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189-571B: Higher Algebra II

Assignment 11

To be returned: Wednesday, April 4.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) Let R be a ring and let M and N be R-modules. In each of the following cases, compute Ext1(M,N). Having determined its cardinality n, write down the n distinct isomorphism types of extnesions of M by N in the category of R-modules corresponding to the distinct elements of Ext1(M,N).

a) R=Z/pZ, M=N=Z/pZ. (p a prime number).

b) R=Z, M=N=Z/pZ, (p a prime number).

c) R=F[e]/(e2), with F a field. M=N=F, with the obvious R-module structure in which e acts as 0.

There will be no problems to do in class.