22.34 Composition of derived tensor products
We encourage the reader to skip this section.
Let R be a ring. Let (A, \text{d}), (B, \text{d}), and (C, \text{d}) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule. Let N' be a differential graded (B, C)-module. We denote N_ B the bimodule N viewed as a differential graded B-module (forgetting about the A-structure). There is a canonical map
22.34.0.1
\begin{equation} \label{dga-equation-plain-versus-derived} N_ B \otimes _ B^\mathbf {L} N' \longrightarrow (N \otimes _ B N')_ C \end{equation}
in D(C, \text{d}). Here (N \otimes _ B N')_ C denotes the (A, C)-bimodule N \otimes _ B N' viewed as a differential graded C-module. Namely, this map comes from the fact that the derived tensor product always maps to the plain tensor product (as it is a left derived functor).
Lemma 22.34.1. Let R be a ring. Let (A, \text{d}), (B, \text{d}), and (C, \text{d}) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule. Let N' be a differential graded (B, C)-module. Assume (22.34.0.1) is an isomorphism. Then the composition
\xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes _ A^\mathbf {L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes _ B^\mathbf {L} N'} & & D(C, \text{d}) }
is isomorphic to - \otimes _ A^\mathbf {L} N'' with N'' = N \otimes _ B N' viewed as (A, C)-bimodule.
Proof.
Let us define a transformation of functors
(- \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} N' \longrightarrow - \otimes _ A^\mathbf {L} N''
To do this, let M be a differential graded A-module with property (P). According to the construction of the functor - \otimes _ A^\mathbf {L} N'' of the proof of Lemma 22.33.2 the plain tensor product M \otimes _ A N'' represents M \otimes _ A^\mathbf {L} N'' in D(C, \text{d}). Then we write
M \otimes _ A N'' = M \otimes _ A (N \otimes _ B N') = (M \otimes _ A N) \otimes _ B N'
The module M \otimes _ A N represents M \otimes _ A^\mathbf {L} N in D(B, \text{d}). Choose a quasi-isomorphism Q \to M \otimes _ A N where Q is a differential graded B-module with property (P). Then Q \otimes _ B N' represents (M \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} N' in D(C, \text{d}). Thus we can define our map via
(M \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} N' = Q \otimes _ B N' \to M \otimes _ A N \otimes _ B N' = M \otimes _ A^\mathbf {L} N''
The construction of this map is functorial in M and compatible with distinguished triangles and direct sums; we omit the details. Consider the property T of objects M of D(A, \text{d}) expressing that this map is an isomorphism. Then
if T holds for M_ i then T holds for \bigoplus M_ i,
if T holds for 2-out-of-3 in a distinguished triangle, then it holds for the third, and
T holds for A[k] because here we obtain a shift of the map (22.34.0.1) which we have assumed is an isomorphism.
Thus by Remark 22.22.5 property T always holds and the proof is complete.
\square
Let R be a ring. Let (A, \text{d}), (B, \text{d}), and (C, \text{d}) be differential graded R-algebras. We temporarily denote (A \otimes _ R B)_ B the differential graded algebra A \otimes _ R B viewed as a (right) differential graded B-module, and {}_ B(B \otimes _ R C)_ C the differential graded algebra B \otimes _ R C viewed as a differential graded (B, C)-bimodule. Then there is a canonical map
22.34.1.1
\begin{equation} \label{dga-equation-plain-versus-derived-algebras} (A \otimes _ R B)_ B \otimes _ B^\mathbf {L} {}_ B(B \otimes _ R C)_ C \longrightarrow (A \otimes _ R B \otimes _ R C)_ C \end{equation}
in D(C, \text{d}) where (A \otimes _ R B \otimes _ R C)_ C denotes the differential graded R-algebra A \otimes _ R B \otimes _ R C viewed as a (right) differential graded C-module. Namely, this map comes from the identification
(A \otimes _ R B)_ B \otimes _ B {}_ B(B \otimes _ R C)_ C = (A \otimes _ R B \otimes _ R C)_ C
and the fact that the derived tensor product always maps to the plain tensor product (as it is a left derived functor).
Lemma 22.34.2. Let R be a ring. Let (A, \text{d}), (B, \text{d}), and (C, \text{d}) be differential graded R-algebras. Assume that (22.34.1.1) is an isomorphism. Let N be a differential graded (A, B)-bimodule. Let N' be a differential graded (B, C)-bimodule. Then the composition
\xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes _ A^\mathbf {L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes _ B^\mathbf {L} N'} & & D(C, \text{d}) }
is isomorphic to - \otimes _ A^\mathbf {L} N'' for a differential graded (A, C)-bimodule N'' described in the proof.
Proof.
