Non-commutative desingularization of determinantal varieties. I
Let \(K\) be a field and \(X=(x_{ij})\) a \((m\times n)\)-matrix of indeterminates over \(K\), \(n\geq m\). With \(S=K[x_{ij}]\), \(X\) determines the generic \(S\)-linear map \(\phi:S^n\rightarrow S^m.\) Let \(\text{Spec}R\) be the locus in \(\text{Spec}S\) where \(\phi\) has non-maximal rank: \(R\) is the quotient of \(S\) given by the maximal minors of \(X\), and is the generic determinantal variety. The classical \(R\)-modules \(M_a=\text{cok}\bigwedge^a_S\phi\) are maximal Cohen-Macaulay and are resolved by the Buchsbaum-Rim complex. In this article the authors prove that the \((M_a)_a\) yields a kind of non-commutative desingularization of the singular variety \(\text{Spec} R\): For \(1\leq a\leq m\) put \(M_a=\text{cok}\bigwedge^a_S\phi\) and \(M=\bigoplus_a M_a\). Then \(E=\text{End}_R(M)\) is maximal Cohen-Macaulay as an \(R\)-module with finite global dimension. That is, \(E\) is a non-commutative desingularization of \(\text{Spec} R\). If \(m=n\) then \(R\) is the hypersurface \(R=S/(\det\phi)\) and so \(R\) is Gorenstein and the non-commutative desingularization is an example of a non-commutative crepant resolution. The authors give a description by generators and relations of the non-commutative resolution \(E\) by stating that \(E\) as a \(K\)-algebra is isomorphic to the path algebra \(K\tilde Q\) of some quiver \(\tilde Q\). The results above are purely algebraic, but are proved by relating them to algebraic geometry. The classical fact that \(\text{Spec} R\) has a Springer type resolution of singularities is frequently used: Define the incidence variety \[ \mathcal Z=\{([\lambda],\theta)\in\mathbb P^{m-1}(K)\times M_{m\times n}(K)|\lambda\theta=0\} \] with projections \(p^\prime:\mathcal Z\rightarrow\mathbb P^{m-1}\) and \(q^\prime:\mathbb Z\rightarrow\text{Spec} R\). The key geometric facts then include: The scheme \(\mathcal Z\) is projective over \(\text{Spec} R\), which is of finite type over \(K\). The \(\mathcal O_{\mathcal Z}\)-module \[ \mathcal T := p^{\prime\ast}\left(\bigoplus^m_{a=1}\left(\bigwedge^{a-1}\Omega_{\mathbb P^{m-1}}\right)(a)\right) \] is a classical tilting bundle on \(\mathcal Z\) , i.e. (1) \(\mathcal T\) is a locally free sheaf, in particular, a perfect complex on \(\mathcal Z\), (2) \(\mathcal T\) generates the derived category \(\mathcal D(\text{Qch}(\mathcal Z))\); \(\text{Ext}^\bullet_{\mathcal O_{\mathcal Z}}(\mathcal T, C)=0\) for a complex \(C\) in \(\mathcal D(\text{Qch}(\mathcal Z))\) implies \(C\cong 0\), (3) \(\text{Hom}_{\mathcal O_{\mathcal Z}}(\mathcal T,\mathcal T[i])=0\) for \(i\neq 0\), (4) \(M\cong \mathbf{R}q^\prime_\ast\mathcal T\) and (5) \(E\cong\text{End}_{\mathcal Z}(\mathcal T)\). These geometric considerations leads to an interesting and important result stating that the variety \(\mathcal Z\) is the fine moduli space for the \(\tilde Q\)-representations \(W\) of dimension vector \((1,m-1,\left(\begin{smallmatrix} m-1\\2\end{smallmatrix}\right),\dots,1)\) that are generated by the last component \(W_m\). The proofs of the results depends mostly on the explicit computation of the cohomology of certain homogeneous bundles on \(\mathcal P^{m-1}\), determination of higher direct images of twisted bundles of homomorphisms between the modules of differential forms and other technical results. The article is more or less self contained, containing e.g. the construction of the projective tautological Koszul complex. In addition, of interest in itself is a construction of projective resolutions from sparse spectral sequences. This is then used in to construct the non-commutative desingularization \(E\) above, with algebra structure given by the quiverized Clifford algebra and its presentation. Particularly nice is the treatment of the noncommutative desingularization as a moduli space for representations. It is really interesting to notice that the points in \(\mathcal Z\) corresponding to the simple representations in \(W\) as those lying over the non-singular locus of \(\text{Spec} R\). The article is strongly recommended to anyone who will understand this level of representation theory in the algebraic geometric view. Be prepared to use a lot of effort to go through all proofs in detail.
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