Generalized $n$-series and de Rham complexes

Abstract

The goal of this article is to study some basic algebraic and combinatorial properties of "generalized $n$-series" over a commutative ring $R$, which are functions $s:{\mathbf{Z}}_{\ge 0}\to R$ satisfying a mild condition. A special example of generalized $n$-series is given by the $q$-integers $\frac{q^n-1}{q-1}\in \mathbf{Z}[[q-1]]$. Given a generalized $n$-series $s$, one can define $s$-analogues of factorials (via $n{!}_s={\prod }_{i=1}^{n}s(n)$) and binomial coefficients. We prove that Pascal's identity, the binomial identity, Lucas' theorem, and the Vandermonde identity admit $s$-analogues; each of these specialize to their appropriate $q$-analogue in the case of the $q$-integer generalized $n$-series. We also study the growth rates of generalized $n$-series defined over the integers. Finally, we define an $s$-analogue of the ($q$-)derivative, and prove $s$-analogues of the Poincaré lemma and the Cartier isomorphism for the affine line, as well as a pullback square due to Bhatt-Lurie.

51 pages. Comments very welcome!