Exercises
Exercise 1. Let X, Y be cochain complexes.
Assume that f, g : X → Y are homotopic cochain maps.
Show that the mapping cones Conef and Coneg are isomorphic
cochain complexes.
Exercise 2. Two R-modules M and M 0 are called projectively equivalent if there are two projective R-modules Q and Q0 such that M ⊕Q0 ∼
=
0
M ⊕ Q.
Let
(∗) 0 → K → Pn−1 → . . . → P0 → M → 0,
be an exact sequence with all Pi projective modules.
Show that the equivalence class of K depends only on M and not on
the choice of the sequence (∗).
(Hint: Use Schanuel’s Lemma)
Exercise 3. Let r ∈ Z(R) be a central element of a ring R and let A
be a right R-module.
ṙ
(1) Prove that the right multiplication by r, A → A, x 7→ xr is an
R-linear homomorphism.
(2) Assume that R is a commutative ring. Show that ṙ induces the
multiplication by r on Ext∗R (A, B) and Ext∗R (B, A).
Exercise 4. Let G be a finite abelian group. Using the previous exercise prove that there is an isomorphism Ext1Z (G, Z) ∼
= HomZ (G, Q/Z).
Exercise 5. For every R-module M , f.d.M and p.d.M denote the flat
and the projective dimension of M .
r.weak.dim R (resp. l.weak.dim R) denotes the supremum of the flat
dimension of right (resp. left) R-modules.
r.gl.dim R (resp. l.gl.dim R) denotes the supremum of the projective
dimension of right (resp. left) R-modules.
(1) Prove that f.d.M ≤ p.d.M for every R-module (left or right).
(2) Prove that r.weak.dim R = l.weak.dim R.
(3) Prove that weak.dim R ≤ min{l.gl.dim R, r.gl.dim R}
(This explain way the global flat dimension of a ring is called
the weak dimension.)
Exercise 6. Given a family of right R-modules (Mi )i∈I , prove that
M
p.d.(
Mi ) = sup{p.d.Mi }.
i∈I
i∈I
Conclude that if r.gl.dimR = ∞, then there exists a right module M
such that p.d.M = ∞.
Exercise 7. Let M be an R-module with p.d.M = n. Prove that there
is a free R-module F such that ExtnR (M, F ) 6= 0
(Hint: every module is an epimorphic image of a free module)
Exercise 8. Prove that every module over a ring R is a direct (filtered)
limit of finitely presented modules.
Exercise 9. Let M be a finitely presented module and φ : F → M an
epimorphism with F a finitely generated module.
Prove that Kerφ is finitely generated.
Exercise 10. A submodule A of a right R-module B is said to be pure
in B if for every left (finitely presented) R-module M
0 → A ⊗R M → B ⊗R M is exact.
Let 0 → A → B → C → 0 be a short exact sequence of right
R-modules with B flat.
(1) Prove that C is flat if and only if for every left ideal I of R
AI = BI ∩ A.
(2) Prove that C is flat if and only if A is pure in B.