Homework 4

Problem 1

Recall the notion of pullbacks, which for the sake of this exercise we will only consider in the category $Set$ .

Show that the functor which assigns to each diagram of the form $X \overset{f}{\to} Z \overset{g}{\leftarrow} Y$ in $Set$ the pullback $X \times_{Z} Y$ is a right adjoint of another functor. Describe the unit and counit of the adjunction.

Note

You don't need to check every tiny detail for this one. Define the pullback as a functor (giving the maps on objects and arrows), and then explicitly define the set map that should be a natural bijection between the appropriate hom-sets.

Hints

Let $J$ be the category with three objects and two non-identity arrows, visualized as $∙ \to ∙ \leftarrow ∙$ . Functors $J \to Set$ then correspond to diagrams in $Set$ of shape $J$ ; i.e., diagrams of the form $X \overset{f}{\to} Z \overset{g}{\leftarrow} Y$ in $Set$ . Let ${Set}^{J}$ denote the category of functors $J \to Set$ and let $Δ : Set \to {Set}^{J}$ denote the diagonal (or constant) functor. Show the pullback functor is a right adjoint of $Δ$ .

Problem 2

Let $R$ be an integral domain and $M$ be a finitely-generated torsion $R$ -module. Prove that the annihilator of $M$ in $R$ is nontrivial.

Problem 3

Prove that quotients of cyclic modules are cyclic.

Problem 4

Suppose $N$ is a submodule of an $R$ -module $M$ , and suppose that both $N$ and $M / N$ are finitely generated. Prove that $M$ is finitely generated.

Problem 5

Prove that any direct sum of free $R$ -modules is free.