for the graduate students and should be elementary.
In the second half, we discuss the asymptotic behavior of Castelnuovo-Mumford regularity of powers of ideals, and that of Ext and Tor, with respect to the homological degree, over graded complete intersection rings. We derive from a theorem of Gulliksen, a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behavior could be pretty hectic when the latter condition is not satisfied.
This talk is based on our joint papers with Marc Chardin, Navid Nemati and Tony Puthenpurakal.
Let (A,m) be a CM local ring and let I be a normal m-primary ideal with e_3(I) = 0.
Then G_I(A) is CM.
(note Itoh only conjectured it for Gorenstein local rings)
Toric varieties are popular objects in algebraic geometry, as they can be modelled on polytopes and polyhedral fans. This is mainly because there is a dictionary between their geometric properties and the combinatorial invariants of their polytopes. This dictionary can be extended from toric varieties to arbitrary varieties through toric degenerations.
In this talk, I will introduce the notion of toric degenerations which generalizes the fruitful correspondence between toric varieties and polytopes, to arbitrary varieties. There are prototypic examples of toric degenerations (of Grassmannians) which are related to Young tableaux and Gelfand-Cetlin polytopes. I will describe how to obtain such degenerations using the theory of Gröbner fans and tropical geometry, and how their associated polytopes are connected by mutations.
Chairperson - Jürgen Herzog
In the first part of this talk, we explore graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequences. In the second part of this talk, we give an application of graded Bourbaki sequences to Hilbert functions of m-primary ideals. We give the inequality of the first three Hilbert coefficients for ideals of reduction number two.
This talk is based on the joint work with J. Herzog and D. I. Stamate.
Chairperson - Jürgen Herzog
The study of toric ideals of graphs lies in the intersection of commutative algebra, algebraic geometry, and combinatorics. Formally, if $G = (V,E)$ is a finite simple graph with edge set $E =\{e_1,\ldots,e_s\}$ and vertex set $V = \{x_1,\ldots,x_n\},$ then the toric ideal of $G$ is the kernel of the ring homomorphism $\varphi:k[e_1,\ldots,e_s] \rightarrow k[x_1,\ldots,x_n]$ where $\varphi(e_i) = x_jx_k$ if the edge $e_i = \{x_j,x_k\}$. Ideally, one would like to understand how the homological invariants (e.g. graded Betti numbers) of $I_G$ are related to the graph $G$. In this talk I will survey some results connected to this theme, with an emphasis on the Castelnuovo-Mumford regularity of these ideals.
Chairperson - Takayuki Hibi
Blickle-Bhatt-Lyubeznik-Singh-Zhang proved that if $X$ is a projective variety over a field $k$ of characteristic zero with isolated complete intersection singularities, then the Kodaira vanishing theorem holds for all thickenings of $X$. What if $k$ is of positive characteristic? Kodaira vanishing can fail in positive characteristic, but it still holds for Frobenius split varieties. In this talk, I will discuss Kodaira vanishing for thickenings of globally $F$-regular varieties, a special class of Frobenius split varieties. This talk is based on joint work with Kenta Sato.
Chairperson - Keiichi Watanabe
The classes of regular, Gorenstein and Cohen-Macaulay rings are among the most important classes of rings in commutative algebra and algebraic geometry. In this talk we recall the definitions and basic properties of these classes, and then explain how to generalize each of them to derived commutative algebra, in the context of commutative differential graded algebras. We further explain how each of these generalizations arise naturallyin various algebraic geometry contexts and discuss some applications.
Graph domination problems are ubiquitous in graph theory. In the broadest terms, they ask how one can ‘observe’ an entire graph by designating a certain list of vertices, following a proscribed list of rules. An example of this is the vertex covering problem which happens to describe the irredundant irreducible decomposition of the edge ideal of a graph. In this talk, we will survey recent work with various collaborators on other monomial ideal constructions that arise from other graph domination problems, including one coming from electrical engineering.
Chaiperson - Siamak Yassemi
Lattice polytopes are the combinatorial backbone of toric varieties. Many important properties of these varieties admit purely combinatorial description in terms of the underlying polytopes. These include normality and projective normality. On the other hand, there are geometric properties of polytopes of integer programming/discrete optimization origin, which can be used to deduce the aforementioned combinatorial properties: existence of unimodular triangulations or unimodular covers. In this talk we present the following recent results: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in R^3 has a unimodular cover, and (3) for every d at least 5, there are ellipsoids in R^d, such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (3) answers a question of Bruns, Michalek, and the speaker.
