Commutative Algebra Seminar

The n-th power of an ideal is 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 improved. 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.

In this talk we will deal with the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we show a criterion that determines when a finite map can be deformed to a one--to--one map. We use this general result that holds for all dimensions, to construct new surfaces of general type with birational canonical map, for different $c_1^2$ and $\chi$ (the canonical map of the surfaces we construct is in fact a finite, birational morphism), addressing a question of Enriques posed in 1944. All known families until now were complete intersections or divisors in three folds. Our results enable us to describe some new components of the moduli of surfaces of general type. We find infinitely many moduli spaces $\mathcal M_{(x',0,y)}$ having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree $2$ morphism, the so called hyperelliptic components. In recent weeks we have proved analogues of these results for varieties of general type in all dimensions.This is part of joint work with F. J. Gallego and M. Gonzalez.

During this talk, we will discuss a theory of cohomological support defined for pairs of DG modules over a Koszul complex. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring. Moreover, the support sets introduced in this talk can be used to define a cohomological support for pairs of complexes over an arbitrary local ring. I will try to remark on what kinds of ring theoretic information are encoded in certain supports.

Abstract - In this talk we will give a brief introduction to the topic of big Cohen-Macaulay test ideals, a mixed characteristic analogue of the multiplier/test ideals defined in equal/positive characteristic. We will provide a formula to compute the BCM test ideal in the case of monomial ideals of mixed characteristic toric schemes. Of particular interest is that this formula is consistent with the formula for the multiplier ideal in equal characteristic and test ideal in positive characteristic.

Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elements generating the ideal form a regular sequence, generalizing a theorem of Bergh and Oppermann. It follows that in this case the Ext algebra is noetherian, providing a partial answer to a question of Kirkman, Kuzmanovich and Zhang.

The canonical Frobenius map defined for any prime characteristic ring can both reveal and obscure an incredible amount of data about the ring. In this talk, we will first discuss a closure operation (Frobenius closure) on ideals defined in terms of the Frobenius map, then outline a similar map (a Frobenius action) on local cohomology which can be used to define a variety of singularities in prime characteristic, and finally discuss connections between the Frobenius closure of parameter ideals and the Frobenius action on local cohomology.

Resolutions possessing DG-algebra structures are powerful tools. In 2019, Avramov, Iyengar, Nasseh, and Sather-Wagstaff show how the existence of a DG-algebra structure on a minimal resolution can be used to prove a strong Tor-rigidity result. We investigate when minimal resolutions do in fact possess DG algebra structures. In particular, we give an explicit construction of DG algebra resolutions for certain products of ideals and give sufficient conditions for this construction to be minimal.

The category FI and its variants have been of great interest recently. Being a finitely generated FI-module implies many desirable properties about sequences of symmetric group representations, in particular representation stability. We define a new combinatorial category analogous to FI for the 0-Hecke algebra, denoted by H, indexing sequences of representations of H_n(0) as n varies. We then provide examples of H-modules and use these to discuss some properties finitely generated H-modules possess, including a new form of representation stability and eventually polynomial growth.

Let R be a local ring of positive characteristic and X a complex with finitely generated nonzero homology and finite injective dimension. We prove that if derived base change of X via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then R is Gorenstein.

For a symmetric monoidal category C, the invertible objects with respect to the symmetric monoidal product often give rise to a group called the Picard group of C. This group contains important information about C, in particular, about its self-equivalences. An important symmetric monoidal category in homotopy theory and derived algebraic geometry is the stable homotopy category. Chromatic homotopy theory studies this category through the lenses of higher analogues of K-theory. In this talk, I will give some background on chromatic homotopy theory and discuss examples of Picard groups that arise in this context, along with tools that have been developed to compute them.

Classically, Bézout’s Theorem counts the number of intersections of two generic algebraic curves over an algebraically closed field. Using tools from A^1-homotopy theory and building on the work of Jesse Kass and Kirsten Wickelgren, we give an enrichment of Bézout’s Theorem over any perfect field. Over algebraically closed fields, this enrichment recovers the classical version of the theorem. Over non-algebraically closed fields, this enrichment imposes a relation on the tangent directions of the curves at their intersection points. As concrete examples, we will discuss Bézout’s Theorem over the reals and finite fields.

We extend Fedder's criterion for F-purity to singular ambient rings. As applications, we discuss several generalizations of Grifo-Huneke's result on symbolic power containments (Harbourne's conjecture) in regular rings to singular rings, either when the ideal has finite projective dimension or when we multiply by a power of the Jacobian ideal. We also give some examples showing the sharpness of these results. This is joint work with Eloisa Grifo and Karl Schwede.

For any polarized hyperkahler manifold of K3 type whose dimension is divisible by 8, we produce a Lagrangian submanifold which is Fano arising as a connected component of the fixed locus of an involution on the hyperkahler manifold. This is joint work with E. Macrì, K. O’Grady, and G. Sacca.

Closure operations are important tools in commutative algebra. The most well-known closure operation is tight closure introduced by Hochster and Huneke, which turns out to be very powerful and fruitful. Other examples are epf and r1f closures defined by Heitmann for rings of mixed characteristics. Dietz and R.G. axiomatized some nice properties of tight closure and its relation to big Cohen-Macaulay algebras/modules. In this talk, I will introduce their axioms and define a new closure operation called “weak epf closure” in mixed characteristic. This closure operation satisfies all axioms mentioned above, which suggests that it may be a good analog of tight closure. Other related results and open questions will be discussed.

The Hilbert-Kunz multiplicity is a prime characteristic numerical invariant that detects regularity. In this talk, I discuss a type of Bertini theorem for equidimensional (quasi)projective schemes which says that the measure of singularity given by Hilbert-Kunz does not get worse upon taking general hyperplane sections. The key technical tool is a uniform convergence result for multiplicities in families which is of independent algebraic interest. This is joint work with Rankeya Datta.