# 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 be smooth and proper of dim n (smooth = geometrically regular and flat). Then if is locally free sheaf of finite rank on , then there exists natural perfect parings

Ogus proved this in the case 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 (, R Noeth) be projective, then there exists a canonical pair where the first term is in the derived category of quasicoherent complexes with coherent cohomology, and the trace map is from . This pair has the property that for all the composition

is an isomoprhism.

Later, Ogus proved that if is smooth of dim d then .

### Tor Stuff

The notation is A is a ring, M,N are A-modules and for a projective resolution . A handy fact is that flat modules N are acyclic. And Tor can be computed by taking a projective resolution of either term. Here’s a fun fact:

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

- M is free
- M is projective
- M is flat

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

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

Tensor with the residue field to get a long exact sequence

It follows that and Nakayama’s lemma says .QED

Here’s another result

Thm: If is projective and then 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.

## About this entry

You’re currently reading “Serre Duality and Tor Stuff,” an entry on Math Meandering

- Published:
- August 11, 2009 / 1:10 am

- Category:
- alg. geo., Ogus Excerpts

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