We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of k-dimensional subspaces in a vector space of dimension n. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when n and k are coprime these decompositions coincide and are minimal. In general, we conjecture minimality of the second decomposition.Exceptional collections can be considered as a kind of basis for the triangulated category: in this situation every object admits a unique filtration with the i-th quotient isomorphic to a direct sum of shifts of the i-th object in the collection.The simplest example of a variety with a full exceptional collection is a projective space. A. Beilinson showed in [Beȋ78] that the collection (O P n , O P n (1), . . . O P n (n)) is full and exceptional in D b (P n ).One can slightly generalize the notion of an exceptional collection and consider subcategories instead of single objects. For a full triangulated subcategory A of a triangulated category T the right orthogonal (resp. left orthogonal) to A in T is the full triangulated subcategory A ⊥ (resp. ⊥ A) consisting of all the objects T ∈ T such that Hom T (A, T ) = 0 (resp. Hom T (T, A) = 0) for all A ∈ A.

We prove that the derived category $D(C)$ of a generic curve of genus greater than one embeds into the derived category $D(M)$ of the moduli space $M$ of rank two stable bundles on $C$ with fixed determinant of odd degree.Comment: 20 pages; v2: exposition improved, some references adde

We show fullness of the exceptional collections of maximal length constructed by Kuznetsov and Polishchuk in the bounded derived categories of coherent sheaves on Lagrangian Grassmannians.