Math 797. Derived Categories
TuTh 10:00AM - 11:15AM, LGRT 177.

Professor: Jenia Tevelev
Office: LGRT 1235E.
Office hours: Wednesday 3-4 and by appointment. To schedule an appointment, e-mail me and suggest date and time either on M between 3 and 6 or on Tu or W between 4 and 7.

E-mail: tevelev(at)math.umass.edu

Class webpage: http://people.math.umass.edu/~tevelev/797_2018/

Books

[H] D. Huybrechts, Fourier Mukai Transforms. USA: Oxford University Press, 2006. Book on derived categories for algebraic geometers. Available on-line via UMass e-library.

[GM] S. Gelfand, Yu. Manin, Homological algebra. Springer, 1999. A classic book on triangulated categories.

[Y] A. Yekutieli, Derived Categories. A draft of the book on DG-enhancement of derived categories.

Surveys and lecture notes

A. Kuznetsov, Derived categories view on rationality problems, arXiv
A. Caldararu, Derived categories of sheaves: A skimming. arXiv.
R. Thomas, Derived categories for the working mathematician. arXiv.

My old lecture notes on homological methods, which you may find useful, although our class will have a different focus.

Topics of student projects

Beilinson spectral sequence (Huybrechts 8.3)
Orlov's blow-up formula (Huybrechts 11.2)
Derived equivalence of K3 surfaces (Huybrechts 10.2)
Bridgeland stability conditions on an elliptic curve (e.g. Paul's notes)
Windows into derived categories [DHL]



  Date    Topic     Assignments  
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[1] 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 Extra problems:
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 5
Apr 10
Apr 12
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. Additional problems:
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:
E⊗RHom(E,F)→F→LEF→
and
REF→F→E⊗RHom(F,E)*
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.

Course Grade

Your course grade will be based on two components, the homework and the project.

Homework

There will be 5 biweekly homework sets. Problems will be worth certain number of points depending on their difficulty with a total of 30 points for each homework.

To get an A+ you have to accumulate 100 points by the end of the semester.

Homework problems can be presented in two ways. You can come to my office and explain your solution orally. This can be done during office hours or by scheduling an appointment (see above). There will be no penalty for a wrong solution, in fact I will probably give you a hint. You can of course just turn in your solution in a written form. I expect all homeworks to be submitted by the deadline but in extenuating circumstances I may give an extension.

Project

The second component of your grade is the student presentation. I will announce possible topics during the second week of classes. My expectation is that you will work in teams of two students. Each presentation should be accompanied by a paper about your topic (about 15 pages). I will be happy to assist with projects (choosing literature, scope, etc.)