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Bimodule tensor totalization respects differentials and homotopies

Statement

Let $F, F^{'}$ be bounded cochain complexes of graded $(B, A)$ -bimodules and let $G, G^{'}$ be bounded cochain complexes of graded $(A, C)$ -bimodules, with degree-zero internal differentials as in Bounded graded bimodule complexes and signed tensor totalization. The signed tensor differential on $F \otimes_{A} G$ descends to the balanced tensor, preserves internal degree, commutes with the outer $B$ - and $C$ -actions, and squares to zero. If $ϕ : F \to F^{'}$ and $ψ : G \to G^{'}$ are internal-degree zero chain maps that are bimodule-linear, then $ϕ \otimes_{A} ψ$ is a chain map. These assignments preserve identities and composition, so tensoring is a bifunctor on the categories of bounded complexes and chain maps.

Use cochain homotopies of internal degree zero. Thus a homotopy $h : G \to G^{'}$ of cochain degree $- 1$ from $ψ_{0}$ to $ψ_{1}$ satisfies $ψ_{0} - ψ_{1} = d_{G^{'}} h + h d_{G}$ , and a homotopy $k : F \to F^{'}$ from $ϕ_{0}$ to $ϕ_{1}$ satisfies $ϕ_{0} - ϕ_{1} = d_{F^{'}} k + k d_{F}$ . Then the induced maps are homotopic in either variable. On a summand $F^{p} \otimes_{A} G^{q}$ , the total homotopies are

H (f \otimes g) = (- 1)^{p} f \otimes h (g), K (f \otimes g) = k (f) \otimes g .

Consequently the tensor bifunctor descends to homotopy classes in both variables.

Facts & Assumptions

Given: Bounded complexes $F, F^{'}$ of graded $(B, A)$ -bimodules and $G, G^{'}$ of graded $(A, C)$ -bimodules; their differentials, maps, and homotopies preserve internal degree and are linear for the applicable bimodule actions.

[F1]

The totalization has summands $F^{p} \otimes_{A} G^{q}$ in degree $p + q$ and differential $d_{F} \otimes 1 + (- 1)^{p} 1 \otimes d_{G}$ (Bounded graded bimodule complexes and signed tensor totalization).

[F2]

In the ordinary right-left module case the signed tensor differential is balanced and squares to zero (The tensor-total differential is balanced, well defined, and squares to zero).

[F3]

A chain homotopy $s$ satisfies $f_{n} - g_{n} = d_{n + 1}^{D} s_{n} + s_{n - 1} d_{n}^{C}$ (A chain homotopy). Reindexing chain degree $n = - p$ gives the cochain formula $f^{p} - g^{p} = d_{D}^{p - 1} s^{p} + s^{p + 1} d_{C}^{p}$ , with $s^{p} : C^{p} \to D^{p - 1}$ .

[F4]

The outer actions on a balanced tensor product descend by $s (m \otimes n) = (s m) \otimes n$ and $(m \otimes n) t = m \otimes (n t)$ ; when both are present they commute (A commuting outer scalar action descends to a tensor product).

Proof

Proof technique: direct sign calculation on elementary tensors, extended linearly to the bounded total modules.

1.1givenF1algebra

For $a \in A$ , right $A$ -linearity of $d_{F}$ and left $A$ -linearity of $d_{G}$ give $d ((f a) \otimes g) = d_{F} (f) a \otimes g + (- 1)^{p} f a \otimes d_{G} (g) = d_{F} (f) \otimes a g + (- 1)^{p} f \otimes a d_{G} (g) = d (f \otimes a g)$ ; hence the differential descends to the balanced tensor, and additivity covers zero summands.

1.2givenF1F4algebra

For $b \in B$ and $c \in C$ , bimodule-linearity gives $d ((b f) \otimes g) = b d (f \otimes g)$ and $d (f \otimes (g c)) = d (f \otimes g) c$ , so the outer actions commute with $d$ ; each differential preserves internal degree and the sign $(- 1)^{p}$ depends only on cochain degree, while [F4] supplies the descended commuting outer actions.

