Solutions to The Rising Sea (Vakil)

Chapter 1

Section 1.2: Categories and Functors

Exercise 1.2.A:
A category in which each morphism is an isomorphism is called a groupoid. (This notion is not important in what we will discuss. The point of this exercise is to give you some practice with categories, by relating them to an object you know well.)
  1. A perverse definition of a group is: a groupoid with one object. Make sense of this.
  2. Describe a groupoid that is not a group

Exercise 1.2.B:
If \( \mathcal{A}\) is an object in a category \(\CC\), show that the invertible elements of \(\Mor_{\CC}(A, A) \) form a group (called the automorphism group of \(\mathcal{A}\), denoted \(\textrm{Aut}(\mathcal{A})\)). What are the automorphism groups of the objects in Examples 1.2.2 and 1.2.3? Show that two isomorphis objects have isomorphic automorphism groups. (For readers with a topological background: if \(X\) is a topological space, then the fundamental groupoid is the category where the objects are points of \(X\), and the morphisms \(x \to y\) are the paths from \(x\) to \(y\), up to homotopy. Then the automorphism group of \(x_0\) is the (pointed) fundamental group \(\pi_1(X, x_0)\). In the case where \(X\) is connected, and \(\pi_1(X)\) is not abelian, this illustrates the fact that for a connected groupoid — whose definition you can guess — the automorphism groups of the objects are all isomorphic, but not canonically isomorphic).

Exercise 1.2.C:
Let \( (\cdot)^{\vee \vee} : f.d.\textbf{Vec}_k \to f.d. \textbf{Vec}_k \) be the double dual functor from the category of finite-dimensional vector spaces over \(k\) to itself. Show that \((\cdot)^{\vee \vee} \) is naturally isomorphic to the identity functor on \( f.d.\textbf{Vec}_k \). (Without the finite-dimensionality hypothesis, we only get a natural transformation of functors from \(id\) to \( (\cdot)^{\vee \vee} \) .)

Exercise 1.2.D:
Show that \( \mathscr{V} \to f.d.\textrm{Vec}_k \) gives an equivalence of categories by describing an "inverse" functor. (Recal that we are being cavalier about set-theoretic assumptions, so feel free to simultaneously choose bases for each vector space in \(f.d.\textrm{Vec}_k\). To make this precise, you will need to use Gödel-Bernays set theory or selse replace \(f.d.\textrm{Vec}_k\) with a very similar small category, but we won't worry about this.)

Section 1.3: Universal Properties

Exercise 1.3.A:
Show that any two initial objects are uniquely isomorphic. Show that any two final objects are uniquely isomorphic.

Exercise 1.3.B:
What are the initial and final objects in \( \textbf{Set}, \textbf{Ring},\) and \(\textbf{Top}\) (if they exist)? How about the two examples of 1.2.9?

Exercise 1.3.C:
Show that \(A \mapsto S^{-1}A \) is injective if and only if \(S\) contains no zerodivisors. (A \(\textbf{zerodivisor} \) of a ring \(A\) is an element \(a\) such that there is a nonzero element \(b\) with \(ab =0\). The other elements of \(A\) are called \(\textbf{non-zerodivisors} \). For example, an invertible element is never a zerodivisor. Counter-intuitively, \(0\) is a zerodivisor in every ring but the \(0\)-ring. More generally, if \(M\) is an \(A\)-module, then \(a \in A\) is a \( \textbf{zerodivisor}\ \textbf{for}\ M \) if there is a non-zero \(m \in M\) with \(am = 0 \). The oter elements of \(A\) are called \(\textbf{non-zerodivisors}\ \textbf{for}\ M\).)

Exercise 1.3.D:
Verify that \(A \to S^{-1}A\) satisfies the following universal property: \(S^{-1}A\) is initial among \(A\)-algebras \(B\) where every element of \(S\) is sent to an invertible element in \(B\). (Recall: the data of "an \(A\)-algebra \(B\)" and "a ring map \(A \to B \)" are the same.) Translation: any map \(A \to B\) where every element of \(S\) is sent to an invertible element must factor uniquely through \(A \to S^{-1}A \). Another translation: a ring map out of \(S^{-1}A\) is the same thing as a ring map from \(A\) that sends every element of \(S\) to an invertible element. Furthermore, an \(S^{-1}A\)-modue is the same thing as an \(A\)-module for which \(s \times \cdot : M \to M\) is an \(A\)-module isomorphism for all \(s \in S\).

