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The Lumenary · First-year graduate

Abelian categories

Name: ______________________

Date: ______________________

You will learn to

Define additive and abelian categories, kernels, cokernels, images and exact sequences, and exact functors
Verify that $R\text{-}\cat{Mod}$ and sheaves of abelian groups on a space are abelian, and explain why free abelian groups, Banach spaces and vector bundles are additive but not abelian
Prove that in an abelian category a monic epimorphism is an isomorphism
Prove the five lemma and the snake lemma by diagram chasing, and explain how they transfer to every abelian category
State the Freyd–Mitchell embedding theorem and explain what it does and does not allow

1.
Which of these categories are abelian? (Choose all that apply.)
- A.Finite abelian groups
- B.Finite-dimensional vector spaces over a field
- C.Sheaves of abelian groups on a topological space
- D.Free abelian groups
- E.Banach spaces and bounded linear maps
- F.All groups
2.
In the proof that a monic epimorphism $f : X \to Y$ in an abelian category is an isomorphism, we write $f = \ker g$ for some $g : Y \to Z$ and conclude $g = 0$ . Which fact gives $g = 0$ ?
- A. $gf = 0 = 0 \circ f$ and $f$ is an epimorphism
- B. $f$ is a monomorphism
- C.Hom-sets are abelian groups
- D.Kernels are unique up to isomorphism
3.
A morphism of short exact sequences of finite-dimensional vector spaces has $\dim \ker\alpha = 1$ , $\dim\ker\beta = 2$ , $\dim\ker\gamma = 3$ , $\dim\operatorname{coker}\alpha = 2$ and $\dim\operatorname{coker}\beta = 1$ . What is $\dim\operatorname{coker}\gamma$ ?
4.
In the five lemma (rows $A_1 \to \cdots \to A_5$ and $B_1 \to \cdots \to B_5$ , vertical maps $h_1, \dots, h_5$ ), which hypotheses does the proof that $h_3$ is injective use?
- A. $h_2$ and $h_4$ injective, $h_1$ surjective
- B. $h_2$ and $h_4$ surjective, $h_5$ injective
- C.All four outer maps must be isomorphisms
- D.Only $h_1$ surjective and $h_5$ injective
5.
Which of the following are legitimate uses of the Freyd–Mitchell embedding theorem? (Choose all that apply.)
- A.Proving the snake lemma in $\mathrm{Sh}(X; \mathbf{Ab})$ by chasing elements
- B.Proving the four lemma in any abelian category
- C.Concluding that $\mathrm{Sh}(X; \mathbf{Ab})$ has enough projectives, because module categories do
- D.Computing $\mathrm{Ext}$ in $\mathcal{A}$ by computing $\mathrm{Ext}$ of the images in $R$ - $\mathbf{Mod}$
- E.Concluding that infinite products in $\mathcal{A}$ are exact, since they are in $R$ - $\mathbf{Mod}$
6.
On $X = \mathbb{C} \setminus \{0\}$ consider $0 \to \underline{\mathbb{Z}} \to \mathcal{O} \xrightarrow{\exp} \mathcal{O}^\times \to 0$ . Which statement is true?
- A.It is exact in sheaves, but $\exp : \mathcal{O}(X) \to \mathcal{O}^\times(X)$ is not surjective
- B.It is not exact, because $z$ has no logarithm on $X$
- C.It is exact, and so is the sequence of global sections
- D.The kernel of $\exp$ is the constant sheaf $\underline{\mathbb{C}}$
7.
The Lean theorem short_five_injective concludes Function.Injective β. Which set of hypotheses (besides commutativity) does it use?
- A.Function.Exact f g, Function.Injective f', Function.Injective α, Function.Injective γ
- B.Function.Surjective g, Function.Exact f' g', Function.Surjective α, Function.Surjective γ
- C.α and γ are linear equivalences
- D.Function.Injective α, Function.Injective γ only
8.
For an integer matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , viewed as a map $\mathbb{Z}^2 \to \mathbb{Z}^2$ , the cokernel in $\mathbf{Ab}$ is $\mathbb{Z}/d_1 \oplus \mathbb{Z}/d_2$ (with $\mathbb{Z}/0 = \mathbb{Z}$ ), where $d_1 = \gcd(a, b, c, d)$ and $d_1 d_2 = |\det M|$ (if $M = 0$ , the cokernel is $\mathbb{Z}^2$ ). Write cokernel_2x2(a, b, c, d) returning (torsion, free_rank): the sorted list of torsion orders greater than $1$ , and the rank of the free part.
Computational — Practice

