Definition 15.59.1. Let $R$ be a ring. A complex $K^\bullet $ is called K-flat if for every acyclic complex $M^\bullet $ the total complex $\text{Tot}(M^\bullet \otimes _ R K^\bullet )$ is acyclic.
15.59 Derived tensor product
We can construct the derived tensor product in greater generality. In fact, it turns out that the boundedness assumptions are not necessary, provided we choose K-flat resolutions.
Lemma 15.59.2. Let $R$ be a ring. Let $K^\bullet $ be a K-flat complex. Then the functor transforms quasi-isomorphisms into quasi-isomorphisms.
\[ K(R) \longrightarrow K(R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes _ R K^\bullet ) \]
Proof. Follows from Lemma 15.58.4 and the fact that quasi-isomorphisms in $K(R)$ are characterized by having acyclic cones. $\square$
Lemma 15.59.3. Let $R \to R'$ be a ring map. If $K^\bullet $ is a K-flat complex of $R$-modules, then $K^\bullet \otimes _ R R'$ is a K-flat complex of $R'$-modules.
Proof. Follows from the definitions and the fact that $(K^\bullet \otimes _ R R') \otimes _{R'} L^\bullet = K^\bullet \otimes _ R L^\bullet $ for any complex $L^\bullet $ of $R'$-modules. $\square$
Lemma 15.59.4. Let $R$ be a ring. If $K^\bullet $, $L^\bullet $ are K-flat complexes of $R$-modules, then $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$ is a K-flat complex of $R$-modules.
Proof. Follows from the isomorphism
\[ \text{Tot}(M^\bullet \otimes _ R \text{Tot}(K^\bullet \otimes _ R L^\bullet )) = \text{Tot}(\text{Tot}(M^\bullet \otimes _ R K^\bullet ) \otimes _ R L^\bullet ) \]
and the definition. $\square$
Lemma 15.59.5. Let $R$ be a ring. Let $(K_1^\bullet , K_2^\bullet , K_3^\bullet )$ be a distinguished triangle in $K(R)$. If two out of three of $K_ i^\bullet $ are K-flat, so is the third.
Proof. Follows from Lemma 15.58.4 and the fact that in a distinguished triangle in $K(R)$ if two out of three are acyclic, so is the third. $\square$
Lemma 15.59.6. Let $R$ be a ring. Let $0 \to K_1^\bullet \to K_2^\bullet \to K_3^\bullet \to 0$ be a short exact sequence of complexes. If $K_3^ n$ is flat for all $n \in \mathbf{Z}$ and two out of three of $K_ i^\bullet $ are K-flat, so is the third.
Proof. Let $L^\bullet $ be a complex of $R$-modules. Then
\[ 0 \to \text{Tot}(L^\bullet \otimes _ R K_1^\bullet ) \to \text{Tot}(L^\bullet \otimes _ R K_2^\bullet ) \to \text{Tot}(L^\bullet \otimes _ R K_3^\bullet ) \to 0 \]
is a short exact sequence of complexes. Namely, for each $n, m$ the sequence of modules $0 \to L^ n \otimes _ R K_1^ m \to L^ n \otimes _ R K_2^ m \to L^ n \otimes _ R K_3^ m \to 0$ is exact by Algebra, Lemma 10.39.12 and the sequence of complexes is a direct sum of these. Thus the lemma follows from this and the fact that in a short exact sequence of complexes if two out of three are acyclic, so is the third. $\square$
Lemma 15.59.7. Let $R$ be a ring. Let $P^\bullet $ be a bounded above complex of flat $R$-modules. Then $P^\bullet $ is K-flat.
Proof. Let $L^\bullet $ be an acyclic complex of $R$-modules. Let $\xi \in H^ n(\text{Tot}(L^\bullet \otimes _ R P^\bullet ))$. We have to show that $\xi = 0$. Since $\text{Tot}^ n(L^\bullet \otimes _ R P^\bullet )$ is a direct sum with terms $L^ a \otimes _ R P^ b$ we see that $\xi $ comes from an element in $H^ n(\text{Tot}(\tau _{\leq m}L^\bullet \otimes _ R P^\bullet ))$ for some $m \in \mathbf{Z}$. Since $\tau _{\leq m}L^\bullet $ is also acyclic we may replace $L^\bullet $ by $\tau _{\leq m}L^\bullet $. Hence we may assume that $L^\bullet $ is bounded above. In this case the spectral sequence of Homology, Lemma 12.25.3 has
\[ {}'E_1^{p, q} = H^ p(L^\bullet \otimes _ R P^ q) \]
which is zero as $P^ q$ is flat and $L^\bullet $ acyclic. Hence $H^*(\text{Tot}(L^\bullet \otimes _ R P^\bullet )) = 0$. $\square$
In the following lemma by a colimit of a system of complexes we mean the termwise colimit.
Lemma 15.59.8. Let $R$ be a ring. Let $K_1^\bullet \to K_2^\bullet \to \ldots $ be a system of K-flat complexes. Then $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet $ is K-flat. More generally any filtered colimit of K-flat complexes is K-flat.
