I’d like to make something more explicit than I have. An effective Cartier divisor on a scheme is a closed subscheme locally cut out by one function, and that function is not a zero-divisor. (Translation: the zero-set does not contain any associated points.) PicX = group of line bundles = Cartier divisors modulo linear equivalence = Cartier divisors modulo principal Cartier divisors. We get a m...

I have one update from last time, and this is aimed more at the experts. Rob pointed out that there was no reason that we know that a Cartier divisor can be expressed as a difference (or quotient) of effective Cartier divisors. More precisely, a Cartier divisor can be described cohomologically as follows. Let X be a scheme. We have a sheaf O of invertible functions. There is another sheaf K tha...

We fix an integer N ≥ 1 and an elliptic curve E over Z[1/6N ], given by an equation y = f(x) with f(x) a cubic in Z[1/6N ] whose discriminant is invertible in Z[1/6N ]. On E we have the differential of the first kind ω = dx/y and the differential of the second kind η = xdx/y. For each prime p not dividing 6N , we look at this data mod p, and apply the Cartier operator Cp. We get quantities αp, ...

Let G = SpecA be an affine functor of monoids. We prove that A∗ is the enveloping functor of algebras of G and that the category of Gmodules is equivalent to the category of A∗-modules. Moreover, we prove that the category of affine functors of monoids is anti-equivalent to the category of functors of affine bialgebras. Applications of these results include Cartier duality, neutral Tannakian du...

We improve an algorithm originally due to Chudnovsky and Chudnovsky for computing one selected term in a linear recurrent sequence with polynomial coefficients. Using baby-steps / giant-steps techniques, the nth term in such a sequence can be computed in time proportional to √ n, instead of n for a naive approach. As an intermediate result, we give a fast algorithm for computing the values take...

Consider a local one-parameter family of Noetherian schemes. More precisely, let f : X → S be a flat map of Noetherian schemes, where S stands for the spectrum of a discrete valuation ring. Let s and η denote the special and generic points of S; putXs := f (s) andXη := f (η). Assume thatXs is of pure dimension and has no embedded components. Let D be an effective Cartier divisor of X. View it a...

We provide a quiver setting for quasi-Hopf algebras, generalizing the Hopf quiver theory. As applications we obtain some general structure theorems, in particular the quasi-Hopf analogue of the Cartier theorem and the Cartier-Gabriel decomposition theorem.

The goal of this paper is to offer a new construction the de Rham-Witt complex smooth varieties over perfect fields characteristic $p$. We introduce category cochain complexes equipped with an endomorphism $F$ (of underlying graded abelian groups) satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object abstract analog Cartier isomorphism, elementary process ass...