Serre Duality and Tor Stuff

This post contains mostly statements of theorems, and essentially no proofs.  I guess I’m getting lazy.

Thm(Serre): Let X/k be smooth and proper of dim n (smooth = geometrically regular and flat).  Then if E is locally free sheaf of finite rank on X, then there exists natural perfect parings 

H^i(X,E) \times H^{n-1}(X, E^\vee\otimes \Omega^n_{X/k}) \to k

Ogus proved this in the case X/k is projective.  In fact Ogus went on to discuss duality in the derived category.  Along the way this definition came up: a complex of R-modules is strictly perfect if its bounded and all its terms are projective and finitely generated.  Then perfect is defined to be quasi isomrophic to a strictly perfect complex.  

Ogus talked about a dualzing sheaf and a trace map much like Harthshrone and proved the following thm called the general duality thm:

Thm: Let X/S (S = \mbox{Spec} R, R Noeth)  be projective, then there exists a canonical pair (\omega^\bullet_{X/S}, \mbox{tr}_{X/S}) where the first term is in the derived category D^+(X/S) of quasicoherent complexes with coherent cohomology, and the trace map is from R\Gamma(X, \omega^\bullet_{X/S}) \to R.  This pair has the property that for all E \in D^b(X) the composition 

R\hom(E, \omega^\bullet) \to R\hom(R\Gamma(E), R\Gamma(\omega^\bullet)) \to R\hom(R\Gamma(E), R)

is an isomoprhism.

Later, Ogus proved that if X/S is smooth of dim d then \omega_{X/S} \cong \Omega^d_{X/S}.

Tor Stuff

The notation is A is a ring, M,N are A-modules and for P^\bullet \to N a projective resolution Tor_i(M, N ) = Tor^{-i} (M, N ) = h^{-1}(M \otimes P^\bullet) .  A handy fact is that flat modules N are Tor(M, - ) acyclic.  And Tor can be computed by taking a projective resolution of either term.  Here’s a fun fact:

Prop: R is a Noeth. local ring and M is fin. gen. R -module.  TFAE

  1.  M is free
  2. M is projective
  3. M is flat
  4. Tor_1(M, R/m = k) = 0

proof: 1 implies 2 implies 3 implies 4 is straightforward.  For the last step, choose a basis to get an isomorhpism 

k^d \cong M \otimes k = M/mM

Nakayama’s lemma says that lifting the basis in M/mM to M gives a generating set, so there is a short exact sequence

0 \to K \to R^d \to M \to 0

Tensor with the residue field to get a long exact sequence

\to 0 = Tor_1(M, k) \to K\otimes k \to k^d \cong M\otimes k \to 0

It follows that K \otimes k = 0 and Nakayama’s lemma says K = 0.QED

Here’s another result

Thm: If X/R is projective and E \in Coh(X) then R\Gamma(X,E) \in D^b(R) is a perfect complex and E is flat over R.

This result is used to prove basic results usually termed under cohomology with base change.  A nice treatment is done in Mumford’s book ‘Abelian Varieties.’  I think at some point I’ll put up a post about it.

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