A note on the algebraic de Rham universal classes

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 $K [\bar{c}_1, \dotsc , \bar{c}_r ]$ and $K [\bar{c}_1, \bar{c}_2, \dotsc ]$ respectively, for the Chern classes $\bar{c}_p$ 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 $X^2 - X$).

Keywords

algebraic de Rham cohomology, Grassmannian, idempotent matrix, universal Chern class

2010 Mathematics Subject Classification

Primary 14F40, 55R40. Secondary 19A49.

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This research was partially supported by MEC-FEDER grant MTM2013-41768-P and JA grants FQM-213.

Received 20 January 2017

Received revised 22 February 2017

Published 15 November 2017