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The twist complexes R_i and R_i^{-1}

Definition

Fix $m \geq 1$ , let $A_{m}$ be the Khovanov–Seidel type A algebra, let $U_{i} = P_{i} \otimes_{Z}_{i} P$ be the graded $(A_{m}, A_{m})$ -bimodule of The two-sided projective bimodules U_i and their tensor functors and let $β_{i} : U_{i} \to A_{m}$ and $γ_{i} : A_{m} \to U_{i} {- 1}$ be the degree-zero bimodule maps of The Khovanov–Seidel bimodule maps β_i and γ_i. Write $C_{m} = K^{b} ({proj}^{g r} A_{m})$ for the bounded homotopy category of The bounded projective homotopy category C_m and the two shifts, and view $U_{i}$ , $A_{m}$ and $U_{i} {- 1}$ as complexes concentrated in homological degree $0$ .

The positive twist. Let $R_{i} : = [U_{i} \overset{β_{i}}{\to} A_{m}]$ be the two-term cochain complex of graded $(A_{m}, A_{m})$ -bimodules with $R_{i}^{- 1} = U_{i}$ , $R_{i}^{0} = A_{m}$ and zero in all other homological degrees, so that $A_{m}$ sits in homological degree $0$ and the differential is $β_{i}$ . That is, $R_{i}$ is the mapping cone $Cone (β_{i})$ of The mapping cone of a chain map of the map $β_{i}$ between complexes concentrated in degree $0$ , as verified below.

The negative twist. Let $R_{i}^{- 1} : = [A_{m} \overset{γ_{i}}{\to} U_{i} {- 1}]$ be the two-term cochain complex of graded $(A_{m}, A_{m})$ -bimodules with $(R_{i}^{- 1})^{0} = A_{m}$ , $(R_{i}^{- 1})^{1} = U_{i} {- 1}$ and zero elsewhere, so that here $A_{m}$ again sits in homological degree $0$ , the term $U_{i} {- 1}$ sits in homological degree $1$ , and the differential is $γ_{i}$ . This is the source's presentation of the negative twist: the mapping cone $Cone (- γ_{i})$ of $- γ_{i}$ between complexes concentrated in degree $0$ , shifted by $[- 1]$ , which has $Cone (- γ_{i})^{- 1} = U_{i} {- 1}^{- 1} \oplus A_{m}^{0} = A_{m}$ , $Cone (- γ_{i})^{0} = U_{i} {- 1}^{0} \oplus A_{m}^{1} = U_{i} {- 1}$ and differential $- γ_{i}$ , so that the shift convention $d_{X [- 1]} = - d_{X}$ returns the terms and the differential $γ_{i}$ displayed above, as verified below. The internal shift ${- 1}$ in the second term is internal and not homological: it moves internal degrees only, and the differential $γ_{i}$ is degree zero for the internal grading because the shift absorbs the degree one of $γ_{i} (1)$ .

Claims proved below. Both $R_{i}$ and $R_{i}^{- 1}$ are bounded complexes of graded $(A_{m}, A_{m})$ -bimodules whose differentials are degree-zero bimodule maps, every term of either complex is a finitely generated graded projective $A_{m}$ -module on the left and on the right, and consequently the signed totalizations $R_{i} \otimes_{A_{m}} -$ and $R_{i}^{- 1} \otimes_{A_{m}} -$ of Signed totalization of graded A_m-bimodule actions are exact endofunctors of $C_{m}$ , identified with the derived tensor products, by Bounded two-sided projective bimodule complexes act on C_m.

Scope. This item defines the two complexes and their actions and nothing else: no inverse, braid-relation or equivalence claim is made here. That $R_{i}$ and $R_{i}^{- 1}$ are mutually inverse equivalences is the source's Proposition 2.4 and belongs to the later stage of the construction.

Facts & Assumptions

Given: An integer $m \geq 1$ , the algebra $A_{m}$ with internal grading and the internal shift ${r}$ , the bimodules $U_{i}$ and maps $β_{i}, γ_{i}$ , and the category $C_{m}$ with its homological shift $[1]$ and its cones.

