Schedule for: 18w5133 - New Trends in Syzygies

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

I will discuss the basic theory of Koszul modules, which were originally introduced by Papadima and Suciu as a tool to study topological invariants of groups. A special instance of Koszul modules had previously appeared in David Eisenbud's ``Green's conjecture: an orientation for algebraists", where he proposed several programs for proving the Green conjecture for generic canonical curves. I will explain a vanishing theorem for Koszul modules that completes one of these programs, providing an alternative approach to the original proof of Voisin for the generic Green conjecture. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.

In ongoing joint work with Christine Berkesch and Daniel Erman we study the minimal resolution conjecture up to scaling. For Hilbert functions corresponding to modules of low regularity there always exist corresponding Betti tables with no consecutive cancellations up to scaling. For Hilbert functions of many naturally occurring modules, like coordinate rings of Veronese varieties, the Betti table can be semi-pure, even though the region of Hilbert functions corresponding to such tables is a tiny part of the cone of Hilbert functions.

We construct an explicit local duality map for codimension 2 toric ideals, thanks in part to the explicit free resolutions of Peeva--Sturmfels for such ideals. We then combine this with our work on the parametric behavior of the series solutions of an A-hypergeometric system to explain how local cohomology causes rank jumps.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Inspired by a question raised by Eisenbud-Musta\c{t}\u{a}-Stillman regarding the injectivity of maps from Ext modules to local cohomology modules, we introduce a class of rings which we call cohomologically full rings. In positive characteristic, this notion coincides with that of F-full rings studied by Pham and Ma, while in characteristic $0$, they include Du Bois singularities. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras. Joint work with Alessandro De Stefani and Linquan Ma.

A well-known formula expresses the socle of an Artinian complete intersection of characteristic zero in terms of a Jacobian determinant. We generalize this formula to rings that are neither Artinian nor complete intersections. We deduce our formula from an explicit description of the Dedekind complementary module of residual intersections. This is a report on joint work with David Eisenbud.

In 1891, Poincaré asked if it is possible to bound the degree of a projective plane curve that is left invariant by a vector field in terms of the degree of the vector field. In joint work with Chardin, Hassenzadeh, Simis, and Ulrich we address this question. The question can be restated as a problem about the initial degree of the module of derivations of the coordinate ring R of the curve modulo the Euler derivation in terms of invariants of R. We exhibit lower and upper bounds for this initial degree and in several instances we are able to determine the initial degree. Examples will be given to illustrate the situation.

Ananyan-Hochster's recent proof of Stillman's conjecture crucially uses the notion of strength of a polynomial. Inspired by this, we present two results which capture the idea that collections of polynomials of sufficiently high strength behave like regular sequences. Both of these results state that suitable limits (ultraproduct and inverse limit, respectively) of polynomial rings are themselves polynomial rings. I will discuss how the first result leads to a different proof of Stillman's conjecture. Time permitting, I will also discuss the inverse limit version and how it connects to recent work of Draisma on GL_infinity-noetherianity of polynomial functors.

We study rational maps from a projective space of dimension two to another of dimension three, both over the same field. We will start by giving the general framework and first results obtained on this question by Botbol, Busé and myself. Then I will turn to questions concerning the fibers of dimension one that such a map can have and present two way to address this question, the first by Tran Quang Hoa, and the second by the same author together with Dale Cutkosky and myself. Examples show that our estimates are pretty sharp, but leave possibilities for improvement.

Let $S = k[x_1, \ldots, x_n]$ be a polynomial ring and $R = S/P$ a complete intersection defined by pure powers of the variables. In this talk we discuss upper bounds for the Betti numbers of ideals of $R$ with fixed Hilbert function or Hilbert polynomial. We will consider both finite free resolutions over $S$ and infinite free resolutions over $R$. This is a joint work with Giulio Caviglia.

Let $k$ be a field and let $R$ be a quadratic standard graded $k$-algebra with $\dim_{k}R_2\le 3$. We construct a graded surjective Golod homomorphism $\varphi \colon P\to R$ such that $P$ is a complete intersection of codimension at most $3$. Furthermore, we show that $R$ is absolutely Koszul (that is, every finitely generated $R$-module has finite linearity defect) if and only if $R$ is Koszul if and only if $R$ is not a trivial fiber extension of a non-Koszul and non-Artinian quadratic algebra of embedding dimension $3$. In particular, we recover earlier results on the Koszul property of Backelin, Conca and D'Al\`i. This is joint work with R. Maleki.

Residual intersections of licci ideals behave well in a sense that residual intersection of ideals of 2x2 minors of a 2xn matrix, for n at least 4, do not. Recently Craig Huneke and Bernd Ulrich and I have been studying a way in which these residual intersections, and a certain wider class of ideals whose powers have linear presentations do behave well after all. I'll review the background and discuss what we've learned.

We present a generalization of Macaulay’s Inverse System to higher dimensions. To date a general structure for Gorenstein $k$-algebras of any dimension (and codimension) is not understood. We extend Macaulay's correspondence characterizing the submodules of the divided powers ring in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. We present effective methods for constructing Gorenstein graded rings with given numerical invariants fixed by their minimal free resolution. Recent generalizations by S. Masuti, M. Schulze and L. Tozzo will be presented. Possible applications are discussed.

