Frobenius and Cartier algebras of Stanley-Reisner rings
In this nice article, the authors prove that the ring of Frobenius operators on the injective hull of a Stanley-Reisner ring of prime characteristic is either principally generated or infinitely generated. Throughout, \(R=S/I\) where \(S=kx_1, \ldots, x_n\) is the power series ring over a field of characteristic \(p>0\) and \(I\) is a square-free monomial ideal. In this case, \(I=P_1 \cap P_2 \cap \cdots \cap P_s\), where \(P_i\) are the minimal primes of \(R\). Since \(\mathcal{F}(E_r) \cong \bigoplus_{e \geq 0}\;(I^{[p^e]}:_S I)/I^{[p^e]}\), their determination of the generators for \(\mathcal{F}(E_R)\) depend on computing \((I^{[p^e]}:_S I)\). In fact, they show that \[ (I^{[p^e]}:_S I)=(P_1^{[p^e]}:_S P_1)\cap (P_2^{[p^e]}:_S P_2) \cap \cdots \cap (P_s^{[p^e]}:_S P_s). \] In the case where \(R\) is Gorenstein or when \((I^{[p^e]}:_S I)=I^{[p^e]}+(x_1x_2\cdots x_n)^{p^e-1},\) \(\mathcal{F}(E_r)\) is principally generated. Otherwise, they use a result of \textit{M. Katzman} [J. Lond. Math. Soc., II Ser. 81, No. 3, 589--607 (2010; Zbl 1197.13006)], to show that \(\mathcal{F}(E_r)\) is infinitely generated. The authors have included some nice examples of equidimensional Stanley-Reisner rings of low dimension to illustrate the principal generation of \(\mathcal{F}(E_r)\) is not tied to how close the ring is to being Gorenstein. If \(I\) is of pure height \(n-1\) or the intersection of all face ideals of height \(k\), the authors show \(\mathcal{F}(E_r)\) is principally generated. As an application, the authors use the \(R\)-Cartier algebra \(\mathcal{C}(R)\) which is the opposite ring of \(\mathcal{F}(E_r)\) to extend work of \textit{M. Blickle} [``Test ideals via algebras of \(p^{-e}\)-linear maps'', \url{arXiv:0912.2255}] to show that the \(F\)-jumping numbers of the generalized test ideals \(\tau(R,\mathfrak{a}^t)\) are a discrete set.
scientific article