MA3204 Homological Algebra Fall 2020

The course description can be found here.

Lectures will be Wednesdays, 8:15 to 10:00, and Fridays, 14:15 to 16:00, in room EL4.

There will be weekly exercises, which will be announced and discussed during the lectures.

Steffen Oppermann; room 844 Sentralbygg II; steffen.oppermann@ntnu.no

Show that a morphism is split mono and epi if and only if it is iso.

In the category of rings, show that \( \mathbb{Z} \to \mathbb{Q} \) is both mono and epi, but not iso.

Let \( G \) be a group. Convince yourself that we can consider \( G \) as a category with only one object. Describe the functors form \( G \) (as a category) to \( \mathbf{Set} \), and the natural transformations between two such functors.

Consider the forgetful functors \( \mathbf{Ab} \to \mathbf{Set} \) and \( \mathbf{Top} \to \mathbf{Set} \). Do these functors have left and or right adjoints? If so, describe these adjoints.

Let \( \eta \) and \( \varepsilon \) be the unit and counit of an adjunction between functors \( F \) and \( G \). Show that \( \varepsilon_{F-} \circ F \eta = \operatorname{id}_F \) and \( G \varepsilon \circ \eta_{G-} = \operatorname{id}_G \).

Given \( D \leftarrow A \to B \to C \), assume that there is a pushout \( P_1 \) of \( D \leftarrow A \to B \) and a pushout \( P_2 \) of \( P_1 \leftarrow B \to C \). Show that \( P_2 \) is also a pushout of the composition \( D \leftarrow A \to C \).

Let \( (F, G) \) be an adjoint pair of functors between categories \( \mathcal{C} \) and \( \mathcal{D} \), and \( X \) be a poset. Show that the induced functors \( F_{\operatorname{Fun}} \colon \operatorname{Fun}(X, C) \to \operatorname{Fun}(X, D) \) and \( G_{\operatorname{Fun}} \colon \operatorname{Fun}(X, D) \to \operatorname{Fun}(X, C) \) also form an adjoint pair.

In a category with \( 0 \), consider a set of objects \( \{ X_i \}_{i \in I} \) which has a product and a coproduct. Show that there is a unique map \( f \colon \coprod_{i \in I} X_i \to \prod_{i \in I} X_i \), such that \( \pi_i f \iota_i = \operatorname{id}_{X_i} \) and \( \pi_i f \iota_j = 0 \) whenever \( i \neq j \). (Here the \( \pi_i \colon \prod_{i \in I} X_i \to X_i \) and \( \iota_i \colon X_i \to \coprod_{i \in I} X_i \) denote the structure maps coming with the product and coproduct, respectively.)

Let \( f \colon A \to B \) ba a morphism in an additive category. Assume that \( f \) has a kernel \( k \colon K \to A \). Show that \( k \) is a monomorphism.

In an abelian category, let \( f \colon A \to B \) and \( g \colon B \to C \) be monomorphisms. Show that there is a short exact sequence \( 0 \to \operatorname{Cok} f \to \operatorname{Cok}(g \circ f) \to \operatorname{Cok} g \to 0 \).

Let \( X \) be a finite poset, and \( F \in \operatorname{Fun}(X, \mathcal{A}) \) for some additive category \( \mathcal{A} \). Show that the limit \( \varprojlim F \) exists if and only if the map \( \bigoplus_{x \in X} F(x) \to \bigoplus_{y \lneq z \in X} F(z) \), given on components by \( \begin{cases} \operatorname{id}_{F(x)} & \text{for } x = z \\ - F(x \leq y) & \text{for } x = y \\ 0 & \text{otherwise} \end{cases} \), has a kernel, and in this case the two are the same.

Reprove the second exercise from last week using the snake lemma.

Let \( X = \{1 \leq 2\} \). Convince yourself that \( \operatorname{Ker} \) gives a functor \( \operatorname{Fun}(X, \mathcal{A}) \to \mathcal{A} \). Show that this functor is left exact, but typically not right exact.

In \( \operatorname{Mod} R \), let \( A \overset{a}{\to} B \overset{b}{\to} C \overset{c}{\to} D \) be an exact sequence, and \( x \colon X \to B \) and \( y \colon C \to Y \) such that \( ybx = 0 \). Show that \( \frac{\operatorname{Ker} yb}{\operatorname{Im}(x) + \operatorname{Im}(a)} \cong \frac{\operatorname{Ker}(c) \cap \operatorname{Ker}(y)}{\operatorname{Im} bx} \). Use this result to give a new proof of the middle part of the exact sequence of the snake lemma.

