STNB 2015 (29th year)

Contributions to the study of Cartier algebras and differential operators


Alberto Fernandez Boix


Introduced by M. Blickle and K. Schwede (see [BS13] and the references therein), the so-called Cartier algebras play an important role in the study of, on one hand, singularities in prime characteristic and, on the other hand, differential operators. Roughly speaking, a Cartier algebra is just a non-commutative ring on which one collects certain homogeneous functions of degree $1/p^e=p^{-e}$, where $e\geq 0$ is any non-negative integer.

From now on, let $\mathbb{K}$ be any finite field of prime characteristic $p$, let $S=\mathbb{K} [x_1,\ldots ,x_d]$ be the ring of polynomials in $d$ variables with coefficients in the field $\mathbb{K}$, let $\mathcal{C}^S$ be the Cartier algebra attached to $S$, and let $\mathcal{D}_S$ be the ring of differential operators associated to $S$. The purpose of this talk is to introduce two algorithms related with Cartier algebras and differential operators. First of all, we describe a method which calculates all the ideals of $S$ contained in $\langle x_1,\ldots ,x_d\rangle$ fixed with respect to a subalgebra of $\mathcal{C}^S$ generated by one homogeneous element. On the other hand, we provide a procedure which produces a differential operator $\delta\in\mathcal{D}_S$ such that $\delta (1/f)=1/f^p$, i.e. a differential operator that acts as the Frobenius homomorphism on $1/f$; as a byproduct of this method, we describe a new characterization of ordinary and supersingular elliptic curves over $\mathbb{F}_p$. Moreover, we also explore the case of homogeneous quadrics (aka quadratic forms).

The content of this talk is based, on one hand, in a joint work with Mordechai Katzman (see [BK14] and [BK13]) and, on the other hand, in a joint work with Alessandro De Stefani and Davide Vanzo (see [BDSV]).


Download article.