Exercise 1.3.E:
Show that \( \phi : M \to S^{-1}M \) exists, by constructing something that satisfying the universal property. Hint: define elements of \(S^{-1}M\) to be of the form \(m/s \) where \( m \in M \) and \(s \in S\), and \(m_1 /s_1 = m_2 / s_2\) if and only if, for some \( s \in S \), \(s (s_2m_1 - s_1m_2) = 0 \). Define the additive structure by \( (m_1 / s_1) + (m_2 / s_2) = ( s_2m_1 + s_1m_2 )/ (s_1s_2) \), and the \( S^{-1}A \)-module structure (and hence \(A\)-module structure) is given by \( (a_1 / s_1) \cdot (m_2 / s_2) = (a_1m_2 / s_1s_2) \).

Exercise 1.3.F:
  1. 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) \(S^{-1} (M_1 \times \dots \times M_n) \to S^{-1}M_1 \times \dots \times S^{-1}M_n\).
  2. Show that localization commutes with arbitrary direct sums.
  3. 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: \( (1, 1/2, 1/3, 1/4, \dots) \in \Q \times \Q \times \dots \) )

Exercise 1.3.G:
Show that \( \Z / (10) \otimes_{\Z} \Z/ (12) \cong \Z/(2) \). (This exercise is intended to give some hands-on-practice with tensor products.)

Exercise 1.3.H:
Show that \( (\cdot) \otimes_A N \) gives a covariant functor \( \Mod{A} \to \Mod{A} \). Show that \( (\cdot) \otimes_A N \) is a right-exact functor, i.e. if $$ \require{amscd} \begin{CD} 0 @>>> M^\prime @>>{g}> M @>>{f}> M^{\prime\prime} @>>> 0 \end{CD} $$ is an exact sequence of \(A\)-modules (which means \(f : M \to M^{\prime\prime} \) is surjective, and \( M^\prime\) surjects onto the kernel of \(f\); see §1.6), then the induced sequence $$ \require{amscd} \begin{CD} 0 @>>> M^\prime \otimes_A N @>>{g}> M \otimes_A N @>>{f}> M^{\prime\prime} \otimes_A N @>>> 0 \end{CD} $$ is also exact. This exercies is repeated in Exercise 1.6.F but you may get a lot out of doing it right now. (You will be reminded of the definition of right-exactness in §1.6.5).

Exercise 1.3.I:
Show that \( (T, t : M \times N \to T) \) is unique up to unique isomorphism. Hint: first figure out what "unique up to unique isomorphism" means for such pairs, using a category of pairs \((T, t)\). The follow the analogous argument for the product.

Exercise 1.3.J:
Show that the construction of §1.3.5 satisfies the universal property of tensor product.

Exercise 1.3.K:
  1. 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 \( \Mod{A} \to \Mod{B} \)
  2. 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 \( (b_1 \otimes c_1)(b_2 \otimes c_2) = (b_1b_2) \otimes (c_1c_2) \)

Exercise 1.3.L:
If \(S\) is a multiplicative subset of \(A\) and \(M\) is an \(A\) module, describe a natural isomorphism \( (S^{-1}A) \otimes_A M \cong S^{-1}M \) (as \(S^{-1}A\)-modules and as \(A\) modules ).

Exercise 1.3.M:
Show that tensor products commute with arbitrary direct sums: if \( M \) and \( \{N_i \}_{i \in I} \) are all \(A\)-modules, describe an isomorphism $$ \require{amscd} \begin{CD} M \otimes (\oplus_{i \in I} N_i) @>{\sim}>> \oplus_{i \in I} (M \otimes N_i) \end{CD} $$

Exercise 1.3.N:
Show that in \( \textbf{Set} \), $$ X \times_Z Y := \{ (x, y) \in X \times Y : \alpha(x) = \beta(y) \} $$ More precisely, show that the right side, equipped with its evident maps to \(X\) and \(Y\), satisfies the universal property of the fibred product. (This will help you build intuition for fibred products.)

Exercise 1.3.O:
If \(X\) is a topological space, show that the fibred products always exist in the category of open sets of \(X\) by describing what a fibred product is. (Hint: it has a one-word description.)

Exercise 1.3.P:
If \(Z\) is the final object in a category \(\CC\), and \(X, Y \in \CC\), show that "\( X \times_Z Y = X \times Y \)": "the" fibred product over \(Z\) is uniquely isomorphic to "the" product. Assume all relevant (fibred) products exist. (This is an exercise about unwinding the definition.)

