Derived category and so on

Question

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from algebraic geometry. I don't want a too rigorous approach, made of a lot of definition and propositions but instead I would like to find an introduction which gives the main ideas and many examples. Thank you

Answer 1

There are two "families" of derived categories people study in algebraic geometry: derived categories of (quasi-)coherent sheaves on a (locally noetherian) scheme and derived categories of constructible complexes on topological spaces/ the etale topology on algebraic varieties. The first case leads one to the world of "derived algebraic geometry" while the second case will lead you to perverse sheaves and the decomposition theorem. In each case you will get the Grothendieck six functor formalism $f^*,f_*,f^!,f_!,Hom,\otimes$ giving rise to the whole yoga of cohomology.

Important Theorems

In either case, there is an important theorem which elucidates what the derived pushforward functors do: it is cohomology and base-change. This states that for a cartesian square $$ \begin{matrix} & g' & \\ Y' & \to & Y \\ f' \downarrow & & \downarrow f\\ X' & \to & X \\ & g & \end{matrix} $$ we have an isomorphism $\mathbf{L}g^* \circ \mathbf{R}f_! \cong \mathbf{R}f'_!\circ \mathbf{L}g'^*$ in the derived category of $X'$. As an illustrative exercise, you should think of a fibration $f:Y\to X$ and $g:\{x\} \to X$ a point to see what this theorem is saying.

The second important theorem is the existence of the Grothendieck Spectral Sequence which states roughly that a composition of derived functors induces a spectral sequences: $$ \mathbf{R}G \circ \mathbf{R}F \Rightarrow \mathbf{R}(G\circ F) $$ The first couple chapters of Dimca's book on sheaves in topology discusses applications of this theorem.

Another important observation is that there always exists a rational morphism from a variety $X$ to $\mathbb{P}^1$. You can use the blow-up of $X$ along the undefined locus and get a pencil of varieties over $\mathbb{P}^1$. One nice "toy example" is given by the Weierstrass family of elliptic curves.

Derived Algebraic Geometry

This area is said to have been discovered through Serre's intersection formula which states that the intersection multiplicity of two subvarieties $V,V'\hookrightarrow X$ can be computing using the euler characteristic of $\mathcal{O}_V\otimes^\mathbf{L}\mathcal{O}_{V'}$ when localization at a connected component $Z \subset V\cap V'$. This formula allows one to construct intersection theory using the bounded derived category of coherent sheaves $D^b(X) := D^b(\text{Coh}(X))$ for a variety $X$ and the grothendieck group $K(X) := K(D^b(X))$. More can be found in

• Riemann-Roch Algebra; Fulton, Lang
• SGA 6

Other applications of derived categories in this land are in deformation theory/cotangent complexes and in derived noncommutative geometry where you consider a derived category as a "space". There is a theorem of Beilinson where you can "compute" the derived category of coherent sheaves for projective space over $\mathbb{C}$.

Perverse Sheaves

The other main application of derived categories is in the theory of perverse sheaves. Essentially, they are a tool which allows you to systematically keep track of intersection cohomology. You can find a good picturesque introduction in MacPhersons "Intersection Homology and Perverse Sheaves". For the construction of perverse sheaves, try looking at Geordie Williamson's notes from a summer school he gave. It is titled "An Illustrated Guide to Perverse Sheaves". Fortunately, these lectures were recorded and can be found on youtube.

Another great place to look for a more in-depth/geometric treatment is in the book

• D-Modules, Perverse Sheaves, and Representation Theory; Hotta Takeuchi Tanisaki

The original book introducing perverse sheaves

• Faisceaux pervers; Beilinson, Bernstein, Deligne Proves one of the central theorems in the field: the decomposition theorem. It states that the derived pushforward of the intersection cohomology sheaf splits (non-canonically) into a direct sum of intersection cohomology sheaves.

In addition, the book

• Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform; Kiehl, Weissauer

discusses the applications of perverse sheaves for proving an analogue of the Weil conjectures for singular varieties.

Answer 2

Gelfand and Manin's "Methods of homological algebra" explains the subject quite nicely, though it takes them some pages to develop the theory (and there are many typos, at least in the first edition). Personally, I also use the first three chapters of Huybrecht's "Fourier-Mukai transforms in algebraic geometry", which is a bit more condensed but has a nice overview of the situation in the algebro-geometric setting. For more advanced material you could have a look at the notes on http://therisingsea.org.

Answer 3

You'll find Bernhard Keller's notes for a short course on the subject in his web page. His exposition is characteristically lucid and clear. His focus is representation theory, so they may not match your interests, though.

I also like a lots the to-the-pointness approach taken by Dieter Happel in his book about triangulated categories.

Answer 4

I like "Sheaves in Topology" by Dimca a lot.

I also second the suggestion of Gelfand/Manin. They actually have two books for some reason, "Homological Algebra" and "Methods of Homological Algebra", which are quite similar but have slightly different focus/applications. Both of them are worthwhile, and I think either one of them could be useful to you.

Answer 5

As already said in a previous answer, Fourier-Mukai transforms in Algebraic Geometry by Huybrechts is a good reference. The first time I studied derived categories (and triangulated categories) I used the first part of Residues and Duality by R. Hartshorne.

Answer 6

If you interested in better understanding the triangulated category structure of the derived category, I suggest you to read chapter 1 of Neeman's "Triangulated Categories". You can find much results explained in a very precise way. A good exercise would be to compare the definitions and results that appear in the above textbook with those in the already suggested Gelfand/Manin book. In this second textbook a very explicit approach for the derived and homotopic categories is taken, and you can surely detect typos by rewriting all computations that are relevant for your studies.