By Luchezar L. Avramov, Mark Green, Craig Huneke, Karen E. Smith, Bernd Sturmfels

ISBN-10: 0511161484

ISBN-13: 9780511161483

ISBN-10: 0521831954

ISBN-13: 9780521831956

This ebook is predicated on lectures through six across the world recognized specialists awarded on the 2002 MSRI introductory workshop on commutative algebra. They specialize in the interplay of commutative algebra with different components of arithmetic, together with algebraic geometry, crew cohomology and illustration conception, and combinatorics, with all worthy heritage supplied. brief complementary papers describing paintings on the study frontier also are incorporated. the bizarre scope and layout make the ebook valuable examining for graduate scholars and researchers attracted to commutative algebra and its quite a few makes use of.

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**Example text**

These include matrix groups over the p-adic integers ∧ such as SL(n, Z p ). The discussion for p-adic Lie groups translates into continuous cohomology the story for virtual duality groups, with the same shift in dimension. The way this works is as follows. 1], if G is a p-adic Lie group then G has a normal open subgroup H for which Hc∗ (H, Fp ) is the exterior algebra on Hc1 (H, Fp ), so that H is a Poincar´e duality group. Furthermore, Hc1 (H, Fp ) is a finite dimensional Fp -vector space whose dimension is equal to the dimension d of G as a p-adic manifold.

Then the nonmaximal homogeneous primes p for which 11 ∗ Ext∗∗ H ∗ (G,k)p (k(p), ExtkG (S, M )p ) = 0 for some simple kG-module S are exactly the primes corresponding to the varieties in VG (M ). The point of this conjecture is that it provides a method for characterizing ∗ VG (M ) just in terms of ExtkG (S, M ), without having to tensor M with the rather mysterious modules κV . 11. Duality Theorems In this section, we describe various spectral sequences which can be interpreted as duality theorems for group cohomology.

2) Hps,t H ∗ (G, k)p =⇒ Ip [d]. This is the Greenlees–Lyubeznik dual localized form of the Greenlees spectral sequence. So for example, taking p to be a minimal prime in H ∗ (G, k), this spectral sequence has only one nonvanishing column, and it follows that H ∗ (G, k)p is Gorenstein. This gives the following theorem. 3. If G is a finite group and k is a field then H ∗ (G, k) is generically Gorenstein. 2), described in [Benson 2001]. 2) of κV . Since the maximal elements of W have codimension one in V G , the cohomology of ˆ ∗ (G, FW ) = H ∗ (G, k)p .

### Trends in Commutative Algebra by Luchezar L. Avramov, Mark Green, Craig Huneke, Karen E. Smith, Bernd Sturmfels

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