Workshop

Frobenius algebras of Stanley-Reisner rings

  • Santiago Zarzuela (Universitat de Barcelona, Spain)
G3 10 (Lecture hall)

Abstract

Let R be a commutative ring of characteristic p>0 and M an R-module. A pe-linear map φe:MM is an additive map that satisfies φe(rm)=rpeφe(m) for all rR,mM. The set of all pe-linear maps Fe(M) is an abelian group. Since composing a pe-linear map and a pe-linear map is a pe+e-linear map, this lead G. Lyubeznik and K. Smith to define in \textit{On the commutation of the test ideal with localization and completion}, Trans. Amer. Math. Soc. 353</b< (2001), 3149--3180, the ring of Frobenius operators of M as the graded, associative, not necessarily commutative ring F(M):=e0Fe(M). They proved that F(R) is principally generated as (F0(R)=R)-algebra, and that if (R,m) is a complete n-dimensional local Cohen-Macaulay ring, then F(Hmn(R)) is also principally generated as (F0(Hmn(R))=R)-algebra. Recently, M. Katzman has given in <i>A non-finitely generated algebra of Frobenius maps, Proc. Amer. Math. Soc, \textbf{138} (2010), 2381--2383, an example showing that this algebra may not be finitely generated, thus answering a question in this sense by Lyubeznik and Smith. The ring R in Katzman's example is a non-Cohen-Macaulay quotient of a formal power series ring in three variables by a square-free monomial ideal, and the module M the injective envelope of the residue field of R.

In this talk we extend Katzman's idea to study the ring of Frobenius operators of the injective envelope of the residue field of any Stanley-Reisner ring. Given a Stanley-Reisner ring R=S / I defined by a square-free monomial ideal Ik[[x1,,xn]], where k is a field of prime characteristic p>0, let ER be the injective envelope of its residue field. Then, we may give a precise description of F(ER) that shows that this algebra can only be principally generated or infinitely generated depending on the minimal primary decomposition of I. Examples will be given showing that one may find both non Cohen-Macaulay ideals with principally generated Frobenius algebra and Cohen-Macaulay ideals with infinitely generated Frobenius algebra. As an application, we will see that independently of the finite or non finite character of the Frobenius algebra F(ER), its Matlis dual algebra C(R) of Cartier operators is always gauge bounded, a notion introduced by M. Bilckle in Test ideals via algebras of pe-linear maps, arXiv/0912.2255, which implies that the set of F-jumping numbers of the corresponding generalized test ideals is always a discrete set.

This is a joint work with Josep Àlvarez Montaner and Alberto Fernàndez Boix.

Max Nitsche

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Jürgen Jost

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Jürgen Stückrad

Universität Leipzig