[L1]

$β_{i} : U_{i} \to A_{m}$ is the multiplication map and a degree-zero map of graded $(A_{m}, A_{m})$ -bimodules with $β_{i} (e_{i} \otimes e_{i}) = e_{i}$ , and $γ_{i} : A_{m} \to U_{i} {- 1}$ is a degree-zero map of graded $(A_{m}, A_{m})$ -bimodules with $γ_{i} (1) = w_{i}$ and $γ_{i} (a) = a \cdot w_{i}$ (The Khovanov–Seidel bimodule maps β_i and γ_i).

[L2]

$U_{i} = P_{i} \otimes_{Z}_{i} P$ is finite graded projective as a left $A_{m}$ -module and as a right $A_{m}$ -module, and $U_{i} \otimes_{A_{m}} -$ is exact on $A_{m} -mod$ and preserves finite graded projectives (The two-sided projective bimodules U_i and their tensor functors).

[L3]

A graded left $A_{m}$ -module is finite graded projective exactly when it is a degree-zero direct summand of a finite direct sum of internal shifts of $A_{m}$ , and shifting such a direct summand internally again gives a degree-zero direct summand of a finite direct sum of shifts; $A_{m}$ itself is free of rank one on $1 = \sum_{j} e_{j}$ (Finite graded projectives are finite shifted-free summands, Finite graded A_m-modules, internal shifts and the vertex projectives).

[L4]

For a chain map $f : X \to Y$ of complexes the cone is $Cone (f)^{n} = Y^{n} \oplus X^{n + 1}$ with $d (y, x) = (d_{Y} y + f x, - d_{X} x)$ , and a bounded complex of graded left $A_{m}$ -modules is an object of $C_{m}$ exactly when every term is a finite graded projective left $A_{m}$ -module, with $[1]$ the homological shift (The mapping cone of a chain map, The bounded projective homotopy category C_m and the two shifts).

[L5]

A bounded complex $R^{∙}$ of graded $(A_{m}, A_{m})$ -bimodules whose terms are finitely generated graded projective on both sides satisfies: $R^{∙} \otimes_{A_{m}} X \in C_{m}$ for every $X \in C_{m}$ ; $R^{∙} \otimes_{A_{m}} -$ is an exact additive functor on $C_{m}$ sending distinguished triangles to distinguished triangles; and it agrees with the derived tensor product, the identity replacements being available (Bounded two-sided projective bimodule complexes act on C_m).

Proof

technique · direct

1.1

$R_{i}$ is a complex with a degree-zero bimodule differential. The complex has exactly two nonzero terms and $β_{i}$ is a degree-zero map of graded $(A_{m}, A_{m})$ -bimodules by [L1]; the composite $β_{i}$ with the zero map $A_{m} \to R_{i}^{1} = 0$ is $0$ , so $d^{0} d^{- 1} = 0$ and the differentials square to zero, and $R_{i}$ is a bounded complex of graded bimodules in the sense of [L4] with differential of internal degree zero.

L1L4

1.2

$R_{i}^{- 1}$ is a complex with a degree-zero bimodule differential. Likewise $(R_{i}^{- 1})^{n}$ is nonzero only for $n = 0, 1$ , where the two terms are $A_{m}$ and $U_{i} {- 1}$ , the differential $γ_{i}$ is a degree-zero bimodule map by [L1] with values in $U_{i} {- 1}$ , and the composite with the zero map $U_{i} {- 1} \to R_{i}^{- 1}^{2} = 0$ is $0$ ; hence $R_{i}^{- 1}$ is a bounded complex of graded bimodules with degree-zero differential.

L1L4

1.3

Every term is finitely generated graded projective on both sides. The regular module $A_{m}$ is free of rank one on each side, hence finitely generated graded projective on the left and on the right by [L3]; the bimodule $U_{i}$ is finitely generated graded projective on both sides by [L2]; and the internal shift $U_{i} {- 1}$ of the finitely generated graded projective $U_{i}$ is again such a direct summand of a finite sum of shifts, hence finitely generated graded projective on both sides by [L3]. So all terms of $R_{i}$ and of $R_{i}^{- 1}$ satisfy the two-sided hypothesis.