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(\lambda)/(1 - \lambda)^d$, where $h_{S/I}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \cdots + h_s\lambda^s$ with $h_s \neq 0$ is the $h$-polynomial of $S/I$. It is known that, when $S/I$ is Cohen--Macaulay, one has $\reg(S/I) = \deg h_{S/I}(\lambda)$, where $\reg(S/I)$ is the (Castelnuovo--Mumford) regularity of $S/I$. In my talk, given arbitrary integers $r$ and $s$ with $r \geq 1$ and $s \geq 1$, a monomial ideal $I$ of $S = K[x_1, \ldots, x_n]$ with $n \gg 0$ for which $\reg(S/I) = r$ and $\deg h_{S/I}(\lambda) = s$ will be constructed. This is a joint work with Kazunori Matsuda.

Let S be a polynomial ring, I a homogeneous ideal and denote by in(I) the initial ideal of I w.r.t. some term order on S. It is well-known that depth(S/I) >= depth(S/in(I)) and reg(S/I) <= reg(S/in(I)), and it is easy to produce examples for which these inequalities are strict. On the other hand, in generic coordinates equalities hold for a degrevlex term order, by a celebrated result of Bayer and Stillman. In a joint paper with Aldo Conca, we prove that the equalities hold as well under the assumption that in(I) is a square-free monomial ideal (for any term order), solving a conjecture of Herzog. In this talk, after discussing where this conjecture came from, I will sketch the proof of its solution.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: --> the Lovasz-Saks-Schrijver ideal defining the d-dimensional orthogonal representations of the graph complementary to G and --> the determinantal ideal of the (d+1)-minors of a generic symmetric with 0s in positions prescribed by the graph G. In characteristic 0 these two ideals turns out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovasz-Saks-Schrijver ideal to the determinantal ideal. For Lovasz-Saks-Schrijver ideals we link these properties to combinatorial properties of G and show that they always hold for d large enough. For specific classes of graph, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovasz-Saks-Schrijver ideals.

Let R be a standard graded Gorenstein algebra over a field presented by quadrics. Conca-Rossi-Valla showed that such a ring is Koszul if reg (R)<= 2 or if reg(R)= 3 and codim(R)<= 4, and asked if this is true for reg(R)= 3 in general. We give a negative answer to their question by finding suitable conditions on a non-Koszul quadratic Cohen-Macaulay ring R that guarantee the Nagata idealization of R with the (twisted) canonical module is a non-Koszul quadratic Gorenstein ring.

A $(d-1)$-dimensional simplicial complex is called balanced, if its $1$-skeleton is $d$-colorable. In this talk, I will discuss upper bounds for the graded Betti numbers of the Stanley-Reisner rings of this class of simplicial complexes. Our results include both, bounds for the Cohen-Macaulay case and for the general situation. Previously, upper bounds have been shown by Migliore and Nagel, and Murai for simplicial polytopes, Cohen-Macaulay complexes and normal pseudomanifolds. If time permits, I will also mention, what can be said for balanced normal pseudomanifolds. This is joint work with Lorenzo Venturello.

The Lefschetz properties are desirable algebraic properties of graded artinian algebras inspired by the Hard Lefschetz Theorem for cohomology rings of complex projective varieties. A standard way to create new varieties from old is by forming connected sums. This corresponds at the level of their cohomology rings to an algebraic operation also termed a connected sum, which has recently started to be investigated in commutative algebra by Ananthnarayan-Avramov-Moore. It is natural to ask whether abstract algebraic connected sums of graded Gorenstein artinian algebras enjoy the Lefschetz properties in the absence of any underlying topological information. We investigate this question as well as the analogous question concerning a closely related construction, the fibered product. This is joint work with Chris McDaniel.

For a smooth variety $X$ and a very ample line bundle $\mathcal L$, $\mathcal O_X$ is $m$-regular with respect to $\mathcal L$ if $H^i(X, \mathcal L^{m-i})=0$ for all $i\ge 1$ and $reg_{\mathcal L}(\mathcal O_X):=\{m \mid \mathcal O_X \text{is $m$-regular}\}.$ For an embedded $n$-dimensional smooth variety $X$ in $\Bbb P^{n+e}$, Eisenbud-Goto conjecture tells us that $reg_{\mathcal L}(\mathcal O_X)\le deg(X)-codim(X)$. We will show that this conjecture is true for smooth varieties and classify boundary cases. On the other hand, there are many counterexamples for singular varieties due to McCullough and Peeva so that it would be desirable to understand the dichotomy between singular cases and smooth cases and how hard to show Castelnuovo normality bound for smooth varieties.

Hochster's results tell that homology groups of a simplicial complex have a nice relation to algebraic properties of its Stanley-Reisner ring. On the other hand, it is unknown that how fundamental groups affect to Stanley-Reisner rings. In this talk, we present lower bounds of the second h-number of simplicial complexes in terms of the number of generators of fundamental groups. Our proof is based on recent results about PL Morse inequality and graded Betti numbers.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.