Outline Here is an outline of the mini-course. Introduction to Derived Categories The first lecture will be about the general theory of derived categories. 1. The Homotopy Category Mini-Course at MSRI, Jan-Feb 2013 2. The Derived Category 3. Derived Functors 4. Resolutions Amnon Yekutieli 5. DG Algebras (new section; change of numbering below) Department of Mathematics The second lecture will be on more specialized topics, leaning towards Ben Gurion University noncommutative algebraic geometry. 5. Commutative Dualizing Complexes Notes available at http://www.math.bgu.ac.il/~amyekut/lectures 6. Noncommutative Dualizing Complexes Updated notes , 29 Jan 2013 7. Tilting Complexes and Derived Morita Theory 8. Rigid Dualizing Complexes Due to the time constraint I had to leave out some important topics (such as DG algebras). Amnon Yekutieli (BGU) Derived Categories 1 / 65 Amnon Yekutieli (BGU) Derived Categories 2 / 65 1. The Homotopy Category 1. The Homotopy Category A homomorphism of complexes φ : M → N is a commutative diagram 1. The Homotopy Category −1 0 − dM dM Suppose M is an abelian category . · · · / M 1 / M 0 / M 1 / · · · (1.1) The main examples for us are these: φ−1 φ0 φ1 ◮ M Mod / −1 / 0 / 1 / A is a ring, and = A, the category of left A-modules. · · · N −1 N 0 N · · · dN dN ◮ (X, A) is a ringed space, and M = Mod A, the category of sheaves of left A-modules. in M. C M M A complex in M is a diagram Let us denote by ( ) the category of complexes in . −1 0 It is again an abelian category; but it is also a differential graded (DG) dM dM M = · · · → M −1 −−−→ M 0 −−−→ M 1 → ··· category , as we now explain. M i+1! i in such that dM ◦ dM = 0 . Amnon Yekutieli (BGU) Derived Categories 3 / 65 Amnon Yekutieli (BGU) Derived Categories 4 / 65 1. The Homotopy Category 1. The Homotopy Category Given M, N ∈ C(M) we let Note that the abelian structure of C(M) can be recovered from the DG structure as follows: i j j+i Hom M(M, N ) := Hom M(M , N ) 0 ∈Z Hom C M (M, N ) = Z Hom M(M, N ) , jY ( ) the set of 0-cocycles. ! and i Indeed, for φ : M → N of degree 0 the condition d( φ) = 0 is equivalent Hom M(M, N ) := Hom M(M, N ) . ∈Z to the commutativity of the diagram (1.1). Mi For φ ∈ Hom M(M, N )i we let i d( φ) := d N ◦ φ − (−1) φ ◦ dM . In this way Hom M(M, N ) becomes a complex of abelian groups, i.e. a DG Z-module. Amnon Yekutieli (BGU) Derived Categories 5 / 65 Amnon Yekutieli (BGU) Derived Categories 6 / 65 1. The Homotopy Category 1. The Homotopy Category Next we define the homotopy category K(M). Suppose K is an additive category, with an automorphism T called the translation (or shift, or suspension). Its objects are the complexes in M (same as C(M)), and A triangle in K is a diagram of morphisms of this sort: 0 Hom K(M)(M, N ) = H Hom M(M, N ) . α β γ ! L −→ M −→ N −→ T (L). In other words, these are homotopy classes of homomorphisms The name comes from the alternative typesetting φ : M → N in C(M). N a γ β There is an additive functor C(M) → K(M), which is the identity on ~ objects and surjective on morphisms. L α / M The additive category K(M) is no longer abelian – it is a triangulated A triangulated category structure on K is a set of triangles called category . Let me explain what this means. distinguished triangles , satisfying a list of axioms (that are not so important for us). Details can be found in the references [Ye5], [Sc], [Ha], [We], [KS1], [Ne2] or [LH]. Amnon Yekutieli (BGU) Derived Categories 7 / 65 Amnon Yekutieli (BGU) Derived Categories 8 / 65 1. The Homotopy Category 1. The Homotopy Category The translation T of the category K(M) is defined as follows. A triangle in K(M) is distinguished if it is isomorphic, as a diagram in K M i i+1 ( ), to the triangle On objects we take T (M) := M and dT (M) := −dM . On morphisms i i+1 it is T (φ) := φ . L −→ α M −→ cone( α) −→ T (L) Given a homomorphism α : L → M in C(M), its cone is the complex for some homomorphism α : L → M in C(M). T (L) cone( α) := T (L) ⊕ M = A calculation shows that K(M) is indeed triangulated (i.e. the axioms M that I did not specify are satisfied). M Z with differential (in matrix notation) For M ∈ K( ) and i ∈ we will write i T (d L) 0 M[i] := T (M), d := . " T (α) d M # the i-th translation of M. The relation between