We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group $G$. Our generalisation, which we call the $G$-centre, is designed to control the endomorphism category of the grading shift functors. We show that the $G$-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory.

We exhibit a Cremona transformation of $\mathbb{P}^{4}$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel–Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.

We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai and Atiyah flops (of type A).

In this paper we introduce a generalization of Picard groups to derived categories of algebras. First we study general properties of this group. Then we consider easy particular algebras such as commutative algebras, where we reduce to the classical case. Finally, we define and study a homomorphism of the braid group to the Picard group of the derived category of a Brauer tree algebra. In the smallest case we show that this homomorphism is injective and that its image is of finite index.