Exercise 1.3.Q:
If the two squares in the following commutative diagram are Cartesian diagrams, show that the "outside rectangle" (involving \(U, V, Y\) and \(Z\)) is also a Cartesian diagram. $$ \require{amscd} \begin{CD} U @>>> V\\ @VVV @VVV\\ W @>>> X \\ @VVV @VVV \\ Y @>>> Z \end{CD} $$

Exercise 1.3.R:
Given morphisms \(X_1 \to Y\), \(X_2 \to Y\), and \(Y \to Z\), show that there is a natural morphism \( X_1 \times_Y X_2 \to X_1 \times_Z X_2 \), assuming that both fibred products exist. (This is trivial once you figure out what it is saying. The point of this exercise is to see why it is trivial.)

Exercise 1.3.S:
Suppose we are given morphisms \(X_1, X_2 \to Y\) and \(Y \to Z\). Show that the following diagram is a Cartesian square. $$ \require{amscd} \begin{CD} X_1 \times_Y X_2 @>>> X_1 \times_Z X_2\\ @VVV @VVV\\ Y @>>> Y \times_Z Y \\ \end{CD} $$ Assume all relevant (fibred) products exist. This diagram is surprisingly useful — so useful that we will call it the \(\textbf{magic}\ \textbf{diagram}\).

Exercise 1.3.T:
Show that the coproduct for \( \textbf{Set} \) is the disjoint union. This is why we use the notation \( \coprod \) for disjoint union.

Exercise 1.3.U:
Suppose \(A \to B\) and \(A \to C\) are two ring morphisms, so in particular \(B\) and \(C\) are \(A\)-modules. Recall (Exercise 1.3.K) that \(B \otimes_A C\) has a ring structure. Show that there is a natural morphism \(B \to B \otimes_A C \) given by \( b \mapsto b \otimes 1 \). (This is not necessarily an inclusion; see exercise Exercise 1.3.G.) Similarly, there is a natural morphism \(C \to B \otimes_A C\). Show that this gives a fibred product of rings, i.e., that $$ \require{amscd} \begin{CD} B \otimes_A C @<<< C \\ @AAA @AAA\\ B @<<< A\\ \end{CD} $$ satisfies the universal property of the fibred coproduct.

Exercise 1.3.V:
Show that the composition of two monomorphisms is a monomorphism.

Exercise 1.3.W:
Prove that a morphism \( \pi : X \to Y \) is a monomorphism if and only if the fibred product \(X \times_Y X\) exists, and the induced morphism \(X \to X \times_Y X\) is an isomorphism. We may then take this as the definition of a monomorphism. (Mononmorphisms aren't central to future discussions, although they will come up again. This exercise is just good practice.)

Exercise 1.3.X:
We will use the notation of Exercise 1.3.R. Show that if \(Y \to Z\) is a monomorphism, then the morphism \( X_1 \times_Y X_2 \to X_1 \times_Z X_2\) you described in Exercise 1.3.R is an isomorphism. (Hint: for any object \(V\), give a natural bijection between maps from \(V\) to the first and maps from \(V\) to the second. It is also possible to use the magic diagram, Exercise 1.3.S.)

Exercise 1.3.Y:
  1. 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 ( 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 \(u \mapsto g \circ u\).
  2. 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 (, and see where the identity goes. )

Exercise 1.3.Z:
  1. 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 \(B \to A\)
  2. 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?)
  3. \(\mathbf{Yoneda's}\ \mathbf{Lemma} \): 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

Section 1.4: Limits and Colimits

Exercise 1.4.A:
Suppose that the partially ordered set \( \mathscr{I} \) has an initial object \(e\). Show that the limit of any diagram indexed by \( \mathscr{I} \) exists.

Exercise 1.4.B:
Show that in the category \( \mathbf{Set} \), $$ \left\{ (a_i)_{i \in \mathscr{I}} \in \prod_i A_i : \FF(m)(a_j) = a_k \ \textrm{for}\ \textrm{all}\ m \in \Mor_\mathscr{I}(j, k) \right\} $$ along with the obvious projection maps to each \(A_i \), is the limit \( \varprojlim_\mathscr{I} A_i \).

Exercise 1.4.C:
  1. Interpret the statement "\( \Q = \varinjlim \frac{1}{n}\Z \)".
  2. 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.