distinguished triangles and exact sequences will There are canonical homomorphisms M → cone( α) and be mentioned later. cone( α) → T (L) in C(M). Amnon Yekutieli (BGU) Derived Categories 9 / 65 Amnon Yekutieli (BGU) Derived Categories 10 / 65 1. The Homotopy Category 2. The Derived Category Suppose K and K′ are triangulated categories. A triangulated functor F : K → K′ is an additive functor that commutes with the translations, 2. The Derived Category and sends distinguished triangles to distinguished triangles. As before M is an abelian category. M M′ Example 1.2. Let F : → be an additive functor (not Given a complex M ∈ C(M), we can consider its cohomologies necessarily exact) between abelian categories. Hi(M) := ker(d i )/ im(d i−1) ∈ M . Extend F to a functor M M C C M C M′ (F ) : ( ) → ( ) Since the cohomologies are homotopy-invariant, we get additive functors in the obvious way, namely Hi : K(M) → M . C(F )( M)i := F (M i) i K M for a complex M = {M }i∈Z. A morphism ψ : M → N in ( ) is called a quasi-isomorphism if Hi(ψ) are isomorphisms for all i. The functor C(F ) respects homotopies, so we get an additive functor Let us denote by S(M) the set of all quasi-isomorphisms in K(M). K(F ) : K(M) → K(M′). This is a triangulated functor. Amnon Yekutieli (BGU) Derived Categories 11 / 65 Amnon Yekutieli (BGU) Derived Categories 12 / 65 2. The Derived Category 2. The Derived Category M M Clearly S( ) is a multiplicatively closed set, i.e. the composition of two Every morphism χ : M → N in K( )S(M) can be written as quasi-isomorphisms is a quasi-isomorphism. −1 −1 χ = Q(φ1) ◦ Q(ψ1 ) = Q(ψ2 ) ◦ Q(φ2) A calculation shows that S(M) is a left and right denominator set (as in K M S M ring theory). for some φi ∈ ( ) and ψi ∈ ( ). M The category K(M)S M inherits a triangulated structure from K(M), It follows that the Ore localization K( )S(M) exists. This is an additive ( ) category, with object set and the localization functor Q is triangulated. M M There is a universal property: given a triangulated functor Ob( K( )S(M)) = Ob( K( )) . F : K(M) → E There is a functor to a triangulated category E, such that F (ψ) is an isomorphism for K M K M Q : ( ) → ( )S(M) every ψ ∈ S(M), there exists a unique triangulated functor called the localization functor, which is the identity on objects. M E FS(M) : K( )S(M) → such that FS(M) ◦ Q = F. Amnon Yekutieli (BGU) Derived Categories 13 / 65 Amnon Yekutieli (BGU) Derived Categories 14 / 65 2. The Derived Category 2. The Derived Category It turns out that the abelian structure of M can be recovered from this Definition 2.1. The derived category of the abelian category M is the embedding. triangulated category D(M) := K(M)S M . ( ) Proposition 2.3. Consider a sequence α β The derived category was introduced by Grothendieck and Verdier 0 → L −→ M −→ N → 0 around 1960. The first published material is the book “Residues and M Duality” [Ha] from 1966, written by Hartshorne following notes by in . Grothendieck. This sequence is exact iff there is a morphism γ : N → L[1] in D(M) D M 0 D M such that Let ( ) be the full subcategory of ( ) consisting of the complexes α β γ whose cohomology is concentrated in degree 0. L −→ M −→ N −→ L[1] is a distinguished triangle. Proposition 2.2. The obvious functor M → D(M)0 is an equivalence. This allows us to view M as an additive subcategory of D(M). Amnon Yekutieli (BGU) Derived Categories 15 / 65 Amnon Yekutieli (BGU) Derived Categories 16 / 65 3. Derived Functors 3. Derived Functors Definition 3.1. A right derived functor of F is a triangulated functor 3. Derived Functors D M E As before M is an abelian category. Recall the localization functor RF : ( ) → , Q : K(M) → D(M). together with a morphism It is a triangulated functor, which is the indentity on objects, and η : F → RF ◦ Q inverts quasi-isomorphisms. of triangulated functors K(M) → E, Suppose E is some triangulated category, and F : K(M) → E a satisfying this condition: triangulated functor. We now introduce the right and left derived functors of F . These are (∗) The pair (R F, η ) is initial among all such pairs. triangulated functors RF, LF : D(M) → E Being initial means that if (G, η ′) is another such pair, then there is a unique morphism of triangulated functors θ : R F → G s.t. η′ = θ ◦ η.