L2L3

2.1

$R_{i}$ is the cone of $β_{i}$ . Regard $β_{i}$ as a chain map between the complexes $U_{i}$ and $A_{m}$ concentrated in homological degree $0$ . By [L4] the cone has $Cone (β_{i})^{- 1} = A_{m}^{- 1} \oplus U_{i}^{0} = U_{i}$ , $Cone (β_{i})^{0} = A_{m}^{0} \oplus U_{i}^{1} = A_{m}$ and $Cone (β_{i})^{n} = 0$ for $n \neq - 1, 0$ , with differential $d (y, x) = (d_{A_{m}} y + β_{i} x, - d_{U_{i}} x) = (β_{i} x, 0)$ ; this is exactly $R_{i}$ with $A_{m}$ in degree $0$ .

step 1.1L1L4

2.2

$R_{i}^{- 1}$ is the shifted cone of $- γ_{i}$ . Regard $γ_{i}$ as a chain map $A_{m} \to U_{i} {- 1}$ between complexes concentrated in degree $0$ and put $f : = - γ_{i}$ . The cone of $f$ has $Cone (f)^{- 1} = U_{i} {- 1}^{- 1} \oplus A_{m}^{0} = A_{m}$ , $Cone (f)^{0} = U_{i} {- 1}^{0} \oplus A_{m}^{1} = U_{i} {- 1}$ and nothing else, with differential $d (y, x) = (f (x), 0)$ ; shifting by $[- 1]$ , whose differentials are the negatives of those of $Cone (f)$ by [L4], places $A_{m}$ in homological degree $0$ and $U_{i} {- 1}$ in homological degree $1$ with differential $- f = γ_{i}$ . This is exactly $R_{i}^{- 1}$ , the shift inside the second term being the internal one, and it is the source's description of the negative twist as the cone of $- γ_{i}$ shifted by $[- 1]$ ; the sign isomorphism $(y, x) \mapsto (y, - x)$ identifies $Cone (γ_{i})$ with $Cone (- γ_{i})$ , so the choice of sign is immaterial.

step 1.2L1L4

2.3

The associated functors. By step 1.3 the terms of $R_{i}$ and of $R_{i}^{- 1}$ are finitely generated graded projective on both sides, so the action lemma [L5] applies to both complexes: for every $X \in C_{m}$ the totalizations $R_{i} \otimes_{A_{m}} X$ and $R_{i}^{- 1} \otimes_{A_{m}} X$ lie in $C_{m}$ , the assignments are exact additive functors on $C_{m}$ carrying distinguished triangles to distinguished triangles, and they agree with the derived tensor products $R_{i} \otimes_{A_{m}}^{L} X$ and $R_{i}^{- 1} \otimes_{A_{m}}^{L} X$ through the identity replacements.

step 1.3L5

3.1

Conclusion. The complexes $R_{i} = [U_{i} \overset{β_{i}}{\to} A_{m}]$ and $R_{i}^{- 1} = [A_{m} \overset{γ_{i}}{\to} U_{i} {- 1}]$ , with $A_{m}$ in homological degree $0$ in both cases, are well-defined bounded complexes of graded $(A_{m}, A_{m})$ -bimodules with degree-zero differentials by steps 1.1 and 1.2, they are the cone of $β_{i}$ and the cone of $- γ_{i}$ shifted by $[- 1]$ by steps 2.1 and 2.2, every term is finitely generated graded projective on both sides by step 1.3, and the resulting functors $R_{i} \otimes_{A_{m}} -$ and $R_{i}^{- 1} \otimes_{A_{m}} -$ are exact endofunctors of $C_{m}$ agreeing with derived tensor by step 2.3. The two shifts occurring here are the internal shift inside a term and the homological degree of that term, and they are kept distinct; no inverse or braid relation is asserted, and no choice principle is used.

step 2.1step 2.2step 1.3step 2.3∎

Depends on

Used by

Totalizing a two-term twist action Example

Dependency tree · two levels

41 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, §2d, printed pp. 11-12 (standard reference, not scraped)