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The bounded projective homotopy category C_m and the two shifts

Definition

Fix $m \geq 1$ , let $A_{m}$ be the Khovanov–Seidel type A algebra and let $A_{m} -mod$ be the abelian category of finitely generated graded left $A_{m}$ -modules and degree-zero maps of Finite graded A_m-modules, internal shifts and the vertex projectives. Write ${proj}^{g r} A_{m} \subseteq A_{m} -mod$ for the full subcategory of finite graded projective modules.

The category $C_{m}$ . Let $C_{m} : = K^{b} ({proj}^{g r} A_{m})$ be the full subcategory of the homotopy category $K (A_{m} -mod)$ of cochain complexes of The homotopy category of chain complexes whose objects are the bounded complexes $P$ with every term $P^{n}$ a finite graded projective left $A_{m}$ -module. Its morphisms are the homotopy classes of chain maps, and by the bounded projective comparison theorem of The bounded projective comparison for the derived category the canonical functor $Θ : C_{m} \to D^{b} (A_{m} -mod)$ into the bounded derived category $D^{b} (A_{m} -mod)$ of Finite graded A_m-modules, internal shifts and the vertex projectives is fully faithful, and every bounded complex of $A_{m}$ -modules is isomorphic in $D^{b} (A_{m} -mod)$ to the image of an object of $C_{m}$ built from the explicit finite resolutions of Finite homological dimension of the finite graded Khovanov-Seidel module category. Claims that $C_{m}$ is a triangulated category, that its shift, cone and Hom-collections are those inherited from $K (A_{m} -mod)$ , and that the internal shift acts on it as a functor are proved below; so each notation denotes.

The two shifts. The homological shift is the shift $(X [1])^{n} = X^{n + 1}, d_{X [1]} = - d_{X}$ of The shift of a chain complex, with inverse $X \mapsto X [- 1]$ . The internal shift is defined termwise by $(X {r})^{n} : = X^{n} {r}, d_{X {r}} : = d_{X},$ using the internal shift $M {r}$ of $A_{m} -mod$ , whose components satisfy $(M {r})_{d} = M_{d - r}$ . Both shifts are functors on $C_{m}$ and are automorphisms of it, and they are different functors: the homological shift moves the homological position of every term, while the internal shift leaves every homological position fixed and moves internal degrees. Concretely the complex $P_{i}$ concentrated in homological degree $0$ has its single nonzero term in degree $0$ while $P_{i} [1]$ has its single nonzero term in degree $- 1$ , whereas $P_{i} {1}$ again has its single nonzero term in homological degree $0$ ; the two shifts therefore cannot be identified, and every consumer of this page uses $[1]$ for the triangulated shift of $C_{m}$ and ${1}$ for the internal shift.

Cones. For a chain map $f : X \to Y$ of $C_{m}$ the cone is the cone of The mapping cone of a chain map, $Cone (f)^{n} = Y^{n} \oplus X^{n + 1}, d (y, x) = (d_{Y} y + f x, - d_{X} x),$ which is again an object of $C_{m}$ and carries the distinguished triangles of $C_{m}$ .

Facts & Assumptions

Given: An integer $m \geq 1$ , the algebra $A_{m}$ , the abelian category $A_{m} -mod$ of finitely generated graded left $A_{m}$ -modules, and the full subcategory ${proj}^{g r} A_{m}$ of finite graded projectives.

[F1]

$K (A)$ has the cochain complexes of an additive category $A$ as objects and homotopy classes of chain maps as morphisms, and if $A$ is abelian then $K (A)$ is triangulated with the shift $C [1]_{n} = C_{n - 1}$ , $d_{n}^{C [1]} = (- 1)^{k} d_{n - k}^{C}$ for $k = 1$ , and the distinguished cone triangles (The homotopy category of chain complexes, The shift of a chain complex, The homotopy category of an abelian category is triangulated, Triangulated category).

[F2]

$Cone (f)^{n} = Y^{n} \oplus X^{n + 1}$ with $d (y, x) = (d_{Y} y + f x, - d_{X} x)$ , and $Cone (f)$ is the third vertex of a distinguished triangle $X \overset{f}{\to} Y \to Cone (f) \to X [1]$ (The mapping cone of a chain map, Triangulated category).

[L3]

$Θ : K^{b} ({proj}^{g r} A_{m}) \to D^{b} (A_{m} -mod)$ is full and faithful, every bounded complex of $A_{m}$ -modules is isomorphic in $D^{b} (A_{m} -mod)$ to $Θ (P)$ for one of the explicitly constructed bounded complexes $P$ of finite graded projectives, and the construction uses finitely many choices only (The bounded projective comparison for the derived category).

[L4]

Every object of $A_{m} -mod$ has a finite graded projective resolution, and $pd M \leq 2 m + 1$ uniformly; the resolutions come from the explicit staircase complexes (Finite homological dimension of the finite graded Khovanov-Seidel module category).

[L5]

$A_{m} -mod$ is abelian, its internal shift $M {r}$ , $(M {r})_{d} = M_{d - r}$ , is an automorphism of it, kernels and cokernels are computed degreewise, and exactness is degreewise (Finite graded A_m-modules, internal shifts and the vertex projectives).

[L6]

A graded left $A$ -module $P$ is finite graded projective if and only if it is a degree-zero direct summand of a finite direct sum of internal shifts $A {r_{1}} \oplus \dots \oplus A {r_{n}}$ ; in particular a finite direct sum of finite graded projectives is finite graded projective (Finite graded projectives are finite shifted-free summands).

