Math 423/502, Spring 2008
Final Exam
For simplicity, all rings are commutative with unit.
Problem 1.
Let Abe an abelian category.
(a) Explain what a chain complex in Ais.
(b) Explain what the homology of a chain complex is.
(c) Explain what a homomorphism of chain complexes is.
(d) Explain what a chain homotopy is.
(e) Prove that chain homotopic homomorphisms induce indentical homomorphisms
on homology.
Problem 2.
Let Rbe a ring and M,Ntwo R-modules. Explain how the R-modules Exti
R(M, N )
are constructed.
Problem 3.
Let Rbe a ring.
(a) Explain what a non zero divisor in Ris.
(b) Define the term projective dimension of an R-module M.
(c) Suppose xis a non zero divisor in R. Prove that R/xR has projective dimen-
sion 1.
(d) Give an example of a ring Rand a module M, such that the projectiv dimen-
sion of Mis infinite.
Problem 4.
Consider a ring R.
(a) Define the term global dimension of R.
(b) Explain why the global dimension of Zis 1.
(c) Give an example of a ring with infinitie global dimension.
Problem 5.
Suppose that f:X→Yis a ‘fibration’ of topological spaces, with fibre F. Suppose
further, that sufficient hypotheses are satisfied, such that the Leray spectral sequence
of freads
Ep,q
2=Hp(Y, Q)⊗Hq(F, Q) =⇒Hp+q(X, Q)
(a) Suppose that Hi(Y, Q) = Q, for i= 0,2,4, and 0 otherwise. Suppose that
Hi(F, Q) = Q, for i= 0,3, and 0 otherwise. Display graphically the E2-term
of this Leray spectral sequence in this case.
(b) What can you conclude about the cohomology of X, under these assumptions?