Chaiperson - Ravi Rao
This talk will be a somewhat historical one, concerning three problems dealing with the idea of torsion. The three problems are those on symbolic powers, the Huneke-Wiegand conjecture, and Berger's conjecture. Besides talking about my own memories, we will focus on torsion in tensor products.
The classical Lech's inequality can be viewed as a uniform, independent of an ideal, upper bound on the ratio of the multiplicity and the colength of an m-primary ideal of a local ring. It was also observed by Lech that, if the dimension is at least two, it is not sharp for any given ideal. Recently, we were able to show more: most of the time, it is possible to improve Lech's upper bound so that it works for all ideals. I will present the proof of this result and all required background in multiplicity theory.
Let $L$ be a finite distributive lattice. By Birkhoff's fundamental structure theorem, $L$ is the ideal lattice of its subposet $P$ of join-irreducible elements. Write $P=\{p_1,\ldots,p_n\}$ and let $K[t,z_1,\ldots,z_n]$ be a polynomial ring in $n+1$ variables over a field $K.$ The {\em Hibi ring} associated with $L$ is the subring of $K[t,z_1,\ldots,z_n]$ generated by the monomials $u_{\alpha}=t\prod_{p_i\in \alpha}z_i$ where $\alpha\in L$. In this talk, we show that a Hibi ring satisfies property $N_4$ if and only if it is a polynomial ring or it has a linear resolution. We also discuss a few results about the property $N_p$ of Hibi rings for $p=2$ and 3. For example, we show that if a Hibi ring satisfies property $N_2$, then its Segre product with a polynomial ring in finitely many variables also satisfies property $N_2$.
Chairperson - Manoj Kummini
The renowned Quillen–Suslin Theorem is closely associated to the Affine Horrocks’ Theorem on algebraic vector bundles. It says : If $R$ is any commutative ring and $E$ is a vector bundle on $\mathbb{A}_{R}^1$ and $E$ extends to a vector bundle on $\mathbb{P}^1_R,$ then $E$ is extended from $Spec(R).$ This is also known as "Monic inversion principle" for projective modules. Here we discuss about analogue of the Monic inversion principle for local complete intersection ideals of height $n$ in $R[T],$ where $R$ is a regular domain of dimension $d,$ which is essentially of finite type over an infinite perfect field of characteristic unequal to $2,$ and $2n \geq d + 3.$ This is a joint work with Mrinal Kanti Das and Md. Ali Zinna.
The Hilbert scheme of points Hilb^n(A^2), parametrizing finite subschemes of the plane of degree n, is a well studied and well behaved parameter space. A classical theorem of Fogarty states that it is a smooth variety of dimension 2n. By contrast, the nested Hilbert scheme Hilb^(n_1,n_2)(A^2), parametrizing nested pairs of subschemes of degrees n_1 and n_2, are usually singular, and very little is known about their singularities. Using techniques from commutative algebra, we prove that the nested Hilbert scheme Hilb^(n,2)(A^2) has rational singularities. This is a joint work with Ritvik Ramkumar.
Let R be a graded direct summand of a positively graded polynomial ring over the p-adic integers. We exhibit an explicit constant D such that, for any positive integer n and any homogeneous prime ideal Q of R, the Dn-th symbolic power of Q is contained in the n-th power of the homogeneous maximal ideal (p)R + R_+. The strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on R. The talk will be based on joint work with E. Grifo and J. Jeffries.
The naive lifting for dg modules is the new concept introduced by M.Ono, S.Nasseh and myself for the purpose of unifying the ideas of lifting and weak lifting for modules over commutative rings. In this talk I will show how we get the obstruction class of naive liftings, which in fact coincides with the Atiyah class that has been introduced by Buchweitz-Flenner. This is a joint work with Saeed Nasseh and Maiko Ono.
This talk is about a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over the ring $\mathbb{C}[x,y]/(x^k).$ This yields an infinite rank analog of the Grassmannian cluster categories introduced by Jensen, King, and Su. In the special case, $k=2,$ $\text{Spec}(\mathbb{C}[x,y]/(x^2))$ is a type $A$-infinity singularity and the ungraded version of the category of maximal Cohen-Macaulay modules over $\mathbb{C}[x,y]/(x^2))$ has been studied by Buchweitz, Greuel, and Schreyer in the 1980s. We demonstrate that his category has infinite type $A$ cluster combinatorics. In particular, we show that it has cluster-tilting subcategories modeled by certain triangulations of the (completed) infinity-gon and we can also interpret certain mutations of the category in this model. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.
