The moduli of representations with Borel mold (Q2878767)

Let \(X\) be a scheme and let \(M_n(\mathcal{O}_X)\) be the sheaf of \(n\times n\) matrices on \(X\). A subsheaf \(\mathcal{A}\subset M_n(\mathcal{O}_X)\) is defined in this paper to be a degree \(n\) \textit{mold} if both \(\mathcal{A}\) and \(M_n(\mathcal{O}_X)/\mathcal{A}\) are locally free sheaves on \(X\). A degree \(n\) mold \(\mathcal{A}\) is said to be \textit{Borel} if \(\mathcal{A}\) is locally equivalent to \(\mathcal{B}_n\otimes_{\mathbb{Z}}\mathcal{O}_X\) where \(\mathcal{B}_n:=\{(b_{ij})\in M_n(\mathbb{Z})\;|\;b_{ij}=0,\;i>j\}.\) More generally, and in a similar fashion, \textit{parabolic molds} are also defined. In the first section of this paper, the moduli space of degree \(n\) and rank \(d\) molds is constructed, and it is shown that it contains a clopen subscheme of parabolic molds.NEWLINENEWLINEThe remainder of the paper is devoted to representations, and their moduli. The second section begins by reminding the reader that for a finitely generated group \(\Gamma\), a representation of degree \(n\) on \(X\) is a group homomorphism \(\rho:\Gamma\to \mathrm{GL}_n(\mathcal{O}_X(X))\). For such a representation, the author defines \(\rho\) to be a \textit{representation with mold} \(\mathcal{A}\) if the subsheaf \(\mathcal{O}_X[\rho(\Gamma)]\) of \(\mathcal{O}_X\)-algebras of \(M_n(\mathcal{O}_X)\) is locally equivalent to \(\mathcal{A}\); and so a \textit{representation with Borel mold} is one having \(\mathcal{O}_X[\rho(\Gamma)]\) a Borel mold. From this, the scheme \(\mathrm{Rep}_n(\Gamma)_B\) of degree \(n\) representations with Borel mold is defined, as is its moduli scheme \(\mathrm{Ch}_n(\Gamma)_B\). It is noted that the canonical morphism \(\mathrm{Rep}_n(\Gamma)_B\to \mathrm{Ch}_n(\Gamma)_B\) is a principal fiber bundle with fiber \(\mathrm{PGL}_n\), locally trivial in the Zariski topology, and that \(\mathrm{Ch}_n(\Gamma)_B\) is a universal geometric quotient. The main theorem here states \(\mathrm{Ch}_n(\Gamma)_B\) is a separated scheme over \(\mathbb{Z}\) of finite type.NEWLINENEWLINEIn the third section further basic results are given along these lines, and in the fourth section the degree 2 case is explored in more detail.