Varieties of modules for \(\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z\) (Q2470379)

Let \(k\) be an algebraically closed field of characteristic \(p\). The restricted nilpotent commuting variety, \({\mathcal C}^{nil}_1\), is the set of pairs of \(n\times n\) matrices \((A,B)\) such that \(A^p=B^p=[A,B]=0\). The variety \({\mathcal C}^{nil}_1\) can be identified with the ``variety of \(n\)-dimensional modules'' for the truncated polynomial ring \(k[X,Y]/(X^p, Y^p)\). As \(k[X,Y]/(X^p, Y^p)\) is isomorphic to the group algebra of \({\mathbb Z}/p{\mathbb Z}\times {\mathbb Z}/p{\mathbb Z}\), one can also speak about the variety of \(n\)-dimensional modules for \({\mathbb Z}/p{\mathbb Z}\times {\mathbb Z}/p{\mathbb Z}\). The main result of the article is the following Theorem. Suppose \(p=2\). (a) If \(n=2m\), then \({\mathcal C}^{nil}_1\) has \([m/2]+1\) irreducible components, each of dimension \(3m^2\). (b) If \(n=2m+1\), then \({\mathcal C}^{nil}_1\) has \(m+1\) irreducible components, each of dimension \(3m(m+1)\). Each irreducible component of \({\mathcal C}^{nil}_1\) is expressed as a direct sum of indecomposable components of varieties of \(k[X,Y]/(X^2, Y^2)\)-modules. It is also shown that, for general \(p\), \({\mathcal C}^{nil}_1\) is not necessarily equidimensional.