9 September 2022, 6:30 pm IST (joining time 6:20 pm IST)
Mircea Mustata, University of Michigan, USA - An estimate for the F-pure threshold via the roots of the Bernstein-Sato polynomial
Given a smooth complex algebraic variety X and a nonzero regular function f on X, I will describe an estimate for the difference between the log canonical threshold of f and the F-pure threshold of a reduction mod p of f, in terms of the roots of the Bernstein-Sato polynomial bf of f. This is based on some old work with S. Takagi and K.-i. Watanabe on one hand, and with W. Zhang on the other hand, plus one simple observation. Most of the talk will be devoted to an introduction to the invariants of singularities that feature in the result.
30 September 2022, 5:30 pm IST (joining time 5:20 pm IST)
Joachim Jelisiejew, University of Warsaw, Poland - When is a homogeneous ideal a limit of saturated ones?
Let I be a homogeneous ideal in a polynomial ring S. If the Hilbert function of S/I is admissible, for example (1,n,n,n,...) is it natural to ask whether I is a limit of homogeneous ideals: does there exist a ideal F in S[t] such that F(t = 0) is equal to I, while F(t = lambda) is a saturated homogeneous ideal for lambda general. Examples of such limits (for the above Hilbert function) can be constructed e.g. by degenerating I(Gamma), where Gamma is a tuple of n general points on the projective space associated to S. However, to decide whether a given ideal I is a limit is very much nontrivial. This problem very recently became of key interest for applications in the theory of tensors: proving that certain ideals are not limits would improve best known lower bounds on border ranks of certain important tensors.
In the talk I will report how surprisingly little is known and present some recent results and some challenges, both theoretical and computational. All this is a joint work with Tomasz Mandziuk.
I’ll discuss the Buchweitz-Greuel-Schreyer Conjecture on the minimal size of a matrix factorization, and my recent proof that the conjecture holds for generic polynomials.
Let k be a field, m a positive integer, V an affine subvariety of $A^{m+3}$ defined by a linear relation of the form $x_1^{ r_1} · · · x_r^{r_m} y = F(x_1, . . . , x_m, z, t),$ A the coordinate ring of V and $G = X_1^{ r_1} · · · X_r^{r_m} Y − F(X_1, . . . , X_m, Z, T).$ We exhibit several necessary and sufficient conditions for V to be isomorphic $A^{m+2}$ and G to be a coordinate in $k[X_1, . . . , X_m, Y, Z, T],$ under a certain hypothesis on F. Our main result immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds.
We also describe the isomorphism classes and automorphisms of integral domains of the type A under certain conditions. These results show that for each integer d ⩾ 3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic.
This is a joint work with Neena Gupta.
In joint work with T\`ai Huy H\`a and Susan Morey we introduced the notion of initially regular sequences on $R/I$, where $I$ is any homogeneous ideal in a polynomial ring $R$. We will discuss this notion, and show how we can construct certain types of (initially) regular sequences on $R/I$ that give effective bounds on the depth of $R/I$. Moreover, we will discuss when these sequences remain (initially) regular sequences on $R/I^t$ and give lower bounds on $\depth R/I^t$ for $t\ge 2$.
All perfect ideals of codimension two are in the linkage class of a complete intersection (licci), but in codimension three and beyond this is no longer the case. I will share some ongoing work, joint with Lorenzo Guerrieri and Jerzy Weyman, which illustrates how the theory of "higher structure maps" originating from Weyman's generic ring may be used to distinguish licci ideals within the broader class of perfect ideals of codimension three.
In the non noetherian situation, valuation rings often behave like regular rings. We will discuss several such results which are classically known to be true for regular rings, but also true for valuation rings. We then focus on Brauer groups. It is well known that Br(R) injects into Br(K) provided R is a regular domain and K=qt(R). We observe that the same is true for valuation rings. In fact, we will discuss a more general result in the setting of etale cohomology.
In this talk, we discuss about the Castelnuovo-Mumford regularity (or regularity) of Rees algebras and symbolic Rees algebras of certain ideals associated to finite simple graphs and we give various combinatorial upper bounds. Also we study upper bounds for symbolic and ordinary powers of edge and vertex cover ideals of simple graphs.
This talk will be about the minimal free resolution of quadratic monomial ideals. It is well known that a quadratic monomial ideal I in the polynomial ring K[x1, . . . , xn], K a field, has a linear resolution if and only if I is the edge ideal of the complement of a chordal graph, and this is equivalent to the linearity of the resolution of all powers of I.
In this talk we will discuss the case that the resolution of a quadratic monomial ideal I is linear up to the homological degree t with t ≥ pd(I) − 2, where pd(I) denotes the projective dimension of I. As an outcome, we give a combinatorial classification of such ideals and also check whether their high powers have a linear resolution.
