Orbit closures and rank schemes
Let \(k\) be an algebraically closed field, and let \(A\) be a finitely generated associative \(k\)-algebra with unit. For each \(d\geq1\) we define the module scheme mod\(_{A}^{d}\) by mod\(_{A}^{d}\left( R\right) =\text{Hom} _{k\text{-alg}}\left( A,\mathbb{M}_{d}\left( R\right) \right) ,\) where \(\mathbb{M}_{d}\left( R\right) \) is the set of \(d\times d\) matrices with entries in \(R\). In particular, mod\(_{A}^{d}\left( k\right) \) can be identified with \(A\)-module structures on \(k^{d};\) furthermore, mod\(_{A}^{d}\) is an affine scheme, say mod\(_{A}^{d}=\text{Spec}\left( k\left[ \text{mod}_{A}^{d}\right] \right)\). The group scheme \(\text{GL} _{d}\) acts on mod\(_{A}^{d}\) by conjugation on its points -- let \(\mathcal{O} _{M}\) denote the \(\text{GL}_{d}\left( k\right) \)-orbit of a fixed \(M\in\)mod\(_{A}^{d}\left( k\right) .\) One can view \(\mathcal{O}_{M}\) as the \(A\)-module structures on \(k^{d}\) isomorphic to \(M.\) \ Understanding the closure of \(\mathcal{O}_{M}\) \ has proved difficult in general. For \(N\in\)mod\(_{A}^{d},\) given a \(p\times q\) matrix \(\underline{a}\) with coefficients in \(A\) one can naturally construct a \(pd\times qd\) matrix \(N\left( \underline{a}\right) \). Let \(\mathcal{I}_{M}\subset k\left[ \text{mod}_{A}^{d}\right] \) be the ideal generated by the minors of such matrices of size \(1+\)rk\ \(M\left( \underline{a}\right) ,\) and let \(\mathcal{C}_{M}=\text{Spec}\left( k\left[ \text{mod}_{A} ^{d}\right] /\mathcal{I}_{M}\right) \) -- this is a closed subscheme of mod\(_{A}^{d}\) containing \(\mathcal{\bar{O}}_{M}\) since these minors vanish. In the work under review, the authors study the properties of this scheme \(\mathcal{C}_{M}.\) Comparisons are made with schemes which arise from quiver representations. As an example, if \(Q\) us an equioriented Dynkin quiver os type \(\mathbb{A}\) then \(\mathcal{C}_{M}=\mathcal{\bar{O}}_{M}\) for \(M\) a representation in rep\(_{Q}^{\mathbf{d}}\left( k\right) \): this is a reformulation of a result from Lakshmibai and Magyar . Using \(\mathcal{C}_{M}\) instead of the orbit closure allows for a module-theoretic interpretation of the tangent space at some \(N\in\) \(\mathcal{\bar{O}}_{M}.\) This allows for a characterization of the singular locus of \(\mathcal{C}_{M}\) when \(A\) is representation-finite. This is useful when trying to describe the singular locus of \(\mathcal{O}_{M}.\)
scientific article; zbMATH DE number 6141063