- A perverse definition of a
*group*is: a groupoid with one object. Make sense of this. - Describe a groupoid that is not a group

- Show that localization commutes with finite products, or equivalently, with finite direct sums. In other words, if \(M_1, \dots, M_n\) are \(A\)-modules, describe an isomorphism (of \(A\)-modules, and of \(S^{-1}A\)-modules) \(S^{-1} (M_1 \times \dots \times M_n) \to S^{-1}M_1 \times \dots \times S^{-1}M_n\).
- Show that localization commutes with
*arbitrary*direct sums. - Show that "localization does not necessarily commute with infinite products": the obvious map \( S^{-1} \left( \prod_i M_i \right) \to \prod_i S^{-1}M_i \) induced by the universal property of localization is not always an isomorphism. (Hint: \( (1, 1/2, 1/3, 1/4, \dots) \in \Q \times \Q \times \dots \) )

- If \(M\) is an \(A\)-module, and \(A \to B\) is a morphism of rings, give \( B \otimes_A M \) the structure of a \(B\)-module (this is part of the exercise). Show that this describes a functor \( \Mod{A} \to \Mod{B} \)
- If further \(A \to C\) is another morphism of rings, show that \( B \otimes_A C \) has a natural structure of a ring. Hint: multiplication will be given by \( (b_1 \otimes c_1)(b_2 \otimes c_2) = (b_1b_2) \otimes (c_1c_2) \)

- Suppose you have two objects \(A\) and \(A^\prime\) in a category \(\CC\), and morphisms $$ i_C : \Mor(C, A) \to \Mor(C, A^\prime) $$ that commute with the maps (1.3.10.1). Show that the \(i_C\) (as \(C\) ranges over the objects of \(\CC\)) are induced from a unique morphism \(g : A \to A^\prime \). More precisely, show that there is a unique morphism \(g : A \to A^\prime\) such that for all \(C \in \CC\), \(i_C\) is \(u \mapsto g \circ u\).
- If furthermore the \(i_C\) are all bijections, show that the resulting \(g\) is an isomorphism. (Hint for both: this is much easier than it looks. This statement is so general that there are really only a couple of things that you could possibly try. For example, if you're hoping to find a morphism \(A \to A'\), where will you find it? Well, you are looking for an element \( \Mor(A, A')\). So just plug in \(C = A\) to (1.3.10.2), and see where the identity goes. )

- Suppose \(A\) and \(B\) are objects in a category \( \CC \). Give a bijection between the natural transformations \( h^A \to h^B \) of covariant functors \( \CC \to \textbf{Set} \) (see example 1.2.14 for the definition) and the morphisms \(B \to A\)
- State and prove the corresponding fact for contravariant functors \(h_A\) (see Example 1.2.20). Remark: A contravariant functor \(\FF \) from \(\CC \) to \(\textbf{Set}\) is said to be
**representable**if there is a natural isomorphism $$ \require{amscd} \begin{CD} \xi : \FF @>{\sim}>> h^A \end{CD} $$ Thus the representing object \(A\) is determined up to unique isomorphism by the pair \( (\FF , \xi) \). There is a similar definition for covariant functors. (We will revisit this in §6.6, and this problem will appear again as Exercise 6.6.C. The element \( \xi^{-1}(id_A) \in \FF(A)\) is often called the "universal object"; do you see why?) - \(\mathbf{Yoneda's}\ \mathbf{Lemma} \): Suppose \( \FF \) is a covariant functor \( \CC \to \textbf{Set} \), and \(A \in \CC\). Give a bijection between the natural transformations \( h^A \to \FF\) and \( \FF(A)\). The corresponding fact for contravariant functors is essentially Exercise 9.1.C

- Interpret the statement "\( \Q = \varinjlim \frac{1}{n}\Z \)".
- Interpret the union of some subsets of a given set as a colimit. (Dually, the intersection can be interpreted as a limit.) The objects of the category in question are the subsets of the given set.

- Show that localization of \(A\)-modules \( \Mod{A} \to \Mod{S^{-1}A} \) is an exact covariant functor.
- Show that \( (\cdot) \otimes_A M \) is a right-exact covariant functor \( \Mod{A} \to \Mod{A} \). (This is a repeat of Exercise 1.3.H.)
- Show that \( \Hom (M, \cdot) \) is a left-exact covariant functor \( \Mod{A} \to \Mod{A} \). If \( \CC \) is any abelian category, and \(C \in \CC\), show that \( \Hom (C, \cdot) \) is a left-exact covariant functor \( \CC \to Ab \).
- Show that \( \Hom (\cdot, M) \) is a left-exact contravariant functor \( \Mod{A} \to \Mod{A} \). If \( \CC \) is any abelian category, and \(C \in \CC\), show that \( \Hom (\cdot, C) \) is a left-exact contravariant functor \( \CC \to Ab \).

- ( \( \FF \)
*right-exact yields*\( \FF H^\bullet \to H^\bullet \FF \) ) If \( \FF \) is right-exact, describe a natural morphism \( \FF H^\bullet \to H^\bullet \FF \). (More precisely, for each \( i\), the left side is \( \FF \) applied to the cohomology at piece \(i\) of \(C^\bullet\), while the right side is the cohomology at piece \(i\) of \( \FF C^\bullet \).) - ( \( \FF\)
*left-exact yields*\( \FF H^\bullet \leftarrow H^\bullet \FF \) ) If \( \FF \) is left-exact, describe a natural morphism \( H^\bullet \FF \to \FF H^\bullet \) - ( \( \FF \)
*exact yields*\( \FF H^\bullet \leftrightarrow H^\bullet \FF \) ) If \( \FF \) is exact, show that the morphisms of (a) and (b) are inverses and thus isomorphisms

Thanks for reading! 😁