Show that \( \mathbb{Q} / \mathbb{Z} \) is injective in the category of abelian groups. (Let \( A \to B \) be a monomorphism, and \( \varphi \colon A \to \mathbb{Q} / \mathbb{Z} \) be any morphism. Use the axiom of choice to find a maximal submodule \( B' \) of \( B \) to which \( \varphi \) extends. Then show that \( B' = B \).

Let \( R \) be any ring. Use the previous exercise to show that \( \operatorname{Hom}_{\mathbb{Z}}(R, \mathbb{Q} / \mathbb{Z} ) \) is injective in \( \operatorname{Mod} R \).

For an abelian group \( A \), let \( \overline{A} = (\mathbb{Z} \setminus \{ 0 \}) \times A / \sim \), where \( (p, a) \sim (q, b) \) if and only if there is a non-zero \( n \) such that \( nqa = npb \). (That is we consider fractions with enumerators in \( A \) and denominators being non-zero numbers, with the usual arithmetic.) Show that \( \mathbb{Q} \otimes_{\mathbb{Z}} A \cong \overline{A} \). Show that \( \mathbb{Q} \) is flat but not projective as a \( \mathbb{Z} \)-module.

Consider the complex \( 0 \to k[x,y] \to k[x,y]^2 \to k[x,y] \to 0 \) where the two maps are given as \( \left( \begin{smallmatrix} y^2 \\ xy \end{smallmatrix} \right) \) and \( ( -x^2 \; xy ) \), respectively. Calculate all homologies of this complex.

Consider \( \mathbb{Z}^2 \) as a poset by pointwise comparison. Consider the full subcategory \( \mathcal{D} \) of \( \operatorname{Fun}(\mathbb{Z}^2, \mathcal{A}) \) given by the functors \( F \) such that \( F(m,n) = 0 \) unless \( m \in \{n, n+1\} \). Show that \( \mathcal{B} \) is equivalent to the category of complexes over \( \mathcal{A} \).

Let \( \mathcal{A} \) be an abelian category. Convince yourself that sending an object \( A \) to the complex \( 0 \to A \to 0 \) (with \( A \) in degree 0) defines a functor \( \mathcal{A} \to \mathbf{C}(\mathcal{A}) \). Show that this functor has a left and a right adjoint, and describe them.

Consider the category of \( k[x,y] \)-modules. Find a projective resolution of \( k \). (Here \( k \) is a \( k[x,y] \)-module by \( x \) and \( y \) acting as zero.)

Let \( \mathcal{A} \) be an abelian category. Show that a complex is isomorphic to \( 0 \) in the homotopy category if and only if it is isomorphic (in the category of complexes) to a complex of the form \[ \cdots \to B^0 \oplus B^{-1} \to B^1 \oplus B^0 \to B^2 \oplus B_1 \to \cdots \] where all maps are in matrix form \( \left[ \begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix} \right] \).

Let \( f \colon A^{\bullet} \to B^{\bullet} \) be a morphism of complexes. Show that \( B \to \operatorname{Cone}(f) \) is a weak cokernel of \( f \) in the homotopy category. (A weak cokernel is a morphism such that the composition is zero, and any other morphism which composes with \( f \) to zero factors through the weak cokernel, but not necessarily uniquely.)

In the category of \( k[x,y] \)-modules, consider \( k \) as in the first exercise of last week. Calculate \( \operatorname{Ext}^n(-, k)(k) \) for all \( n \).

Let \( \mathcal{A} \) be an abelian category with enough projectives and enough injectives. Use dimension shift to show that \( \operatorname{Ext}^1(X, -)(Y) = \operatorname{Ext}^1(-, Y)(X) \).

Let \( \mathcal{A} \) be a category with enough injectives. Calculate the right derived functors of the kernel functor. (Note: kernel is a functor \( \operatorname{Fun}(\{1 \leq 2\}, \mathcal{A}) \to \mathcal{A} \), so we need to use injective resolutions in the functor category.)