Answer key — Abelian categories

1.
Abelian: finite abelian groups, finite-dimensional vector spaces, and sheaves of abelian groups. Not abelian: free abelian groups and Banach spaces (each has a monic epimorphism that is not an isomorphism: $\cdot 2 : \mathbb{Z} \to \mathbb{Z}$ , and $\ell^1 \hookrightarrow \ell^2$ ), and groups (not additive).
2.
$g \circ f = 0 = 0 \circ f$ , and $f$ is an epimorphism, so $g = 0$ . Then $\mathrm{id}_Y$ satisfies $g \circ \mathrm{id}_Y = 0$ and factors through the kernel $f$ : $\mathrm{id}_Y = f h$ . Finally $f h f = f$ and $f$ monic give $h f = \mathrm{id}_X$ .
3.
The snake lemma gives an exact sequence $0 \to \ker\alpha \to \ker\beta \to \ker\gamma \to \operatorname{coker}\alpha \to \operatorname{coker}\beta \to \operatorname{coker}\gamma \to 0$ , so $1 - 2 + 3 - 2 + 1 - d = 0$ and $d = 1$ . (The experiment abelian_categories_snake checks this alternating-sum identity in 300 random examples.)
4.
From $h_3(x) = 0$ : $h_4 f_3 (x) = 0$ needs $h_4$ injective; lifting $h_2(y) = g_1(z')$ to $z' = h_1(z)$ needs $h_1$ surjective; and $h_2 f_1(z) = h_2(y) \Rightarrow f_1(z) = y$ needs $h_2$ injective. So injectivity of $h_3$ uses $h_2, h_4$ injective and $h_1$ surjective.
5.
Freyd–Mitchell transfers statements about finite diagrams — commutativity, exactness, monos, epis, isomorphisms and the existence of morphisms — so the snake and four lemmas transfer. It does not preserve projectives or injectives, infinite limits or colimits, or derived functors such as $\mathrm{Ext}$ .
6.
Stalkwise the sequence is exact (local logarithms), so it is exact in $\mathrm{Sh}(X; \mathbf{Ab})$ . But a global logarithm of $z$ would give $\oint_{|z|=1} dz/z = 0$ , whereas this integral is $2\pi i$ . So global sections are not right exact; the obstruction lives in $H^1(X, \underline{\mathbb{Z}}) \cong \mathbb{Z}$ and is the winding number.
7.
γ (g b) = g' (β b) = 0 and γ injective give g b = 0; exactness Function.Exact f g gives b = f a; f' (α a) = β (f a) = 0 and f' injective give α a = 0; α injective gives a = 0. So the hypotheses are exactness of the top row at B and injectivity of f', α and γ.
8.
```
from math import gcd


def cokernel_2x2(a, b, c, d):
    if a == b == c == d == 0:
        return ([], 2)
    d1 = gcd(gcd(a, b), gcd(c, d))
    det = abs(a * d - b * c)
    if det == 0:
        return ([d1] if d1 > 1 else [], 1)
    d2 = det // d1
    return (sorted(x for x in (d1, d2) if x > 1), 0)
```
In the category of free abelian groups the cokernel is only the free part, so the torsion found here is exactly what that category cannot see; whenever the torsion list is nonempty, the canonical map from coimage to image there fails to be an isomorphism. This uses the determinantal-divisor formula for invariant factors, which the experiment abelian_categories_free_cokernels implements for general matrices.