Algebraic computations in derived categories Original Research Article

This paper presents explicit algorithms for computations over a finite subspectroid of the bounded derived category of a finite spectroid. We will demonstrate methods for the construction of a projective resolution of a module and for finding the quiver of a finite spectroid given in terms of its radical spaces. This enables us to compute the endomorphism algebra of a tilting complex – or, in fact, any finite complex – in the derived category. In order to carry out these computations, we have to restrict to a finite base field or the field of rational numbers. We will show that it is possible to transfer the results to any extension of the base field, in particular to the algebraic closure.