Let be a commutative ring of characteristic and an -module. A -linear map is an additive map that satisfies for all . The set of all -linear maps is an abelian group. Since composing a -linear map and a -linear map is a -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 as the graded, associative, not necessarily commutative ring . They proved that is principally generated as ()-algebra, and that if is a complete -dimensional local Cohen-Macaulay ring, then is also principally generated as ()-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 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 the injective envelope of the residue field of .
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 defined by a square-free monomial ideal , where is a field of prime characteristic , let be the injective envelope of its residue field. Then, we may give a precise description of that shows that this algebra can only be principally generated or infinitely generated depending on the minimal primary decomposition of . 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 , its Matlis dual algebra of Cartier operators is always gauge bounded, a notion introduced by M. Bilckle in Test ideals via algebras of -linear maps, arXiv/0912.2255, which implies that the set of -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.