Emily E. Witt

Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians

Abstract We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2 n × 2 n Pfaffians of a ( 2 n + 1 ) × ( 2 n + 1 ) generic skew-symmetric matrix. As an application, we characterize the Cohen–Macaulay modules of covariants for the action of the special linear group SL ( G ) on G ⊕ m . The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.

Lyubeznik numbers and injective dimension of local cohomology in mixed characteristic

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated

Local cohomology with support in ideals of maximal minors

\mathbb{Z}

Generalized Lyubeznik numbers

-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of

F-Pure thresholds of homogeneous polynomials

\mathbb{Z}

Lyubeznik numbers in mixed characteristic

-algebras, we also exhibit an example where this bound fails to hold. This example settles Lyubezniks question, and illustrates one way that the behavior of local cohomology modules of regular rings of equal characteristic and of mixed characteristic can differ.