Math 711, Fall 2005 Problem Set #5
Due: Thursday, December 22
1. Let Rbe Noetherian and let Gbe a finitely generated module that is locally free.
Suppose that Spec (R) is connected. Prove that the rank of GPover RPdoes not depend
on the prime P.
2. Let Rbe a semilocal Noetherian ring and Ga finitely generated module such that for
every maximal ideal mof R,Gmis free of rank r. Prove that Gis free of rank r.
(The results of 1. and 2. were assumed in the discussion of the trace of an endomorphism
of a module of finite projective dimension.)
3. Let Rbe a Cohen-Macaulay local ring of dimension dthat is a homomorphic image of a
complete regular local ring of dimension n. Let Ebe the injective hull of the residue class
field. Let ω= Extn−d
S(R, S), which is a finitely generated R-module. Consider the full
subcategory Cof R-modules whose objects are the finitely generated R-modules of finite
injective dimension. Are the functors HomR(ω , ) and Extd
R(E, ) isomorphic on C?
4. Let cbe a nonzero element of R. Let I#denote the tight closure of the ideal Iwith
respect to the family {cR}. Show that it is possible that (I#)#contains I#strictly.
5. Show that the canonical element of a local ring (R, m, K) of Krull dimenson dmay be
viewed as an element of TorR
dHd
m(R), K.