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Commutative Algebra Homework 5

This document contains homework problems on commutative algebra concepts: 1) The snake lemma and constructing a long exact sequence from a diagram of modules with exact rows. 2) Showing various equivalences relating projective, flat, and Ext modules. 3) Proving that homology, tensor products, and Tor commute with direct limits. 4) Showing that over a noetherian local ring, finitely generated flat modules are free, and that for finitely generated modules over a noetherian ring, flat is equivalent to being locally free.

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73 views2 pages

Commutative Algebra Homework 5

This document contains homework problems on commutative algebra concepts: 1) The snake lemma and constructing a long exact sequence from a diagram of modules with exact rows. 2) Showing various equivalences relating projective, flat, and Ext modules. 3) Proving that homology, tensor products, and Tor commute with direct limits. 4) Showing that over a noetherian local ring, finitely generated flat modules are free, and that for finitely generated modules over a noetherian ring, flat is equivalent to being locally free.

Uploaded by

Wes Ley
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Homework 5 for Math 215A Commutative Algebra

Burt Totaro
Due: Monday, October 29, 2012

Rings are understood to be commutative, unless stated otherwise.


(1) (Snake lemma) Suppose that

0 / M1 / M2 / M3 / 0
f g h
  
0 / N1 / N2 / N3 / 0

is a commutative diagram of R-modules with exact rows. The snake lemma says
that there is a long exact sequence of R-modules:

0 → ker f → ker g → ker h → coker f → coker g → coker h → 0.

Do the following steps of the proof. (a) Define the “boundary map” ker h →
coker f . (The other maps in the sequence should be clear.) Show that your definition
of the boundary map is independent of choices in your construction. (b) Show that
the sequence is exact at ker h. (You should be able to check exactness of the sequence
at any step, upon request.)
(2) Show that the following are equivalent, for a module M over a ring R.
(1) M is projective. (2) ExtiR (M, N ) = 0 for all R-modules N and all i > 0.
(3) Ext1R (M, N ) = 0 for all R-modules N . Likewise, show that the following are
equivalent, for a module M over a ring R. (1) M is flat. (2) TorR
i (M, N ) = 0 for all
R
R-modules N and all i > 0. (3) Tor1 (M, N ) = 0 for all R-modules N .
(3) (a) Show that homology commutes with direct limits. That is, suppose we
are given a directed system of two-step complexes Bα → Cα → Dα of R-modules, for
α running over a directed set A. (A “directed system of complexes” means a functor
from a directed set A to the category of complexes.) Show that the homology of
the complex lim Bα → lim Cα → lim Dα is isomorphic to the direct limit of the
−→ −→ −→
homology of the original complexes. (Use the universal property of the direct limit
to define a map.)
(b) It is a general result of category theory that any functor which is a left
adjoint (that is, which has a right adjoint) preserves all colimits which exist in
the domain category. Deduce that tensor products commute with direct limits
in each variable. That is, given a directed system of R-modules Nα , show that

1
M ⊗R lim Nα is isomorphic to lim(M ⊗R Nα ). (The same argument shows that
−→ −→
(lim Mα ) ⊗R N ∼= lim(Mα ⊗R N ).)
−→ −→
(c) Deduce that Tor commutes with direct limits in each variable.
(4) (a) Let M be a finitely generated flat module over a noetherian local ring
R. Show that M is free. (Hint: Choose elements x1 , . . . , xn of M which map to
a basis for the R/m-vector space M/mM = M ⊗R R/m. By Nakayama’s lemma,
we know that x1 , . . . , xn generate M as an R-module. Show that the resulting map
R⊕n → M is an isomorphism.)
(b) Let M be a finitely generated module over a noetherian ring R. Show that
the following are equivalent:
(1) M is flat;
(2) Mp is a free Rp -module for all prime ideals p in R;
(3) Mm is a free Rm -module for all maximal ideals m in R.
In short, “flat = locally free”, for finitely generated modules over a noetherian
ring.

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