Exercise sheet 2

Algorithmic homological algebra

Exercise sheet 2

Prof. Dr. Mohamed Barakat, Sebastian Posur

Exercise 1. (Free modules)

Let R be a ring and let I be a set. A pair (F, β :I →F) consisting of a left R-module F and a map β : I →F is called a free module with (unordered) basis β if it satisfies the following universal property: For every left R-module T and every map τ : I → T, there exists exactly one R-module homomorphism u(τ) : F → T such that the following diagram commutes:

I

F T.

β τ

u(τ)

(a) Given a natural number n ∈ N⁰, show that for I = {1, . . . n}, the left R-module R^I (whereR^Idenotes the functions fromI toR) can be equipped with a mapβ :I →R^I such that (R^I, β) is a free module with (unordered) basis β.

(b) Generalize (a) to the case where I is an infinite set. Hint: Consider an appropriate submodule ofR^I.

(c) Given two free modules (F, β : I → F) and (G, γ : I → G), show that there exists exactly one isomorphism ι:F →G such thatι◦β =γ.

Exercise 2. (Images)

Let φ : M → N be a morphism. A triple (I, , ι) consisting of an object I, a morphism : M → I, and a monomorphism ι : I ,→ N such that ι ◦ = φ is called a mono factorization of φ. A given mono factorization (I, , ι) ofφ is called an image of φ if for every other mono factorization (J, e, i) of φ, there exists exactly one morphism u :I →J such that the following diagram commutes:

M N

I J φ

ι

e u i

(a) If φ is an R-module homomorphism, let ι : φ(M) ,→ N be the inclusion of the set- theoretic image of φ into N. Show that the pair (φ(M), ι) can be extended to an image of φ.

Algorithmic homological algebra

(b) Ifφ is an R-module homomorphism, show that the object defined in the lecture im(φ) := ker(CokernelEpi(φ))

can be equipped with morphisms : M → im(φ) and ι : im(φ) → N such that (im(φ), , ι) is an image of φ.

(c) Optional: Can you prove (b) in the context of an arbitrary abelian category?

Exercise 3. (Morphism from coimage to image) Using the notation of Definition 1.2.18, show that

(_κ\φ)/κ =_κ\(φ/κ).

Exercise 4. (Pushout)

Construct the pushout of a given diagramL←−^φ M −→^ψ J using direct sums and cokernels.

This exercise sheet will be discussed on 10.11.2016.