A note on the algebraic de Rham universal classes

摘要

This paper contains the algebraic analog of universal classifying bundles and Chern classes. We imitate the topological counterpart of universal bundles over the Grassmannian to construct some graded commutative differential algebras $hat{Omega}_{*} (hat{K} [X] / (X^2 - X, mathrm{tr}X - r))$ and $hat{Omega}_{*} (hat{K} [X] / (X^2 - X))$, whose corresponding cohomology are polynomial algebras isomorphic to $K [ar{c}_1, dotsc , ar{c}_r ]$ and $K [ar{c}_1, ar{c}_2, dotsc ]$ respectively, for the Chern classes $ar{c}_p$ with $p geqslant 1$, for the field $K = mathbb{Q}$, $mathbb{R}$ or $mathbb{C}$. Here $X$ denotes the infinite matrix $X = [X_{pq}]$, $X^n$ denotes the corresponding matrix obtained from $X$ by setting to zero the entries $X_{pq}$ when $p gt n$ or $q gt n$, and $(X^2 - X, mathrm{tr} X - r)$ (resp. $(X^2 - X)$) denotes the ideal generated by the power series $sum_p X_{pp} - r$ and the entries of the matrix $X^2 - X$ (resp. the entries of $X^2 - X$).