2022 Fall. Derived categories
Lecture notes: https://drive.google.com/drive/folders/1Punl47XnvdA61xoFwHOqgse7gFFdbJye?usp=sharing
Zoom link for the online part 👑🦠:
https://upenn.zoom.us/j/93645842098?pwd=YWZ3QitldEUweUdPTzRaRmxCVGZpQT09
Passcode: meow
Topics for presentations
Students taking this class for a grade should prepare two presentations, each 30-40 min long, or one full lecture, 60-90 min. Here are suggested references:
(Taken) Hartshorne "Residues and Duality". Chapter II, proof of theorem 7.18.
(Taken) Generalities on Kan extensions. Explain "when specialized to posets, the definition of Kan extensions becomes a relatively familiar type of question on constrained optimization". Alternatively (or in addition!), cover Section 10.7 of Mac Lane, titled "All Concepts Are Kan Extensions".
Huybrechts "Fourier-Muaki transforms in algebraic geometry". Prove Corollary 2.68 on page 57 and explain how it gives us that the derived functors in AG defined on the level of D^+ QCoh coincide with the derived functors on the level of D^b Coh.
1985 Kapranov "On the derived category of coherent sheaves on Grassmann manifolds". Requires familiarity with Schur functors and representation theory of GL. You will need to formulate the Borel-Weil-Bott theorem, but don't prove it.
1989 Bondal, Kapranov. "Representable functors, Serre functors, and mutations". Prove Theorem 2.10, state Theorems 2.11, 2.14. Time permitting, prove Theorem 2.14.
1990 Bondal "Representations of associative algebras and coherent sheaves". Two choices:
Prove Theorem 4.1.
Prove Theorem 6.2.
2020 Pirozhkov "Stably semiorthogonally indecomposable varieties". May need more than 1 talk.
2020 Pirozhkov "Admissible subcategories of del Pezzo surfaces". May need more than 1 talk.
You are welcome to choose another relevant topic for a presentation.
Complexes and triangulated categories
8/31
Introduction and motivation. Overview of main examples and what will be covered.
9/7
Abelian categories and complexes. Exact sequences as puzzles. Triangulated categories.
9/12
CANCELED. Optional video 👑🦠
DERIVED CATEGORIES
9/14
INDEPENDENT READING 👑🦠 Homotopy category of complexes is triangulated.
9/19
ONLINE 👑🦠 Multiplicative systems, aka localizing classes. Localizations and roofs. The derived category D*A as the localization of K*A in Qis.
9/21
ONLINE 👑🦠 Understanding morphisms in derived categories. Injective and projective objects in abelian categories.
9/26
Injective resolutions, homotopy category of complexes of injective objects.
9/28
Thick abelian subcategories. Complexes with cohomology in a subcategory.
Derived functors
10/03
Definition of right derived functors as a left Kan extension. Existence of right derived functors.
10/05
Restriction of derived functors to subcategories.
10/10
Deriving Hom. Ext.
setting the stage in algebraic geometry
10/12
QCoh X, Mod O_X have enough injectives. When X locally noetherian, QCoh X has enough (Mod O_X)-injectives.
10/17
Db X = Db (Coh X). When X noetherian, Db X = Db_{Coh} (QCoh X). Derived functors of global sections, pushforward, local Hom.
10/19
Derived functors of dual, tensor product, inverse image. Decomposing complexes with disconnected support. For noetherian X, DbX is indecomposable <=> X is connected.
Semiorthogonal decompositions
10/24
Non-symmetric bilinear forms. Semiorthogonal decompositions. K_0.
10/26
Homological dimension. Derived categories of curves. SOD of Db(P1).
10/31
Most used identities: projection formula, base change, Künneth. Beilinson's collection generates Db(Pn).
11/2
Full exceptional collections. Fourier-Mukai functors. Orlov's theorem: every full embedding between categories of smooth projective varieties is a FM functor for a unique kernel.
11/7
Serre functors and Serre duality. Grothendieck duality.
11/9
Grothendieck duality as "Serre duality on steroids". Derived category remembers dimension.