Frobenius algebras of Stanley-Reisner rings

Let $R$ be a commutative ring of characteristic $p>0$ and $M$ an $R$-module. A $p^e$-linear map $\varphi_e:M\to M$ is an additive map that satisfies $\varphi_e(rm) = r^{p^e}\varphi_e(m)$ for all $r\in R, m\in M$. The set of all $p^e$-linear maps $\mathcal{F}^e (M)$ is an abelian group. Since composing a $p^e$-linear map and a $p^{e'}$-linear map is a $p^{e+e'}$-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 $\mathcal{F}(M):= \bigoplus _{e\geq 0} \mathcal{F}^e(M)$. They proved that $\mathcal{F}(R)$ is principally generated as ($\mathcal{F}^0(R) = R$)-algebra, and that if $(R, \mathfrak{m})$ is a complete $n$-dimensional local Cohen-Macaulay ring, then $\mathcal{F}(H_{\mathfrak{m}}^n(R))$ is also principally generated as ($\mathcal{F}^0(H_{\mathfrak{m}}^n(R)) = R$)-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 $I\subset k[[x_1, \dots , x_n]]$, where $k$ is a field of prime characteristic $p > 0$, let $E_R$ be the injective envelope of its residue field. Then, we may give a precise description of $\mathcal{F}(E_R)$ 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 $\mathcal{F}(E_R)$, its Matlis dual algebra $\mathcal{C}(R)$ of Cartier operators is always gauge bounded, a notion introduced by M. Bilckle in Test ideals via algebras of $p^{-e}$-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.