Chairperson: Siamak Yassemi, University of Tehran, Iran
Watanabe classified all abelian quotient complete intersection singularities. Watanabe defined a special datum in order to classify abelian quotient complete intersection singularities. In this talk, I investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in terms of the special datum. Moreover I give bounds of the multiplicity of abelian quotient complete intersection singularities.
Gorenstein binomial edge ideals have been completely characterized and they are the paths only. There are two interesting generalizations of Gorenstein rings: level rings and pseudo-Gorenstein rings. In the first half, we will talk about the behavior of the levelness and pseudo-Gorensteinness on the decomposable graphs and cone graphs.
In the next half, we will discuss the characterization of Cohen-Macaulay binomial edge ideals of bipartite graphs and then their levelness and pseudo-Gorensteinness.
This talk is based on the joint work with Giancarlo Rinaldo.
16 December 2022, 5:30 pm IST (joining time 5:20 pm IST)
Utsav Chowdhury, Indian Statical Institute, Kolkata, India - Characterisation of the affine plane using A^1 -homotopy theory
Characterisation of the affine n-space is one of the major problem in affine algebraic geometry. Miyanishi showed an affine complex surface X is isomorphic to C^2 if O(X) is a U.F.D., O(X)^∗ = C^∗ and X has a non-trivial Ga-action [3, Theorem 1]. Since the orbits of a Ga-action are affine lines, existence of a non-trivial Ga-action says that there is a non-constant A^1 in X. Ramanujam showed that a smooth complex surface is isomorphic to C^2 if it is topologically contractible and it is simply connected at infinity [5]. Topological contractibility, in particular pathconnectedness says that there are non-constant intervals in X. On the other hand, A^1 -homotopy theory has been developed by F.Morel and V.Voevodsky [4] as a connection between algebra and topology. An algebrogeometric analogue of topological connectedness is A^1 -connectedness. In this talk, using ghost homotopy techniques [2, Section 3] we will prove that if a surface X is A^1 -connected, then there is an open dense subset such that through every point there is a non-constant A^1 in X. As a consequence using the algebraic characterisation, we will prove that C^2 is the only A^1 -contractible smooth complex surface. This answers the conjecture appeared in [1, Conjecture 5.2.3]. We will also see some other useful consequences of this result. This is a joint work with Biman Roy.
References
[1] A. Asok, P. A. Østvær; A 1 -homotopy theory and contractible varieties: a survey, Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects. Lecture Notes in Mathematics, vol 2292. Springer, Cham. https://doi.org/10.1007/978-3-030-78977-05.
[2] C. Balwe, A. Hogadi and A. Sawant; A 1 -connected components of schemes. Adv Math, Volume 282, 2016.
[3] M. Miyanishi; An algebraic characterization of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.
Let k be a field of characteristic p>0. In this talk, we consider the Z/pZ-actions on the affine n-space over k, or equivalently the order p automorphisms of the polynomial ring k[X] in n variables over k. For example, every automorphism induced from a G_a-action is of order p. Hence, the famous automorphism of Nagata is of order p. Such an automorphism is important to study the automorphism group of the k-algebra k[X].
We discuss two topics: (1) classification, and (2) the relation between polynomiality of the invariant ring and principality of the plinth ideal. We also present some conjectures and open problems.
doi: 10.1007/s00031-022-09764-2
Let k be an algebraically closed field of characteristic p > 0, n a positive integer, and V = k^d. Let G be a finite subgroup of GL(V) without pseudoreflections. Let S = Sym V be the symmetric algebra of V, and A = S^G be the ring of invariants. The functor (?)^G gives an equivalence between the category {}^*Ref(G,S), the category of Q-graded S-finite S-reflexive (G,S)-modules and the category {}^*Ref(A), the category of Q-graded A-finite A-reflexive A-modules. As the ring of invariants of the Frobenius pushforward ({}^e S)^G is the Frobenius pushforward {}^eA, the study of the (G,S)-module {}^e S for various e yields good information on {}^eA. Using this principle, we get some results on the properties of A coming from the asymptotic behaviors of {}^eA. In this talk, we talk about the following:
the generalized F-signature of A (with Y. Nakajima and with P. Symonds).
Examples of G and V such that A is F-rational, but not F-regular.
Examples of G and V such that (the completion of) A is not of finite F-representation type (work in progress with A. Singh).
Generalizing finite groups to finite group schemes G, we have that s(A)>0 if and only if G is linearly reductive, and if this is the case, s(A)=1/|G|, where |G| is the dimension of the coordinate ring k[G] of G, provided the action of G on Spec S is ‘small’ (with F. Kobayashi).