In the category of abelian groups, calculate \( \operatorname{YExt}^1(\mathbb{Z}/(3), \mathbb{Z}/(3)) \) and \( \operatorname{Ext}^1(-, \mathbb{Z}/(3))(\mathbb{Z}/(3)) \).

Let \( A \) be a finitely generated abelian group. (Recall the fundamental theorem: \( A \) is of the form \( \mathbb{Z}^n \oplus \mathbb{Z} / (q_1) \oplus \cdots \oplus \mathbb{Z} / (q_r) \) where the \( q_i \) are prime powers.) Calculate \( \operatorname{Hom}(A, \mathbb{Z}) \) and \( \operatorname{Ext}^1(A, \mathbb{Z}) \).

Consider the double complex of abelian groups where all terms are \( \mathbb{Z} / (4) \), and all maps both horizontally and vertically are multiplication by \( 2 \). Check that all columns and rows of this double complex are exact. Is the total complex exact? What about if we use products instead of coproducts in the definition of total complexes?

Let \( R \) be a commutative ring without zero-divisors. Show that \( \operatorname{mod} R \) is hereditary if and only if \( R \) is a principal ideal domain.

Let \( 0 \to A \to B \to C \to 0 \) and \( 0 \to C \to D \to E \to 0 \) be two short exact sequences in a hereditary category. Show that is an object \( X \) and short exact sequences \( 0 \to A \to X \to D \to 0 \) and \( 0 \to B \to X \to E \to 0 \) making a commutative diagram as in Exercise V.10. in the notes.

Let \( A^{\bullet, \bullet} \) be a double complex of abelian groups with exact rows, and assume \( A^{m,n} = 0 \) whenever \( n > 0 \). Show that the total complex of \( A^{\bullet, \bullet} \) is exact.

In a triangulated category, show that any monomorphism is a split monomorphism, and any epimorphism is a split epimorphism. Show that if a triangulated category is simultaniously abelian, then it is semi-simple.

Let \( \mathcal{A} \) be an abelian category. Show that the homotopy category \( \operatorname{K}(\mathcal{A}) \) is abelian if any only if \( \mathcal{A} \) is semi-simple.

Let \( P^{\bullet} \) be a right bounded complex of projectives, and \( X^{\bullet} \) be an acyclic complex. Show that \( \operatorname{Hom}_{\operatorname{K}(\mathcal{A})}(P^{\bullet}, X^{\bullet}) = 0 \).

Let \( A^{\bullet} \) be a complex with \( \operatorname{H}^n(A^{\bullet}) = 0 \; \forall n > 0 \). Show that there is a quasi-isomorphism \( B^{\bullet} \to A^{\bullet} \), where \( B^{\bullet} \) is a complex such that \( B^n = 0 \; \forall n > 0 \).

Let \( A^{\bullet} \) be a complex with \( A^n = 0 \; \forall n > 0 \), and \( B^{\bullet} \) be a complex with \( B^n = 0 \; \forall n \leq 0 \). Show that \( \operatorname{Hom}_{\operatorname{D}(\mathcal{A})}(A^{\bullet}, B^{\bullet}) = 0 \).

(Harder?) Let \( \mathcal{A} \) be a hereditary abelian category. Let \( A^{\bullet} \) be a complex, and denote by \( \operatorname{H}(A^{\bullet}) \) the complex \( \cdots \to \operatorname{H}^{-1}(A^{\bullet}) \to \operatorname{H}^0(A^{\bullet}) \to \operatorname{H}^1(A^{\bullet}) \to \cdots \) where all differentials are zero. Show that there is a complex \( B^{\bullet} \) and a pair of quasi-isomorphisms \( A^{\bullet} \leftarrow B^{\bullet} \to \operatorname{H}(A^{\bullet}) \). In particular \( A^{\bullet} \) and \( \operatorname{H}(A^{\bullet}) \) are isomorphic in the derived category.

We will mainly be following these notes. Please be aware that these likely contain typos and/or mistakes.

These are also good references:

• Joseph J. Rotman, An introduction to Homological Algebra, first edition
• Joseph J. Rotman, An introduction to Homological Algebra, second edition
• Charles A. Weibel, An introduction to Homological Algebra
• Sergei I. Gelfand, Yuri I. Manin, Methods of Homological Algebra , second edition

The final exam will be an oral exam. Everything discussed in class or in the exercises is material for the exam.