Commutative Algebra Seminar

It is elementary and well-known that a polynomial in one variable of degree d with coefficients in a field F has at most d zeros in F. An analogue of this for homogeneous polynomials is that a homogeneous polynomial in two variables of degree d with coefficients in F has at most d non-proportional zeros in F^2 (excluding the origin), or in other words, d projective zeros. We note that d is a "good" bound in the sense that if F has at least d elements, then there are polynomials of degree d that attain this bound. When F is a finite field with q elements, it makes sense to ask similar questions for the number of common zeros of systems of multivariable polynomials of a given degree. A remarkable answer for the general case of r linearly independent polynomials of degree d in m variables over the field with q elements was given by Heijnen and Pellikaan in 1998. The analogous problem for systems of multivariable homogeneous polynomials turned out to be more challenging, and answers were known only in the case of a single polynomial or a system of two linearly independent polynomials, thanks to the work of Serre (1991) and Boguslavsky (1997). For the general case, there was an elaborate conjecture by Boguslavsky and Tsfasman that remained open for almost two decades. In this talk we will outline some recent progress on this conjecture as well as some newer developments. This is based on a joint work with M. Datta, and also with P. Beelen and M. Datta

We discuss the conjectured curious behavior of the Betti numbers of the Frobenius powers of the maximal ideal in hypersurfaces k[x,y,z]/(f), where k is an infinite field of any positive characteristic p, such as 1000000000000066600000000000001. Indeed, if f is chosen generically, then high enough Frobenius powers of the maximal ideal have identical graded Betti numbers up to explicit shifts. The proof is based on our constructions of free resolutions of relatively compressed Artinian graded algebras. This is joint work with Hamid Rahmati and Rebecca R.G.

Let (R, m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. Let e(I,M) denote the Hilbert-Samuel multiplicity of M on the ideal I. Lech's inequality states that the set {length(R/I)/e(I,R)}, as I runs through all m-primary ideals, is bounded below by 1/d!e(m,R). Stückrad and Vogel showed that this set is not in general bounded above. However, they conjectured that whenever the completion of M is equidimensional that {length(M/IM)/e(I,M)} will indeed be bounded above. We prove this conjecture. This talk is based on joint work with Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao.

Suppose that I is a homogeneous ideal of height c in a polynomial ring. Such ideals have at least c minimal generators, and that is about all we can say in this generality. If however we impose bounds on the regularity of the ideal then there are surprising constraints on the number of generators (and other Betti numbers). In this talk I'll describe how Boij-Soederberg Theory gives rise to statements like this and how we use these methods to prove strong lower bounds for the total Betti number of modules with low regularity.

I will discuss an analogue of taking categories of (possibly graded) modules over a polynomial ring as a purely categorical construction. This requires the formalism of derivators, but in return this allows us to expand the construction to other areas of mathematics. These constructions satisfy some good properties also seen for rings.

In the derived category of modules over a noetherian ring a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of X. In this talk I will present some local to global type results for computing this invariant, and also discuss some applications.

In this talk I'll describe how a bicategorical gadget, called the shadow, allows one to extract many interesting algebraic and topological invariants. For example, in algebraic bicategories, one easily recovers group characters, and in certain topological categories, one recovers the Lefschetz number. I'll describe joint work with Kate Ponto generalizing work of Ben-Zvi--Nadler which allows us to simultaneously recover the Lefschetz theorem for DG-algebras due to Lunts and the theory of 2-characters due to Ganter Kapranov, along with many other results. Prerequisites: An appetite for category theory (but I will not assume knowledge of bicategories!). There will be some motivation from stable homotopy theory, but one need only believe in the stable homotopy category, not have knowledge of it.

Let R be a local ring of positive prime characteristic p. The notion of tight closure, developed by Hochster and Huneke, is a powerful tool in understanding Noetherian rings, and is also of intrinsic interest. In particular, we can classify singularities in terms of the “amount” of tight closure over all ideals. The nicest situation is when all ideals in the ring are equal to their own tight closure. Such a ring is called weakly F-regular. Unfortunately, tight closure does not necessarily localize well. In particular, the weak F-regularity property has not been shown to localize. In order to get around this problem, Hochster and Huneke defined strong F-regularity (for F-finite rings), a property which both localizes well and implies weak F-regularity. The converse has only been proven in a limited number of cases — it is known essentially only when R is Q-Gorenstein on the punctured spectrum (which settles the case of dimension at most 3). We show that if R is an excellent local ring and some symbolic power of an anti-canonical ideal has analytic spread two, then weak F-regularity implies strong F-regularity. There is reason to believe that this result combined with results from the MMP will solve the dimension 4 case. The techniques employed differ significantly from any previous methods. As a corollary of the proof, we obtain the result that, in this situation, the F-signature of R may be expressed not just as an infimum of relative Hilbert-Kunz multiplicities, but is achieved as an actual minimum. This work is joint with Thomas Polstra.

I will describe a Linearization Conjecture that identifies the spectral dualizing module of a p-adic analytic group in terms of a representation sphere built from its Lie algebra. We can prove this when the action is restricted to certain small finite subgroups. These results are enough to determine Spanier-Whitehead duals of some chromatically interesting spectra. This is joint work in progress with Beaudry, Goerss, and Hopkins.