|Jan 23||Duality for abelian groups via derived category [Y, p.9-14]|
|Jan 25||Additive and abelian categories||[Huy, 1.1]|
|Jan 30||Adjoint functors||[Huy, 1.1]|
|Feb 1||Triangulated categories||[Huy, 1.2]|
|Feb 6||Semi-orthogonal decompositions||[Huy, 1.2]|
|Feb 8||Exceptional collections. Serre functor.||[Huy, 1.2, 1.4]|
|Feb 13||Homotopy category of an abelian category||[Huy, 2.1]|
|Feb 15||Localization in (triangulated) categories|| [Huy, 2.1]. HW 1 due. All problems are worth 2 points. From Huybrechts, Chapter 1: 1.14(i), 1.17, 1.18, 1.20, 1.36, 1.37, 1.44, 1.62, 1.63. Extra problems:|
I. Show that no locally free sheaves on P1 are projective objects of Coh(P1).
II. Show that the category of finite abelian groups has no non-zero projective objects.
III. Find the left adjoint functors to the fully faithful embeddings of categories of (a) fields into domains; (b) complete metric spaces into metric spaces; (c) abelian groups into groups.
IV. Show that the category of complexes C(A) of an abelian category A is abelian.
V. Show that a complex in C(A) is a projective object if and only if it is a split-exact complex of projective objects in A.
VI. In an abelian category, show that a morphism is a monomorphism (resp. an epimorphism) iff its kernel (resp. cokernel) is trivial.
|Feb 20||Derived categories.||[Huy, 2.1]|
|Feb 22||Bounded subcategories. Derived category of a semisimple category.||[Huy, 2.1]|
|Feb 27||Projective, injective subcategories. Derived category of modules over PID.||[Huy, 2.1]|
|Mar 1||Derived functors.||[Huy, 2.2]|
|Mar 6||Derived functors via adapted subcategories.||[Huy, 2.2].|
|Mar 8||Coherent sheaves on affine schemes||
HW 2 due. All problems are worth 2 points. From Huybrechts, Chapter 2:
2.7, 2.19, 2.20, 2.32, 2.37, 2.43, 2.52, 2.53, 2.54, 2.55. Extra problems:|
I. Let D be an additive category. Show that every morphism A->B in the homotopy category K(D) can be composed with an isomorphism B->C (in K(D)!) such that for each term An->Cn is isomorphic to the inclusion of a direct summand.
II. In class we checked that the homotopy category of an abelian category is triangulated. Show that in fact this is true for an additive category (don't check TR4).
III. Let f:A->B be a map of complexes in the homotopy category of an abelian category. Show that f is a homotopy equivalence if and only if Cone(f) is homotopy equivalent to 0.
IV. Let Q:D->DS be a right Ore localization of an additive category D in a right denominator set S. Let D->C be an additive functor such that F(s) is invertible for every s in S. Show that F uniquely factors through Q.
IV. Let Q:D->DS be a right Ore localization of an additive category D in a right denominator set S. Show that any two morphisms f,g:A->B in DS have a common denominator, i.e. can be written as Q(a)Q(s)-1 and Q(b)Q(s)-1 for some s in S.
V. Consider a sequence of morphisms 0->A->B->C->0 in an abelian category M. Show that it is an exact sequence if and only if it can be extended to a distinguished triangle A->B->C->A in the derived category D(M).
VI. Let D' be a full triangulated subcategory of a triangulated category D. Let D'' be a full subcategory in D of all objects isomorphic to objects in D'. Show that D'' is also triangulated.
|Mar 20||Coherent sheaves on schemes||[Huy, 3.1]|
|Mar 22||Derived functors on Db(X)||[Huy, 3.3].|
|Mar 27||Derived functors on Db(X) - II|
|Mar 29||Derived functors on Db(X) - III|
|Apr 3||Dualizing complex. Gorenstein schemes.|| HW 3 due. All problems are worth 2 points unless stated otherwise. From Huybrechts, Chapter 3: 3.27 (3 points), 3.30, 3.36, 3.39 (3 points), 3.42
I. Let X be a noetherian scheme. Show that D-(Coh X) is equivalent to a full triangulated subcategory in D-(QCoh X) of complexes with coherent cohomology.
II. Assume an abelian category A has enough injectives and let X be a bounded from below complex of injectives in K(A). Show that every quasi-isomorphism X->Y in K(A) is isomorphic to morphism which is a term-wise split injection.
III (3 points). An additive functor F:A->B of abelian categories (assuming A has enough injectives) has cohomological dimension at most n if RiF(X)=0 for i>n and every X in A. Show that in this case the derived functor RF exists in D(A) and its restriction to D+(A) is the usual derived functor.
IV. Let R be a commutative ring. A bounded complex A of R-modules has a Tor-dimension n if n is the smallest integer such that Tori(A,B)=0 for i>n and for every R-module B. Show that Tor-dimension is finite if and only if there is a quasi-isomorphism P->A where P is a bounded complex of flat R-modules.
|Apr 19||Arie. Orlov's blow-up formula.||HW 4 due. All problems are worth 2 points. From Huybrechts, 4.4, 5.5, 8.30, 8.31, 8.32.
I. Let M be a f.g. module over a local Noetherian ring A with residue field k. Show that M has a finite injective dimension if and only if Exti(k,M)=0 for all sufficiently large i.
II. Show that the previous problem also holds if M is a complex in D+(A) with finitely generated cohomology modules.
III. Show that injective sheaves of OX-modules are flasque and that flasque sheaves are acyclic for the functor of global sections.
IV. Let M be a f.g. module over a Noetherian ring R. Show that (sheafy=curly) Hom(M~,N~) is quasi-coherent (resp. coherent) for every (resp. finitely generated) R-module N. V. Show that the previous problem can fail if M is not f.g. (Hint: take M to be a direct sum of countably many copies of k[t]). VI. Let X be a Noetherian scheme. Let F be an object of D-(Coh X) and let G be an object of D+(QCoh X) (resp. D+(Coh X)). Show that (sheafy=curly) RHom(F,G) is an element of D+(QCoh X) (resp. D+(Coh X)).
VII. Show that every coherent sheaf on a regular scheme with an ample sheaf admits a finite resolution by locally free sheaves.
|Apr 24||Bowen. Derived equivalence of K3 surfaces.|
|Apr 26||Andreas. Perverse sheaves.|
|May 1||Sebastian. Derived categories of GIT quotients|| HW 5 due on May 5. All problems are worth 2 points. From Huybrechts, 5.9, 5.16, 8.2, 8.4, 8.5, 8.8, 8.36, 8.37, 10.17, 11.5,
11.11, 11.19. Additional problems:|
Let X be a smooth projective variety and let E be an exceptional object in D(X). For any F in D(X), consider triangles:
Objects LEF and REF are called the left (resp. right) mutations of F with respect to E.
A. Show that mutations of exceptional objects with respect to exceptional objects are themselves exceptional if the pair is exceptional.
B. Let E1, ..., En is an exceptional collection. Show that E1, ...Ei-1, LEiEi+1, Ei, Ei+2, ..., En and E1, ...Ei-1, Ei+1, REi+1Ei, Ei+2, ..., En are also exceptional.
C. Show that formulas of part B define the braid group action on the set of exceptional collections in D(X).
D. Show that two standard exceptional collections on the projective space are related through a sequence of mutations.