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The Lumenary · Research pathways

Triangulated categories

Name: ______________________

Date: ______________________

You will learn to

State Verdier's axioms TR1–TR4 and derive their first consequences, including long exact Hom sequences and the triangulated five lemma
Show that the homotopy category $K(\mathcal{A})$ and the derived category $D(\mathcal{A})$ are triangulated, with mapping cones as distinguished triangles
Prove that every monomorphism in a triangulated category splits, and deduce that $D(\mathrm{Ab})$ is not abelian
Define t-structures and their hearts, and identify $\mathcal{A}$ with the heart of the standard t-structure on $D(\mathcal{A})$
Explain why cones are unique only up to non-unique isomorphism, and how enhancements respond, separating established results from open questions

1.
Suppose $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$ is distinguished. Which triangle is distinguished by the rotation axiom TR2?
- A. $Y \xrightarrow{v} Z \xrightarrow{w} X[1] \xrightarrow{-u[1]} Y[1]$
- B. $Y \xrightarrow{v} Z \xrightarrow{w} X[1] \xrightarrow{u[1]} Y[1]$
- C. $Z \xrightarrow{v} Y \xrightarrow{u} X \xrightarrow{w} Z[1]$
- D. $X[1] \xrightarrow{u[1]} Y[1] \xrightarrow{v[1]} Z[1] \xrightarrow{w[1]} X[2]$
2.
Over a field $k$ , let $f : k^3 \to k^5$ be a linear map of rank $2$ , viewed as a morphism in $D(k)$ between complexes concentrated in degree $0$ . What is $\dim H^{-1}(C(f)) + \dim H^0(C(f))$ for its cone?
3.
In the proof that every monomorphism $u : X \to Y$ in a triangulated category splits, we deduce $w = 0$ from $u[1]\, w = 0$ . Which fact justifies cancelling $u[1]$ ?
- A. $u[1]$ is a monomorphism, because the shift $[1]$ is an equivalence and equivalences preserve monomorphisms
- B. $u[1]$ is an isomorphism
- C.Every map into $X[1]$ is zero
- D.The octahedral axiom
4.
Apply $\operatorname{Hom}(\Z/2, -)$ to the distinguished triangle $\Z \xrightarrow{2} \Z \to \Z/2 \xrightarrow{\delta} \Z[1]$ in $D(\mathrm{Ab})$ . What is the induced map $\delta_* : \operatorname{Hom}(\Z/2, \Z/2) \to \operatorname{Hom}(\Z/2, \Z[1])$ ?
- A.An isomorphism $\Z/2 \to \Z/2$
- B.The zero map
- C.A map $\Z/2 \to 0$ , since $\operatorname{Hom}(\Z/2, \Z[1]) = 0$
- D.Multiplication by $2$
5.
In $D(\mathbb{F}_5)$ take $X = \mathbb{F}_5$ , $Y = \mathbb{F}_5[1]$ and the zero map $X \to Y$ , with cone $Z = Y \oplus X[1]$ as in the lesson. How many morphisms $c : Z \to Z$ make $(\mathrm{id}_X, \mathrm{id}_Y, c)$ a morphism of triangles?
6.
Which of these triangulated categories are also abelian? (Choose all that apply.)
- A. $D(k)$ for a field $k$
- B. $D(\mathrm{Ab})$
- C. $K(\mathrm{Ab})$
- D. $D^b(kQ)$ for the quiver $Q = (1 \to 2)$
7.
For the standard t-structure on $D(\mathcal{A})$ , which subcategory is the heart?
- A.The complexes with $H^i = 0$ for all $i \ne 0$ ; it is equivalent to $\mathcal{A}$ via $X \mapsto H^0(X)$
- B.The acyclic complexes
- C. $D^{\le 0}$ , the complexes with $H^i = 0$ for $i > 0$
- D.The complexes of projective objects
8.
Which Lean statement formalises “in a pretriangulated category, every monomorphism is split”? (C carries the instances of a pretriangulated category.)
- A.theorem t {X Y : C} (f : X ⟶ Y) [Mono f] : IsSplitMono f
- B.theorem t {X Y : C} (f : X ⟶ Y) [IsSplitMono f] : Mono f
- C.theorem t {X Y : C} (f : X ⟶ Y) : Mono f
- D.theorem t {X Y : C} (f : X ⟶ Y) [Mono f] : IsIso f
9.
A linear map $f : \mathbb{Q}^n \to \mathbb{Q}^m$ , viewed in $D(\mathbb{Q})$ in degree $0$ , has cone $C(f) = (\mathbb{Q}^n \xrightarrow{f} \mathbb{Q}^m)$ in degrees $-1, 0$ . Write cone_cohomology(matrix, n) that returns [dim H^-1, dim H^0], where matrix is the list of the $m$ rows of $f$ (each of length n; it may be empty if $m = 0$ ). Use exact rational arithmetic (fractions.Fraction) to compute the rank.
Computational — Practice

