MA3204∶HOMOLOGICAL ALGEBRA - EXERCISE SHEET 5
Exercise 1. Let T be a triangulated category. Show that the opposite category T op is also triangulated.
f
g
h
f
g
h
Exercise 2. Let X Ð→ Y Ð→ Z Ð→ X[1] be a triangle in a triangulated category (T , [1], ∆). Show that
if h = 0, then the triangle splits, i.e. f is split mono and g is split epi.
Exercise 3. Let X Ð→ Y Ð→ Z Ð→ X[1] be a triangle in a triangulated category (T , [1], ∆). Show that
the following are equivalent∶
(i) The map f is an isomorphism.
(ii) Z = 0.
Exercise 4. Let (T , [1], ∆) be a triangulated category and consider two triangles∶
f
g
h
X Ð→ Y Ð→ Z Ð→ X[1] ∈ ∆
and
X ′ Ð→ Y ′ Ð→ Z ′ Ð→ X ′ [1] ∈ ∆
f′
g′
h′
Show that the diagram
( f0 f0′ )
( g 0′ )
( h 0′ )
′ 0 g
′ 0 h
X ⊕ X Ð→ Y ⊕ Y Ð→ Z ⊕ Z Ð→ X ⊕ X ′ [1]
′
is a triangle in T .
Assuming that T has coproducts, generalize the above statement as follows∶ If
Xi Ð→ Yi Ð→ Zi Ð→ Xi [1]
is a family of triangles in T for all i ∈ I, show that
∐ Xi Ð→ ∐ Yi Ð→ ∐ Z Ð→ ∐ Xi [1]
i∈I
i∈I
i∈I
i∈I
is a triangle in T .
Exercise 5. Let A be an abelian category such that the derived category D(A ) exists.
(i) Show that D(A ) is an additive category.
(ii) Show that the quotient functor Q∶ K(A ) Ð→ D(A ) is additive.
Exercise 6. Let A be an abelian category and 0 Ð→ X Ð→ Y Ð→ Z Ð→ 0 a short exact sequence in the
category of complexes C(A ). Show that there is triangle X Ð→ Y Ð→ Z Ð→ X[1] in D(A ).
Exercise 7. Let A be an abelian category with exact coproducts. Show that the derived category D(A )
has coproducts and that the functor Q∶ K(A ) Ð→ D(A ) preserves coproducts.
Exercise 8. Let R be a ring. For any object A in C(Mod-R) show that
HomD(Mod-R) (R, A) ≅ H 0 (A)
Exercise 9. Let A be an abelian category. Show that the canonical functors
Db (A ) Ð→ D(A ), D− (A ) Ð→ D(A ) and D+ (A ) Ð→ D(A )
are fully faithful.
Date: November 26, 2016.
1
2
Exercise 10. Let A be an abelian category.
(i) Assume that A has enough projectives and denote by Proj A the full subcategory of A consisting
of projective objects. Show that there is a triangle equivalence
K− (Proj A )
≃
/ D− (A )
(ii) Assume that A has enough injectives and denote by Inj A the full subcategory of A consisting
of injective objects. Show that there is a triangle equivalence
K+ (Inj A )
≃
/ D+ (A )
Exercise 11. Let A be an abelian category with enough projectives such that gl. dim A < ∞. Show that
Kb (Proj A )
≃
/ Db (A )
Exercise 12. Describe the derived category of a semisimple abelian category.
Exercise 13. Let A be a hereditary abelian category. Show that
D(A ) = ⊔ A [n]
n∈Z
Exercise 14. (Auslander-Reiten quiver, Challenge!) Let k be a field and consider the following finite dimensional k-algebra∶
k 0
Λ=(
)
k k
(i) Show that Λ is hereditary.
(ii) Describe the category mod-Λ of finitely generated left Λ-modules.
(iii) Describe the derived category Db (mod-Λ).
Exercise 15. Exercises VI.1 – VI.6 from the notes.
Chrysostomos Psaroudakis, Department of Mathematical Sciences, Norwegian University of Science and
Technology, 7491 Trondheim, Norway
E-mail address: [email protected]