Consistent algebras and special tilting sequences
We consider a nonsingular projective algebraic variety \(X\) over an algebraic closed field of characteristic 0 and \(\varepsilon = (E_0, \dots, E_N)\) an exceptional sequence of coherent sheaves on \(X\) with the additional conditions, that the \(E_i\) generate the derived category \({\mathcal D}^b(X)\) and there are \(\dim X + 1\) disjoint subsequences, every sheaf \(E_i\) is in one of these subsequences and for all \(E_i\) and \(E_j\) in the same subsequence we have \(\text{Ext}^l(E_i, E_j) = 0\) for all \(l \geq 0\). Then we consider the finite dimensional algebra \({\mathcal A} = \text{End} (\bigoplus E_i)\). We show that \(\bigoplus E_i\) is a tilting sheaf on \(X\) and \({\mathcal A}\) is a Koszul and coKoszul algebra. In the main result we construct an explicit functor \(\varphi : \text{mod-} \widehat {\mathcal B} \to C^b (X)\) from the module category of finite generated modules over the repetive algebra of the priddy dual algebra of \({\mathcal A}\) to the category of bounded complexes of coherent sheaves on \(X\) which induces a triangle equivalence \(\Phi : \underline {\text{mod}}-\widehat {\mathcal B} \to {\mathcal D}^b(X)\) of the corresponding stable module category with the derived category of coherent sheaves on \(X\).
stable module category of a selfinjective algebra
exceptional sequence of coherent sheaves
scientific article; zbMATH DE number 856559