[L8]

The canonical functors $D^{b} (A) \to D (A)$ are fully faithful and exact, with essential image the complexes whose cohomology is bounded on both sides (Bounded derived localizations embed fully faithfully).

Proof

technique · direct

1.1

$C_{m}$ is a full additive triangulated subcategory of $K (A_{m} -mod)$ . By [F1] the category $K (A_{m} -mod)$ is additive and triangulated, and $C_{m}$ is a full subcategory by definition, so it is additive with the inherited addition of homotopy classes and the zero complex. It is closed under the shift $[1]$ : if $P$ is bounded with finite graded projective terms, then $P [1]$ has $(P [1])^{n} = P^{n + 1}$ , so it is bounded with the same terms. It is closed under cones: by [F2] the cone of a chain map $f : P \to Q$ of $C_{m}$ has $Cone (f)^{n} = Q^{n} \oplus P^{n + 1}$ , a finite direct sum of finite graded projectives, hence finite graded projective by [L6], and it vanishes outside the finite interval spanned by the intervals of $P$ and $Q$ , so it is bounded; the distinguished triangles of $C_{m}$ are the distinguished triangles of $K (A_{m} -mod)$ whose three vertices lie in $C_{m}$ , and [F1] transfers TR1, TR2 and TR3 to this full subcategory.

F1F2L6

1.2

The bounded projective comparison. By [L3] the canonical functor $Θ : C_{m} \to D^{b} (A_{m} -mod)$ is full and faithful and every bounded complex of $A_{m}$ -modules is isomorphic in $D^{b} (A_{m} -mod)$ to $Θ (P)$ for an explicitly constructed $P \in C_{m}$ ; by [L4] that construction exists for every bounded complex because every object of $A_{m} -mod$ has a finite graded projective resolution; by [L8] the Hom-collections of $D^{b} (A_{m} -mod)$ and the exactness of its localizations are those of the comparison. Thus $Θ$ is exact and fully faithful, and each target object has a projective replacement; the Hom-collections of $C_{m}$ are sets of homotopy classes of chain maps between sets of complexes.

L3L4L8

2.1

The internal shift acts on $C_{m}$ . Let $X$ be an object of $C_{m}$ and $r \in Z$ . The termwise assignment $(X {r})^{n} : = X^{n} {r}$ with differential $d_{X {r}} : = d_{X}$ is a cochain complex, because $d_{X}^{n} : X^{n} \to X^{n + 1}$ is a degree-zero map between graded modules and the internal shift of a degree-zero map is a degree-zero map of the shifted modules, with $d^{2} = 0$ inherited; it is bounded with finite graded projective terms because each $X^{n} {r}$ is finite graded projective by [L6] and $X^{n}$ vanishes outside the bounded interval. A chain map $f : X \to Y$ induces $f {r} : X {r} \to Y {r}$ termwise, the assignment preserves composition and identity, and it respects homotopies because a homotopy $h^{n} : X^{n} \to Y^{n - 1}$ is a family of degree-zero maps and shifting the terms shifts each $h^{n}$ ; hence ${r}$ is a functor $C_{m} \to C_{m}$ , with inverse ${- r}$ by [L5].

step 1.1L5L6

3.1

The two shifts differ. Let $P_{i} = A_{m} e_{i}$ be a vertex projective concentrated in homological degree $0$ , that is the complex with $P^{0} = P_{i}$ and $P^{n} = 0$ for $n \neq 0$ , an object of $C_{m}$ because $P_{i}$ is finite graded projective by [L5]; the shift $P [1]$ has its single nonzero term in homological degree $- 1$ with differential $d_{P [1]} = - d_{P}$ , while $P {1}$ has its single nonzero term in homological degree $0$ , namely the module $P_{i} {1}$ with $(P_{i} {1})_{d} = (P_{i})_{d - 1}$ ; the two complexes are therefore not equal, and no identification of the two shift functors is available on $C_{m}$ . Both are automorphisms: $[1]$ by step 1.1 with inverse $[- 1]$ , and ${1}$ by step 2.1 with inverse ${- 1}$ .

step 1.1step 2.1L5

4.1

Conclusion. $C_{m} = K^{b} ({proj}^{g r} A_{m})$ is an additive triangulated category whose shift $[1]$ and cones are inherited from $K (A_{m} -mod)$ (step 1.1), whose internal shift ${1}$ is a different functor acting termwise with homological degree preserved (steps 2.1 and 3.1), and which has an exact, fully faithful comparison $Θ$ to $D^{b} (A_{m} -mod)$ with an explicit replacement for each target object (step 1.2). Every term of a complex $X$ in $C_{m}$ is a finite graded projective, so every Hom-set ${Hom}_{C_{m}} (X, Y)$ is the homotopy classes of chain maps between two complexes of finite graded modules, a set; the internal shift is written ${r}$ and the homological shift $[r]$ throughout this page, and they are never identified.

step 1.1step 2.1step 3.1step 1.2∎

Depends on

Used by

The internal and homological shifts are not interchangeable Counterexample
The triangulated K₀ of the Khovanov–Seidel projective category Definition
The twist complexes Rᵢ and Rᵢ⁻¹ Definition
Bounded two-sided projective bimodule complexes act on Cₘ Lemma
Homological and internal shifts on K₀(Cₘ) Lemma

Dependency tree · two levels

57 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §§2a-2c, printed pp. 9-11 (standard reference, not scraped)
Charles Weibel, An Introduction to Homological Algebra, ch. 10 §10.4, pp. 387-390 (standard reference, not scraped)