# Cohomological approach to the graded Berezinian

### Tiffany Covolo

University of Luxembourg, Luxembourg

## Abstract

We develop the theory of linear algebra over a $(Z_{2})_{n}$-commutative algebra ($n∈N$), which includes the well-known super linear algebra as a special case ($n=1$). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra $H$. Following a cohomological approach, we introduce analogues of the notions of trace and determinant. Our construction reduces in the classical commutative case to the coordinate-free description of the determinant by means of the action of invertible matrices on the top exterior power, and in the supercommutative case it coincides with the well-known cohomological interpretation of the Berezinian.

## Cite this article

Tiffany Covolo, Cohomological approach to the graded Berezinian. J. Noncommut. Geom. 9 (2015), no. 2, pp. 543–565

DOI 10.4171/JNCG/200