1.3givenF1F2algebra

Applying $d$ twice gives $d^{2} (f \otimes g) = d_{F}^{2} (f) \otimes g + ((- 1)^{p + 1} + (- 1)^{p}) d_{F} (f) \otimes d_{G} (g) + f \otimes d_{G}^{2} (g) = 0$ : the pure terms vanish by the complex identities and the mixed terms cancel, as in the ordinary calculation [F2]; the same formula covers zero differentials and zero summands.

1.4F1algebra

If $F$ is supported in $[a, b]$ and $G$ in $[c, d]$ , the total complex is supported in $[a + c, b + d]$ with finite diagonals; at the upper endpoint the differential has zero target and below the lower endpoint there is no preceding nonzero degree, so the totalization is bounded at both ends.

1.5givenF1algebra

For internal-degree-zero bimodule chain maps $ϕ : F \to F^{'}$ and $ψ : G \to G^{'}$ , their tensor is balanced and outer-linear, and $d (ϕ (f) \otimes ψ (g)) = ϕ (d_{F} f) \otimes ψ (g) + (- 1)^{p} ϕ (f) \otimes ψ (d_{G} g) = (ϕ \otimes ψ) d (f \otimes g)$ by the chain-map identities; identities and composition agree on elementary tensors and hence on the totalization.

1.6givenF1F3algebra

Let $h : G \to G^{'}$ be an internal-degree-zero bimodule homotopy of cochain degree $- 1$ with $ψ_{0} - ψ_{1} = d_{G^{'}} h + h d_{G}$ ; for $H (f \otimes g) = (- 1)^{p} f \otimes h (g)$ , the mixed terms in $d H + H d$ have coefficients $(- 1)^{p}$ and $(- 1)^{p + 1}$ and cancel, leaving $d H (f \otimes g) + H d (f \otimes g) = f \otimes (d_{G^{'}} h + h d_{G}) (g) = f \otimes (ψ_{0} - ψ_{1}) (g)$ , so $H$ is a homotopy from $1_{F} \otimes ψ_{0}$ to $1_{F} \otimes ψ_{1}$ with the sign forced by [F1].

1.7F1F3algebra

Let $k : F \to F^{'}$ be an internal-degree-zero bimodule homotopy of cochain degree $- 1$ with $ϕ_{0} - ϕ_{1} = d_{F^{'}} k + k d_{F}$ ; for $K (f \otimes g) = k (f) \otimes g$ , the mixed terms in $d K + K d$ have coefficients $(- 1)^{p - 1}$ and $(- 1)^{p}$ and cancel, leaving $d K (f \otimes g) + K d (f \otimes g) = (d_{F^{'}} k + k d_{F}) (f) \otimes g = (ϕ_{0} - ϕ_{1}) (f) \otimes g$ , so $K$ is a homotopy from $ϕ_{0} \otimes 1_{G}$ to $ϕ_{1} \otimes 1_{G}$ .

2.1step 1.6step 1.7algebra∎

Decompose $ϕ_{0} \otimes ψ_{0} - ϕ_{1} \otimes ψ_{1} = (ϕ_{0} - ϕ_{1}) \otimes ψ_{0} + ϕ_{1} \otimes (ψ_{0} - ψ_{1})$ ; postcomposing the homotopy in step 1.7 by $1_{F^{'}} \otimes ψ_{0}$ handles the first summand and precomposing the homotopy in step 1.6 by $ϕ_{1} \otimes 1_{G}$ handles the second, so their sum proves well-definedness on homotopy classes in both variables. If an input is concentrated in one cochain degree the formulas reduce to one summand with no sign from its internal degree.

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Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §2c (standard reference, not scraped)
Stacks Project, Differential Graded Algebra, §22.33, tag 09LP (standard reference, not scraped)