Derived Cat., and some Cohom. results

Here’s a brief rundown of what Ogus said starting the beginning of Feb. 09.

Setup: \mathscr{A} is a n abel. cat. \mathscr{C(A)} is cat. of complexes in \mathscr{A}; \mathscr{K(A)} is the cat. of complexes in \mathscr{A} with homotopy classes of morphisms.  Then the derived category D(\mathscr{A}) can be defined; its objects are again just complexes, but all quasi isomorphisms are declared to be isomorphisms.  In a nutshell, declaring something to be an isomorphism means that if I have a map in one direction C^\bullet \to A^\bullet that is a quasi isomorphism, and a map C^\bullet \to B^\bullet, then there is a morphism A^\bullet \to B^\bullet.  Of course there might not be any set/sheaf-theoretic map between complexes, so the way around this is just declaring a morphism A^\bullet \to B^\bullet to be something consisting of the above data.  So morphisms actually look like “roofs.”   This is explained well in Huybrechts book (Fourier-Mukai Transforms in Alg. Geo).  This also has a lot of the proofs (or at least references to) to the results presented here.

Also for any cat. with objects consisting of complexes, I can refer to the subcategories of bounded below, bounded above, and bounded by using the appropriate superscript \{ +, - , b \}.  I \subset \mathscr{A} is the subcategory consisting of injective objects.

Thm: D^+(\mathscr{A}) is equivalent to \mathscr{K^+(I)} 

Comments:  There’s a natural functor \mathscr{K(I)} \to D^+(\mathscr{A}).  It must be shown its fully faithful and essentially surjective.  This is done cleanly in Huybrechts.  Ogus did it with some technical lemmas like if u \colon A^\bullet \to B^\bullet is a map of complexes and id_{C(u)} \sim 0 meaning the identity morphism on the cone complex is homotopically trivial, then u has a homotopy inverse.

The up shot is there is a picture like this, for a functor F \colon \mathscr{A} \to \mathscr{B}, I have

\begin{array}{ccc} \mathscr{A} & \to & \mathscr{B} \\ \downarrow & \mbox{} & \downarrow \\ \mathscr{C(A)} & \to & \mathscr{C(B)} \\ \downarrow & \mbox{} & \downarrow \\ \mathscr{K(A)} & \xrightarrow{F} & \mathscr{K(B)} \\ \downarrow & \mbox{} & \downarrow \\ D(\mathscr{A}) & \mbox{} & D(\mathscr{B}) \end{array}

Call the bottom two vertical maps Q,Q' respectively.  In general, there is no m \colon D(\mathscr{A}) \to D(\mathscr{B}) making the bottom square commute.  But but the first theorem, there is a backwards map D(\mathscr{A}) \to \mathscr{K(I)} \subset \mathscr{K(A)} which can be used to define the right derived functor RF \colon D(\mathscr{A}) \to D(\mathscr{B}).  It doesn’t make the above square commute, but there is a natural morphism

Q' \circ F \to RF \circ Q

In fact, the RF is universal for this property.  Namley, any for any \widetilde{F} such that there is a morphism

Q' \circ F \xrightarrow{\alpha} \widetilde{F} \circ Q

  There is unique morphism RF \to \widetilde{F} such that \alpha is the composition

Q' \circ F \to RF \circ Q \to \widetilde{F} \circ Q

Then Ogus proceeded to define delta functors and sheaf cohomology in much the same way that is done in Hartshorne.  

Upper Shriek and more propositions

Recall given \theta \colon A \to B, and an A-module M, then there is B-module

\theta^*M = M\otimes_A B

But there’s another B-module

\theta^! M = \hom_A(B,M)

In fact the upper shriek functor can be extended to affine morphisms of certain schemes, vakil talks about this here. Upper shriek is right adjoint to pushforward.  So its left exact.  And its shows \theta_* has left and right adjoint, hence its exact.  

Prop: If \theta_* is exact, then \theta^! carries injective to injectives:

proof: If I is injective, need to show \hom( - , \theta^! (I)) is exact, but by adjointness:

\hom( - , \theta^!(I)) \cong \hom(- , I) \circ \theta_*

which is the composition of two exact functors, hence exact. QED.

Corollary: \theta_* \colon \mbox{Mod}_A \to \mbox{Mod}_B is exact and faithful, if \mbox{Mod}_A has enough injectives then so does \mbox{Mod}_B.

Thm: If X is a ringed space, then \mod{Mod}_X has enough injectives.  (proof similar to Hartshorne)

Thm: if f \colon X \to Y is morphims of ringed spaces and E \in \mbox{Mod}_X, f_*E \in \mbox{Mod}_Y, then

  1. R \Gamma(X,E) \cong R \Gamma(Y, Rf_*E)
  2. H^q(X,E) \cong H^0(Y, R^qf_*E)
  3. H^q(Y, f_*E) \cong H^q (X,E) 
  4. There is a spectral sequence E_i^{p,q} = H^p(Y, R^qf_*E) which converges to H^{p+q}(X,E).

Here’s some more propositions

Prop

  1. Injective objects are F-acyclic
  2. If E' is F-acyclic and 0 \to E' \to E \to E'' \to 0 exact, then F preserves exactness
  3. If E', E are F-acyclic so is E''

And finally, to end this post:

Prop: If \mathscr{A} \xrightarrow{F} \mathscr{B} \xrightarrow{G} \mathscr{C} funtors between ab. cat. with \mathscr{A}, \mathscr{B} having enough injectives, and G left exact, and for every injective I, F_*I is G-acyclic, then

R G \circ F \to R G \circ R F

is iso.

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