189-571B: Higher Algebra II

To be presented: Wednesday, March 21.

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. (*) Reading assignment: Read carefully chapter 6.3 of the handout (The Long Exact Homology Sequence) and master all the steps in the proof of theorem 6.1. You will find it helpful, while reading the proof, to draw the relevant commutative diagram and translate the somewhat cumbersome algebraic formulae into a "diagram chase".

Written assignment:

a) Give an example of a ring R, a surjective morphism

f: C --> C''

in the category of complexes of R-modules, and an integer i for which the morphism Hi(f) is not surjective. Letting C' be the kernel of f, compute the connecting homomorphism from Hi(C'') to Hi-1(C') in this case.

b) Give an example of a ring R, an injective morphism

f: C' --> C

in the category of complexes of R-modules, and an integer i for which the morphism Hi(f) is not injective. Letting C'' be the cokernel of f, compute the connecting homomorphism from Hi+1(C'') to Hi(C') in this case.

2. (Marni Mishna) Let (C,d) be a bounded positive complex of finite-dimensional F-vector spaces (i.e., Ci=0 for i<0 and for all i sufficiently large). Let ri and si denote the dimensions of Ci and Hi(C) respectively. Show that the alternating sum

r0 - r1+r2 -r3 + ...

is equal to

s0 - s1+s2 -s3 + ...

3. (Dan Segal) Let X be a smooth region in Rn. Define a co-chain complex by letting Ck be the R vector space of k-differentials on X, i.e., linear combinations of expressions of the form

f(x1,...,xn) dxi1 dxi2 ... dxik

where the f are infiinitely differentiable functions on Rn, and letting dk :Ck --> Ck+1 be the differential map that arose in advanced calculus during your discussion of Stoke's theorem. (If it did not, or your memory is hazy, come see me...)

a) Show that C is indeed a cochain complex.

b) Compute the cohomology of this complex in the case where X is the punctured unit disk in R2.