We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of isogenous to given surface $X$ terms certain integral sublattices the second rational $\ell$-adic and crystalline cohomology groups $X$. This characteristic analog Huybrechts, extends results author. give applications reduction types surjectivity period morphism. To prove these we describ...

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences ...

The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct set equivalences corresponding to mutations an exceptional collection on the relevant Grassmannian, and give mirror symmetry interpretation, following physical analysis Eager, Hori, Knapp, Romo.

We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel’s theorem which states that each tilting triple yields an equivalence between derived categories. We establish its link with Rickard’s theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalenc...

Given an adjunction F ⊣ G connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calcul...

We extend Orlov’s result on representability of equivalences to schemes projective over a field. We also investigate the quasi-projective case.

We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are “componentwise” tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizi...

For any algebraically closed field k of positive characteristic p and any non negative integer n Külshammer defined ideals TnA ⊥ of the centre of a symmetric k-algebra A. We show that for derived equivalent algebras A and B there is an isomorphism of the centres of A and B mapping TnA ⊥ to TnB ⊥ for all n. Recently Héthelyi, Horváth, Külshammer and Murray showed that this holds for Morita equiv...