Curtis homomorphisms and the integral Bernstein center for GL_n

We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GL_n(F) (that is, the center of the category of smooth W(k)[GL_n(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic l different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m at most n) implies the strong version of the conjecture. In a companion paper [HM] we show that the strong conjecture for n-1 implies the weak conjecture for n; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the integral Bernstein center for GL_n in purely Galois-theoretic terms; previous work of the author shows that such a description implies the conjectural "local Langlands correspondence in families" of Emerton and the author.Comment: Final accepted version. 36 pages. Note that the published version of Helm-Moss, "Converse theorems and the local Langlands correspondence in families" references an earlier version of this paper, and the section numbering has changed; in particular sections 9,10, and 11 of the referenced version correspond to sections 8,9, and 10 of the current version, respectivel

Bass Numbers of Semigroup-Graded Local Cohomology

Given a module M over a ring R which has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology of M at any Q-graded ideal I in terms of Ext modules. This method is used to obtain finiteness results for the local cohomology of graded modules over semigroup rings; in particular we prove that for a semigroup Q whose saturation is simplicial, the Bass numbers of such local cohomology modules are finite. Conversely, if the saturation of Q is not simplicial, one can find a graded ideal I and a graded R-module M whose local cohomology at I in some degree has an infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.Comment: 19 pages LaTeX, 1 figure (.eps) Definition 5.1 corrected; transcription error in Theorem 7.1.3 fixe