Resolutions and cohomologies of toric sheaves: the affine case (Q2855490)

The paper deals with local cohomologies of toric sheaves for affine toric varieties and equivariant resolutions. A. A. Klyachko proved that the category of reflexive toric sheaves is equivalent to the category of vector spaces with a certain family of filtrations. Using this, the author contributes in constructing minimal free resolutions in the smooth case and acyclic resolutions with respect to local cohomologies in general.NEWLINENEWLINESection 1 is an introductory part and gives an exposition of the main results of the paper. In section 2 some general facts on poset representations and graded modules are given. In section 3 the author presents technics for finding resolutions and local cohomologies of graded modules. The work is related to the results of D. Helm and E. Miller on injective resolutions. The following main result is stated (Theorem 3.27): Let \(E\) be a \(\mathbb Z^n\)-graded, finitely generated, reflexive \(K[x_1,\ldots,x_n]\)-module. Then the poset of nonzero graded Betti numbers is determined by the embedding of the poset given by the underlying vector space arrangement \(\mathcal V\) (vector space arrangement in \(E\) generated by the intersections of the filtration subspaces) into \(\mathbb Z^n\). For given \(X\in\mathcal V\), the corresponding Betti number depends only on \(\mathcal V\).NEWLINENEWLINEIn section 4 the developed machinery is used to compute local cohomology, Bass and Betti numbers of graded reflexive \(K[x_1,\ldots,x_n]\)-modules in the case when the associated filtrations generate hyperplane arrangements \(\mathcal V\) (Theorem 4.5 and 4.13). Thus, in sections 3 and 4 for the class of reflexive modules over a polynomial ring whose associated filtrations form hyperplane arrangements the author computes in combinatorial terms the \(\mathbb Z^n\)-graded Betti numbers and local cohomology.NEWLINENEWLINEFinally, in section 5 in the nonsmooth, simplicial case, for dimensions \(d\geq 3\) new examples of indecomposable maximal Cohen-Macaulay modules of rank \(d-1\) over \(K[\sigma_M]\) are constructed explicitly (Theorem 5.9). That is, \(\sigma\) is a simplicial, nonregular (strictly convex rational polyhedral) cone of dimension \(d\) and \(K[\sigma_M]\) is the coordinate ring of the normal affine toric variety \(U_\sigma\).