Exercise 1.4.D:
Suppose \( \mathscr{I} \) is filtered. (We will almost exclusively use the case where \( \mathscr{I} \) is a filtered set.) Recall the symbol \( \coprod \) for disjoint union of sets. Show that any diagram in \( \textbf{Set} \) indexed by \( \mathscr{I} \) has the following, with the obvious maps to it, as a colimit: $$ \left\{ (a_i, i) \in \coprod_{i \in \mathscr{I}} A_i \right\} \Big/ \left( (a_i, i) \sim (a_j, j) \Leftrightarrow \exists f : A_i \to A_k, g : A_j \to A_k, f(a_i) = g(a_j)\right) $$ (You will see that the "filtered" hypothesis is there to ensure that \(\sim\) is an equivalence relation.)

Exercise 1.4.E:
Verify that the \(A\)-module described above is indeed the colimit. (Make sure that you verify that addition is well-defined, i.e., is independent of the choice of representatives \( m_i\) and \(m_j\), the choice of \(l \), and the choice of arrows \(u \) and \(v\). Similarly, make sure that scalar multiplication is well-defined.)

Exercise 1.4.F:
Generalize Exercise 1.4.C(a) to interpret localization of an integral domain as a colimit over a filtered set: suppose \(S\) is a multiplicative set of \(A\), and interpret \( S^{-1}A = \varprojlim \frac{1}{s}A \) where the limit is over \(s \in S\), and in the category of \(A\)-modules.

Exercise 1.4.G:
Suppose you are given a diagram of \(A\)-modules indexed by \(\mathscr{I} \): \( \FF : \mathscr{I} \to \Mod{A} \), where \( M_i = \FF(i) \). Show that the colimit is \( \oplus_{i \in \mathscr{I}} M_i \) modulo the relations \( m_j - \FF(n)(m_i) \) for every \( n : i \to j \) in \( \mathscr{I} \) (i.e. for every arrow in the diagram). (Somewhat more precisely: " modulo " means " quotiented by the submodule generated by").

Section 1.5: Adjoints

Exercise 1.5.A:
Write down what this diagram should be.

Exercise 1.5.B:
Show that the map \( \tau_{AB} \) has the following properties. For each \(A \) there is a map \( \eta_A : A \to \mathscr{G}\FF(A) \) so that for any \( g : \FF(A) \to B \), the corresponding \( \tau_{AB}(g) : A \to \mathscr{G}(B) \) is given by the composition $$ \require{amscd} \begin{CD} A @>{\eta_A}>> \mathscr{G}\FF(A) @>{\mathscr{G}(g)}>> \mathscr{G}(B) \end{CD} $$ Similarly, there is a map \( \epsilon_B : \FF\mathscr{G}(B) \to B \) for each \(B\) so that for any \( f : A \to \mathscr{G}(B) \), the corresponding map \( \tau_{AB}^{-1}(f) : \FF(A) \to B \) is given by the composition $$ \require{amscd} \begin{CD} \FF(A) @>{\FF(f)}>> \FF\mathscr{G}(B) @>{\epsilon_B}>> B \end{CD} $$

Exercise 1.5.C:
Suppose \(M, N\) and \(P\) are \(A\)-modules (where \(A\) is a ring). Describe a bijection $$ \Hom_A(M \otimes_A N, P) \leftrightarrow \Hom_A(M, \Hom_A(N, P) ) $$ (Hint: try to use the universal property of \(\otimes\).)

Exercise 1.5.D:
Show that \( (\cdot) \otimes_A N \) and \( \Hom_A(N, \cdot) \) are adjoint functors.

Exercise 1.5.E:
Suppose \(B \to A \) is a morphism of rings. If \(M\) is an \(A\)-module, you can create a \(B\)-module \(M_B\) by considering it as a \(B\)-module. This gives a functor \( \cdot_B : \Mod{A} \to \Mod{B}\). Show that this functor is right-adjoint to \( \cdot \otimes_B A \). In other words, describe a bijection $$ \Hom_A( N \otimes_B A, M ) \cong \Hom_B(N, M_B) $$ functorial in both arguments. (This adjoint pair is very important, and is the key player in Chapter 16.)

Exercise 1.5.F:
Show that if an abelian semigroup is already a group then the identity morphism is the groupification. (More correct: the identity morphism is a groupification.) Note that you don't need to construct groupification (or even know that it exists in general) to solve this exercise.

