Date: Fri, 5 Jun 2020 12:19:26 +0700
From: Nicolas Addington <adding@uoregon.edu>
Subject: supplemental

One more thing: I promised Ilan I'd send an email about the pairing

	Ext^i(L,M) tensor Ext^j(M,N)  ->  Ext^{i+j}(L,N)

which extends the usual composition

	Hom(L,M) tensor Hom(M,N)  ->  Hom(L,N)

that sends f tensor g to g \circ f.  So here it is.

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You can check that in general, if C^* and D^* are two complexes, then 
there's a natural Künneth map from H^i(C^*) tensor H^j(D^*) to H^{i+j} 
of the total complex of the double complex C^* tensor D^*.

Take a projective resolution

	... -> P^{-1} -> P^0 -> L -> 0

and an injective resolution

	0 -> N -> I^0 -> I^1 -> ...

Then Hom(P^*,M) is a complex whose cohomology is Ext^*(L,M), and 
Hom(M,I^*) is a complex whose cohomology is Ext^*(M,L), so we get a 
Künneth map from

	Ext^i(L,M) tensor Ext^j(M,N)

to H^{i+j} of the total complex of the double complex

	Hom(P^*,M) tensor Hom(M,I^*).

Next, there's a composition map from that double complex to

	Hom(P^*,I^*),

and the cohomology of the total complex of the latter is Ext^*(L,M), as 
we saw in Lecture 25.

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The algebra Ext^*_R(R/m, R/m), where m is a maximal ideal in R, is often 
very interesting.  If R = k[x_1..x_n] and m = (x_1..x_n) then it's the 
exterior algebra on n generators.  If R = k[x_1..x_n]/quadratic form 
then it's related to the Clifford algebra.  This is the beginning of 
Koszul duality; Polishchuk and Positselskii's book "Quadratic Algebras" 
is one place to read.