By Lemma 22.33.3 we may replace N and N' by quasi-isomorphic bimodules. Thus we may assume N, resp. N' has property (P) as differential graded (A, B)-bimodule, resp. (B, C)-bimodule, see Lemma 22.28.4. We claim the lemma holds with the (A, C)-bimodule N'' = N \otimes _ B N'. To prove this, it suffices to show that
N_ B \otimes _ B^\mathbf {L} N' \longrightarrow (N \otimes _ B N')_ C
is an isomorphism in D(C, \text{d}), see Lemma 22.34.1.
Let F_\bullet be the filtration on N as in property (P) for bimodules. By Lemma 22.28.5 there is a short exact sequence
0 \to \bigoplus \nolimits F_ iN \to \bigoplus \nolimits F_ iN \to N \to 0
of differential graded (A, B)-bimodules which is split as a sequence of graded (A, B)-bimodules. A fortiori this is an admissible short exact sequence of differential graded B-modules and this produces a distinguished triangle
\bigoplus \nolimits F_ iN_ B \to \bigoplus \nolimits F_ iN_ B \to N_ B \to \bigoplus \nolimits F_ iN_ B[1]
in D(B, \text{d}). Using that - \otimes _ B^\mathbf {L} N' is an exact functor of triangulated categories and commutes with direct sums and using that - \otimes _ B N' transforms admissible exact sequences into admissible exact sequences and commutes with direct sums we reduce to proving that
(F_ pN)_ B \otimes _ B^\mathbf {L} N' \longrightarrow (F_ pN)_ B \otimes _ B N'
is a quasi-isomorphism for all p. Repeating the argument with the short exact sequences of (A, B)-bimodules
0 \to F_ pN \to F_{p + 1}N \to F_{p + 1}N/F_ pN \to 0
which are split as graded (A, B)-bimodules we reduce to showing the same statement for F_{p + 1}N/F_ pN. Since these modules are direct sums of shifts of (A \otimes _ R B)_ B we reduce to showing that
(A \otimes _ R B)_ B \otimes _ B^\mathbf {L} N' \longrightarrow (A \otimes _ R B)_ B \otimes _ B N'
is a quasi-isomorphism.
Choose a filtration F_\bullet on N' as in property (P) for bimodules. Choose a quasi-isomorphism P \to (A \otimes _ R B)_ B of differential graded B-modules where P has property (P). We have to show that P \otimes _ B N' \to (A \otimes _ R B)_ B \otimes _ B N' is a quasi-isomorphism because P \otimes _ B N' represents (A \otimes _ R B)_ B \otimes _ B^\mathbf {L} N' in D(C, \text{d}) by the construction in Lemma 22.33.2. As N' = \mathop{\mathrm{colim}}\nolimits F_ pN' we find that it suffices to show that P \otimes _ B F_ pN' \to (A \otimes _ R B)_ B \otimes _ B F_ pN' is a quasi-isomorphism. Using the short exact sequences 0 \to F_ pN' \to F_{p + 1}N' \to F_{p + 1}N'/F_ pN' \to 0 which are split as graded (B, C)-bimodules we reduce to showing P \otimes _ B F_{p + 1}N'/F_ pN' \to (A \otimes _ R B)_ B \otimes _ B F_{p + 1}N'/F_ pN' is a quasi-isomorphism for all p. Then finally using that F_{p + 1}N'/F_ pN' is a direct sum of shifts of {}_ B(B \otimes _ R C)_ C we conclude that it suffices to show that
P \otimes _ B {}_ B(B \otimes _ R C)_ C \to (A \otimes _ R B)_ B \otimes _ B {}_ B(B \otimes _ R C)_ C
is a quasi-isomorphism. Since P \to (A \otimes _ R B)_ B is a resolution by a module satisfying property (P) this map of differential graded C-modules represents the morphism (22.34.1.1) in D(C, \text{d}) and the proof is complete.
\square
Lemma 22.34.3. Let R be a ring. Let (A, \text{d}), (B, \text{d}), and (C, \text{d}) be differential graded R-algebras. If C is K-flat as a complex of R-modules, then (22.34.1.1) is an isomorphism and the conclusion of Lemma 22.34.2 is valid.
Proof.
Choose a quasi-isomorphism P \to (A \otimes _ R B)_ B of differential graded B-modules, where P has property (P). Then we have to show that
P \otimes _ B (B \otimes _ R C) \longrightarrow (A \otimes _ R B) \otimes _ B (B \otimes _ R C)
is a quasi-isomorphism. Equivalently we are looking at
P \otimes _ R C \longrightarrow A \otimes _ R B \otimes _ R C
This is a quasi-isomorphism if C is K-flat as a complex of R-modules by More on Algebra, Lemma 15.59.2.
\square
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