Exercise 1.5.G:
Construct the "groupification functor" \(H\) from the category of nonempty abelian semigroups to the category of abelian groups. (One possible construction: given an abelian semigroup \(S\), the elements of its groupification \(H(S)\) are ordered pairs \( (a, b) \in S \times S \), which you may thing of as \(a - b\), with the equivalence that \( (a, b) \sim (c, d) \) if \( a + d + e = b + c + e \) for some \(e \in S\). Describe addition in this group, and show that it satisfies the properties of an abelian group. Describe the abelian semigroup map \( S \mapsto H(S) \). ) Let \( \mathcal{F} \) be the forgetful functor from the category of abelian groups Ab to the category of abelian semigroups. Show that \(H\) is left-adjoint to \( \mathcal{F} \).

Exercise 1.5.H:
The purpose of this exercise is to give you more practice with "adjoints of forgetful functor pairs", the means by which we get groups from semigroups, and sheaves from presheaves. Suppose \(A\) is a ring, and \(S\) a multiplicative set. Then \( S^{-1}A\)-modules are a fully faithful subcategory (§1.2.15) of the category of \(A\)-modules (via the obvious inclusion \( \Mod{S^{-1}A} \hookrightarrow \Mod{A} \) ). Then \( \Mod{A} \to \Mod{S^{-1}A} \) can be interpreted as adjoint to the forgetful functor \( \Mod{S^{-1}A} \to \Mod{A} \). State and prove the correct statements.

Section 1.6: An Introduction to Abelian Categories

Exercise 1.6.A:
Describe exact sequences $$ \require{amscd} \begin{CD} 0 @>>> \im f^i @>>> A^{i+1} @>>> \coker f^i @>>> 0 \end{CD} $$ $$ \require{amscd} \begin{CD} 0 @>>> \textrm{H}^i(A^\bullet) @>>> \coker f^{i-1} @>>> \im f^i @>>> 0 \end{CD} $$

Exercise 1.6.B:
Suppose $$ \require{amscd} \begin{CD} 0 @>>d^0> A^1 @>>d^1> \dots @>>d^{n-1}> A^n @>>{d^n}> 0 \end{CD} $$ is a complex of finite-dimensional \( k \)-vector spaces (often called \(A^\bullet \) for short.) Define \( h^i (A^\bullet) := \dim H^i(A^\bullet) \). Show that \( \sum (-1)^i \dim A^i = \sum h^i(A^\bullet) \). In particular, if \( A^\bullet \) is exact, then \( \sum_i (-1)^i \dim A^i = 0 \). (If you haven't dealt with much cohomology, this will give you some practice. )

Exercise 1.6.C:
Suppose \( \CC \) is an abelian category. Define the category \( \Com{\CC} \) of complexes as follows. The objects are infinite complexes $$ \require{amscd} \begin{CD} A^\bullet: \hspace{7em} \dots @>>> A^{i-1} @>>f^{i-1}> A^i @>>{f^i}> A^{i+1} @>>{f^{i+1}}> \dots \end{CD} $$ in \(\CC\), and the morphisms \( A^\bullet \to B^\bullet \) are commuting diagrams $$ \require{amscd} \begin{CD} \dots @>>> A^{i-1} @>>f^{i-1}> A^i @>>{f^i}> A^{i+1} @>>{f^{i+1}}> \dots \\ \ @VVV @VVV @VVV \ \\ \dots @>>> B^{i-1} @>>g^{i-1}> B^i @>>{g^i}> B^{i+1} @>>{g^{i+1}}> \dots \end{CD} $$ Show that \( \Com{\CC}\) is an abelian category. Feel free to deal with the speacial case of modules over a fixed ring. (Remark for experts: Essentially the same argument shows that \( \CC^\mathscr{I}\) is an abelian category for any small category \( \mathscr{I} \) and any abelian category \( \CC\). This immediately implies that the category of presheaves on a topological space \(X\) with values in an abelian category \( \CC \) is automatically an abelian category, c.f. §2.3.5.)

Exercise 1.6.D:
Show that ( induces a map of homology \( H^i (A^\bullet ) \to H^i(B^\bullet) \). Show furthermore that \(H^i\) is a covariant functor \( \Com{\CC} \). (Again, feel free to deal with the special case \( \Mod{A} \)).

Exercise 1.6.E:
Suppose \( \FF \) is an exact functor. Show that applying \( \FF\) to an exact sequence preserves exactness. For example, if \( \FF \) is covariant, and \( A' \to A \to A'' \) is exact, then \( \FF A' \to \FF A \to \FF A''\) is exact. (This will be generalized in Exercise 1.6.H(c).)