Answer key — Triangulated categories

1.
(a). TR2 says $(u, v, w)$ is distinguished iff $(v, w, -u[1])$ is. Applying rotation three times shows that the shifted triangle $(u[1], v[1], -w[1])$ is distinguished, which is why option (d), without the sign, is wrong in general.
2.
$C(f) = (k^3 \xrightarrow{f} k^5)$ in degrees $-1, 0$ , so $H^{-1} = \ker f$ has dimension $3 - 2 = 1$ and $H^0 = \operatorname{coker} f$ has dimension $5 - 2 = 3$ : total $4$ . In $D(k)$ the cone is $\operatorname{coker} f \oplus (\ker f)[1]$ . (Check against the long exact sequence: $0 \to H^{-1}(C) \to k^3 \xrightarrow{f} k^5 \to H^0(C) \to 0$ .)
3.
(a). Since $[1]$ is an equivalence, it preserves monomorphisms, so $u[1] \circ w = u[1] \circ 0$ implies $w = 0$ . Then $\mathrm{id}_X$ factors through $u$ by exactness of $\operatorname{Hom}(-, X)$ on the rotated triangle, giving the retraction.
4.
(a). The sequence reads $0 = \operatorname{Hom}(\Z/2, \Z) \to \operatorname{Hom}(\Z/2, \Z/2) \xrightarrow{\delta_*} \operatorname{Ext}^1(\Z/2, \Z) \xrightarrow{2} \operatorname{Ext}^1(\Z/2, \Z)$ . Exactness on the left makes $\delta_*$ injective; multiplication by $2$ on $\Z/2$ is zero, so exactness on the right makes $\delta_*$ surjective. Thus $\delta_* : \Z/2 \to \Z/2$ is an isomorphism, sending $\mathrm{id}$ to $\delta$ , the generator of $\operatorname{Ext}^1$ .
5.
Writing $c = \begin{pmatrix} a & b \\ e & d \end{pmatrix}$ , the square with $Y \to Z$ forces the first column to be $(1, 0)$ , and the square with $Z \to X[1]$ forces the second row to be $(0, 1)$ . The entry $b = t \in \operatorname{Hom}(X[1], Y) \cong \mathbb{F}_5$ is free: $5$ fill-ins, all isomorphisms. The lesson's experiment confirms the count $p$ over $\mathbb{F}_p$ by brute force.
6.
Only (a). In (b) and (c), $2 : \Z \to \Z$ fails regularity. In (d), the indecomposable projectives of $kQ$ satisfy $\operatorname{Hom}(P_2, P_1) = k$ and $\operatorname{Hom}(P_1, P_2) = 0$ (as for $\mathrm{mod}\, kQ$ , by full faithfulness), so a non-zero $f : P_2 \to P_1$ is not regular.
7.
(a). $X \in D^{\le 0} \cap D^{\ge 0}$ means $H^i(X) = 0$ for $i \ne 0$ . Then $X \cong H^0(X)[0]$ , and since $\mathcal{A} \to D(\mathcal{A})$ is fully faithful, the heart is equivalent to $\mathcal{A}$ — and in the heart, short exact sequences are exactly distinguished triangles with all three vertices in the heart.
8.
(a). The hypothesis is the instance [Mono f], the conclusion IsSplitMono f. In the lesson it is proved by completing f to a distinguished triangle and showing that its third map vanishes.

from fractions import Fraction

def rank(rows):
    m = [[Fraction(v) for v in r] for r in rows]
    if not m:
        return 0
    r = 0
    for c in range(len(m[0])):
        piv = next((i for i in range(r, len(m)) if m[i][c] != 0), None)
        if piv is None:
            continue
        m[r], m[piv] = m[piv], m[r]
        for i in range(len(m)):
            if i != r and m[i][c] != 0:
                t = m[i][c] / m[r][c]
                m[i] = [a - t * b for a, b in zip(m[i], m[r])]
        r += 1
    return r

def cone_cohomology(matrix, n):
    m = len(matrix)
    r = rank(matrix)
    return [n - r, m - r]

The cone of $f$ is $\operatorname{coker} f \oplus (\ker f)[1]$ in $D(\mathbb{Q})$ : exactly the part of $f$ that is not an isomorphism. In particular $f$ is a quasi-isomorphism iff its cone is acyclic iff cone_cohomology returns [0, 0], i.e. iff $f$ is invertible.