By Jon F. Carlson
The notes during this quantity have been written as part of a Nachdiplom path that I gave on the ETH in the summertime semester of 1995. the purpose of my lectures was once the improvement of a few of the fundamentals of the interplay of homological algebra, or extra particularly the cohomology of teams, and modular illustration thought. each time that I had given this type of path some time past fifteen years, the alternative of the cloth and the order of presentation of the implications have roughly a similar simple development. this type of direction all started with the basics of crew cohomology, after which investigated the constitution of cohomology earrings, and their maximal excellent spectra. Then the diversity of a module was once outlined and on the topic of real module constitution in the course of the rank type. purposes undefined. the normal strategy used to be utilized in my college of Essen Lecture Notes [e1] in 1984. Evens [E] and Benson [B2] have written it up in a lot clearer element and incorporated it as a part of their books at the topic.
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The two sub complexes Dt, D:; of C* 0 C* are isomorphic to (proj) D:;: 0 -----+ 02n(k) EEl (proj) -----+ on(k) -----+ 0 -----+ 0, whose homologies are concentrated in degree 2 and with ( equivalent to (. In particular, we have ima = ker( ~ LC;, so that H 2(Dt) ~ O(Lc;) EEl (proj). Also H 2(D:;) ~ on(Lc;) EEl (proj). 8). 11 can be adapted to show also that if p > 2 and n is odd, then no nonzero element ( E Hn(G, k) has the property that ( annihilates the cohomology of LC;. For p = 2, the situation is far more complicated, and in general the question of when ( annihilates the cohomology of LC; is an open one.
Assume that ( -=I- 0, as the result is obvious otherwise. Then we have an exact sequence o -----+ LC; ~ on(k) ~ k ---+ where ( is a co cycle representing (. Applying &tkdLc;, sponding long exact sequence and a commutative diagram 0 ) we obtain a corre- ... ~ &t;c1(LC;, k) ~ &tk'dLc;,Lc;) ~ &tk'c(Lc;,on(k)) 1, 1, 1, c;. ~ c;. ~ Modules and group algebras 42 where the vertical maps are all multiplication by (. Notice that Extk'G(Lc;:, nn(k)) ~ Ext~Gn(Lc;:, k), and the connecting homomorphisms are likewise multiplications by (.
Next let's look at the dihedral group of order 8, G for p = 2. Let A := x = Ds = (x,y I x 2 = y2 = (xy)4 = 1) + 1, kG = B := Y + 1. Then one computes that k(A, B)/(A 2 , B2, ABAB + BABA) and rad kG = (A, B). Here k(A, B) is the polynomial ring over k in the noncommuting variables A and B. 48 Modules and group algebras Again, our aim is to determine the cohomology ring by looking at diagrams. VA Xl r t • Bt • • • tA At B\! • • VAB\! • • VA tB S • X2 AV • ",B AV • ",B L--t • Bt • At • B\!