Exercise 1.6.F:
Suppose \(A\) is a ring, \(S \subset A\) is a multiplicative subset, and \(M\) is an \(A\)-module.
  1. Show that localization of \(A\)-modules \( \Mod{A} \to \Mod{S^{-1}A} \) is an exact covariant functor.
  2. 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.)
  3. 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 \( \CC \to Ab \).
  4. 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 \( \CC \to Ab \).

Exercise 1.6.G:
Suppose \( M \) is a finitely presented \(A\)-module: \(M\) has a finite number of generators, and with these generators it has a finite number of relations; or usefully equivalently, fits into an exact sequence $$ \require{amscd} \begin{CD} A^{\oplus q} @>>> A^{\oplus p} @>>> M @>>> 0 \end{CD} $$ (You might be able to interpret this in light of a variant of Exercise 1.6.H below, for left-exact contravariant functors rather than right-exact covariant functors.)

Exercise 1.6.H:
This result can take you far, and perhaps for that reason it has sometimes been called the Fernbahnhof (FernbaHnHoF) theorem, notably in Vakil Exercise 1.6.H. Suppose \( \FF : \mathscr{A} \to \mathscr{B} \) is a covariant functor of abelian categories, and \( C^\bullet \) is a complex in \( \mathscr{A} \).

  1. ( \( \FF \) 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 \( \FF C^\bullet \).)
  2. ( \( \FF\) left-exact yields \( \FF H^\bullet \leftarrow H^\bullet \FF \) ) If \( \FF \) is left-exact, describe a natural morphism \( H^\bullet \FF \to \FF H^\bullet \)
  3. ( \( \FF \) 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

Exercise 1.6.I:
Suppose \( \CC\) is an abelian category, and \( a: \II \to \CC \) and \( \II \to \CC \) are two diagrams in \( \CC \) indexed by \( \II \). For convenience, let \( A_i = a(i) \) and \( B_i = b(i) \) be the objects in those two diagrams. Let \( h_i : A_i \to B_i \) be maps commuting with the maps in the diagram. (Translation: \( h\) is a natural transformation of functors \(a \to b\), see §1.2.21.) Then the \( \ker h_i \) form another diagram in \( \CC \) indexed by \( \II \). Describe a canonical isomorphism \( \varprojlim \ker h_i \cong \ker ( \varprojlim A_i \to \varprojlim B_i ) \), assuming the limit exists.

Exercise 1.6.J:
Make sense of the statement "limits commute with limits" in a general category, and prove it. ( Hint: recall that kernels are limits. The previous exercise should be a corollary of this one. )

Exercise 1.6.K:
Show that in \( \Mod{A} \), colimits over filtered index categories are exact. (Your argument will apply without change to any abelian category whose objects can be interpreted as "sets with additional structure".) Right-exactness follows from the above discussion, so the issue is left-exactness. (Possible hint: After you show that localization is exact, Exercise 1.6.F(a), or stalkification is exact, Exercise 2.6.D, in a hands-on way, you will be easily able to prove this. Conversely, if you do this exercise, those two will be easy.)

Exercise 1.6.L:
Show that filtered colimits commute with homology in \( \Mod{A} \). Hint: use the FHHF theorem (Exercise 1.6.H ) and the previous exercise.

Exercise 1.6.M:
Suppose $$ \require{amscd} \begin{CD} @.\vdots @. \vdots @. \vdots \\ @. @VVV @VVV @VVV \\ 0 @>>> A_{n+1} @>>> B_{n+1} @>>> C_{n+1} @>>> 0\\ @. @VVV @VVV @VVV \\ 0 @>>> A_n @>>> B_n @>>> C_n @>>> 0 \\ @. @VVV @VVV @VVV \\ @. \vdots @. \vdots @. \vdots \\ @. @VVV @VVV @VVV \\ 0 @>>> A_0 @>>> B_0 @>>> C_0 @>>> 0 \\ @. @VVV @VVV @VVV \\ @. 0 @. 0 @. 0 \end{CD} $$ is an inverse system of exact sequences of modules over a ring, such that the maps \( A_{n+1} \to A_n \) are surjective. (We say: "transition maps of the left are surjective".) Show that the limit $$ \require{amscd} \begin{CD} 0 @>>> \varprojlim A_n @>>> \varprojlim B_n @>>> \varprojlim C_n @>>> 0 \end{CD} $$ is also exact. (You will need to define the maps.)

Section 1.7*: Spectral Sequences

Thanks for reading! 😁