By Garrett Birkhoff

This vintage, written via younger teachers who turned giants of their box, has formed the certainty of recent algebra for generations of mathematicians and is still a beneficial reference and textual content for self research and faculty classes.

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**Extra resources for Álgebra Moderna**

**Example text**

The latter is a class of groups which includes arithmetic groups [Borel and Serre 1973], mapping class groups of orientable surfaces [Harer 1986] and automorphism groups of free groups of finite rank [Bestvina and Feighn 2000]. A discrete group G is said to be a duality group of dimension d over k (see [Bieri 1976]) if there is a dualizing module. This is defined to be a kG-module I such that there are isomorphisms H i (G, M ) ∼ = Hd−i (G, I ⊗k M ) for all kG-modules M . It turns out that such isomorphisms may be taken to be functorial in M if they exist at all, and in that case, I ∼ = H d (G, kG).

R for H ∗ (G, k) so that ζ1 , . . , ζh lie in p. So ζ1 , . . , ζh is a system of parameters for H ∗ (G, k)p . 1) for ζ1 , . . , ζh to obtain a complex Λ∗ (ζ1 , . . , ζh ) of the form h 0→k→ i=1 h Fζ i → · · · → i=1 Fζ i → 0 and then tensor the answer with the module FW to obtain a complex Λ∗p = Λ∗ (ζ1 , . . , ζh ) ⊗ FW whose cohomology is Ω−h EV ⊗ FW = Ω−h κV concentrated in degree h. 1) ˆ s+t (G, κV ). 2. 1 are isomorphic from the E2 page onwards. 1. 2 clearly holds in the case where H ∗ (G, k) is Cohen–Macaulay, because there is no room for nontrivial differentials or ungrading problems.

1, comes from an idea of Amnon Neeman. 1), and we can ask which nonmaximal prime ideals occur in such a decomposition. Since each Iˆj is coinduced from some injective H ∗ (G, k)-module Ij , we have Hom∗Hˆ ∗ (G,k) (H ∗ (G, k)/p, Iˆj ) ∼ = Hom∗H ∗ (G,k) (H ∗ (G, k)/p, Ij ), and this is nonzero exactly for the primes appearing in this minimal resolution. Since coinduction is exact on injectives away from the maximal ideal, it does not ˆ ∗ (G, k) or H ∗ (G, k). The following conjecture matter whether we resolve over H says that the primes appearing in these minimal resolutions, as S runs over the simple kG-modules, correspond exactly to the varieties in V G (M ).

### Álgebra Moderna by Garrett Birkhoff

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