Algorithmic homological algebra
Exercise sheet 3
Prof. Dr. Mohamed Barakat, Sebastian Posur
Exercise 1. (Equalizer, coequalizer)
Let A be a category, and let α, β : A → B be two morphisms in A (also called parallel morphisms). A pair (E, ι) consisting of an object E ∈ Aand a morphism ι:E → A such that α◦ι=β◦ι is called anequalizer of α and β if for any other pair (T, τ) consisting of an object T ∈ A and a morphism τ : T → A such that α◦τ = β ◦τ, there exists a unique morphism u:T →E such that the following diagram commutes:
E A B.
T
α β ι
u τ
(a) Define the dual notion of a coequalizer. Hint: Use the definition of an equalizer in the opposite category Aop.
(b) Show that if (E, ι) is an equalizer of α and β, then ι is a monomorphism. What is the dual proposition for coequalizers?
(c) Construct the equalizer of α and β using kernels (e.g. in the context of an abelian category). Dually construct the coequalizer ofα and β using cokernels.
(d) We now work in the category URngs, i.e., the category of unital rings with unital ring homomorphisms. Given a ring R and an ideal I of R, can you find a ring S and two morphisms α, β : S → R such that the quotient ring R/I together with the canonical morphism R → R/I is the coequalizer of α and β? Hint: Try S :=
{(r1, r2)∈R×R | r1−r2 ∈I}.
Exercise 2. (Regular monomorphisms, regular epimorphisms)
Let A be a category. A morphism ι : A → B is called a regular monomorphism if it is the equalizer of some parallel morphisms. Dually, a morphism : A → B is called a regular epimorphism if it is the coequalizer of some parallel morphisms.
(a) Give an example of a monomorphism which is not the kernel of its cokernel.
(b) Show that in the category of R-modules, every monomorphism and every epimor- phism is regular.
Exercise 3. (Pre-abelian categories)
Algorithmic homological algebra
(a) LetA be a pre-Abelian category. Prove that for an A-morphism φ:
• φ is mono iff kerφ= 0;
• φ is epi iff cokerφ= 0.
(b) Show that the full subcategory (tfAb)⊂(Ab) of torsion-free Abelian groups is pre- Abelian but not Abelian. For this consider the map Z
,→2 Z and prove that it is mono and epi (!) but not an isomorphism in (tfAb).
Exercise 4. (Direct sums)
Show that the following is true for additive categories:
• For objectsM and N define the morphisms
ιM :={1M,0M N}:M →M ⊕N and ιN :={0N M,1N}:N →M ⊕N.
Then πMιM +πNιN = 1M⊕N.
• It follows that finite coproducts also exist: (M⊕N;ιM, ιN) withιM,ιN as above and the coproduct morphism defined by
hφ, ψi:=πMφ+πNψ :M ⊕N →L, for two morphisms φ:M →L and ψ :N →L.
• ForK −−−→{α,β} L⊕M −−−→hφ,ψi N we have {α, β}hφ, ψi=αφ+βψ.
• In particular, for φ, ψ:M →N we have φ+ψ ={1M,1M}hφ, ψi.
This exercise sheet will be discussed on 24.11.2016.
