STNB 2015 (29è any)
Contributions to the study of Cartier algebras and differential operators
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Resum
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]).
[BDSV] A. F. Boix, A. De Stefani, and D. Vanzo. An algorithm for differential operators in positive characteristic. In preparation. Draft available upon request.
[BK13] A. F. Boix and M. Katzman. FPureAlgorithm.m2: a Macaulay2 package for computing F-pure ideals with respect to principal Cartier algebras. Available at http://atlas.mat.ub.edu/personals/aboix/thesis.html, 2013.
[BK14] A. F. Boix and M. Katzman. An algorithm for producing F-pure ideals. Arch. Math. (Basel), 103(5):421–433, 2014.
[BS13] M. Blickle and K. Schwede. $p^{-1}$-linear maps in algebra and geometry. In Commutative algebra, pages 123–205. Springer, New York, 2013.
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