Proof. Because we are taking termwise colimits we have
\[ \mathop{\mathrm{colim}}\nolimits _ i \text{Tot}(M^\bullet \otimes _ R K_ i^\bullet ) = \text{Tot}(M^\bullet \otimes _ R \mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet ) \]
by Algebra, Lemma 10.12.9. Hence the lemma follows from the fact that filtered colimits are exact, see Algebra, Lemma 10.8.8. $\square$
Lemma 15.59.9. Let $R$ be a ring. Let $K^\bullet $ be a complex of $R$-modules. If $K^\bullet \otimes _ R M$ is acyclic for all finitely presented $R$-modules $M$, then $K^\bullet $ is K-flat.
Proof. We will use repeatedly that tensor product commute with colimits (Algebra, Lemma 10.12.9). Thus we see that $K^\bullet \otimes _ R M$ is acyclic for any $R$-module $M$, because any $R$-module is a filtered colimit of finitely presented $R$-modules $M$, see Algebra, Lemma 10.11.3. Let $M^\bullet $ be an acyclic complex of $R$-modules. We have to show that $\text{Tot}(M^\bullet \otimes _ R K^\bullet )$ is acyclic. Since $M^\bullet = \mathop{\mathrm{colim}}\nolimits \tau _{\leq n} M^\bullet $ (termwise colimit) we have
\[ \text{Tot}(M^\bullet \otimes _ R K^\bullet ) = \mathop{\mathrm{colim}}\nolimits \text{Tot}(\tau _{\leq n} M^\bullet \otimes _ R K^\bullet ) \]
with truncations as in Homology, Section 12.15. As filtered colimits are exact (Algebra, Lemma 10.8.8) we may replace $M^\bullet $ by $\tau _{\leq n}M^\bullet $ and assume that $M^\bullet $ is bounded above. In the bounded above case, we can write $M^\bullet = \mathop{\mathrm{colim}}\nolimits \sigma _{\geq -n} M^\bullet $ where the complexes $\sigma _{\geq -n} M^\bullet $ are bounded but possibly no longer acyclic. Arguing as above we reduce to the case where $M^\bullet $ is a bounded complex. Finally, for a bounded complex $M^ a \to \ldots \to M^ b$ we can argue by induction on the length $b - a$ of the complex. The case $b - a = 1$ we have seen above. For $b - a > 1$ we consider the split short exact sequence of complexes
\[ 0 \to \sigma _{\geq a + 1}M^\bullet \to M^\bullet \to M^ a[-a] \to 0 \]
and we apply Lemma 15.58.4 to do the induction step. Some details omitted. $\square$
Lemma 15.59.10. Let $R$ be a ring. For any complex $M^\bullet $ there exists a K-flat complex $K^\bullet $ whose terms are flat $R$-modules and a quasi-isomorphism $K^\bullet \to M^\bullet $ which is termwise surjective.
Proof. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\text{Mod}_ R)$ be the class of flat $R$-modules. By Derived Categories, Lemma 13.29.1 there exists a system $K_1^\bullet \to K_2^\bullet \to \ldots $ and a diagram
\[ \xymatrix{ K_1^\bullet \ar[d] \ar[r] & K_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}M^\bullet \ar[r] & \tau _{\leq 2}M^\bullet \ar[r] & \ldots } \]
with the properties (1), (2), (3) listed in that lemma. These properties imply each complex $K_ i^\bullet $ is a bounded above complex of flat modules. Hence $K_ i^\bullet $ is K-flat by Lemma 15.59.7. The induced map $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet \to M^\bullet $ is a quasi-isomorphism and termwise surjective by construction. The complex $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet $ is K-flat by Lemma 15.59.8. The terms $\mathop{\mathrm{colim}}\nolimits K_ i^ n$ are flat because filtered colimits of flat modules are flat, see Algebra, Lemma 10.39.3. $\square$
Remark 15.59.11. In fact, we can do better than Lemma 15.59.10. Namely, we can find a quasi-isomorphism $P^\bullet \to M^\bullet $ where $P^\bullet $ is a complex of $R$-modules endowed with a filtration by subcomplexes such that
$P^\bullet = \bigcup F_ pP^\bullet $,
the inclusions $F_ iP^\bullet \to F_{i + 1}P^\bullet $ are termwise split injections,
the quotients $F_{i + 1}P^\bullet /F_ iP^\bullet $ are isomorphic to direct sums of shifts $R[k]$ (as complexes, so differentials are zero).
\[ 0 = F_{-1}P^\bullet \subset F_0P^\bullet \subset F_1P^\bullet \subset \ldots \subset P^\bullet \]
This will be shown in Differential Graded Algebra, Lemma 22.20.4. Moreover, given such a complex we obtain a distinguished triangle
\[ \bigoplus F_ iP^\bullet \to \bigoplus F_ iP^\bullet \to M^\bullet \to \bigoplus F_ iP^\bullet [1] \]
in $D(R)$. Using this we can sometimes reduce statements about general complexes to statements about $R[k]$ (this of course only works if the statement is preserved under taking direct sums). More precisely, let $T$ be a property of objects of $D(R)$. Suppose that
if $K_ i \in D(R)$, $i \in I$ is a family of objects with $T(K_ i)$ for all $i \in I$, then $T(\bigoplus K_ i)$,
if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object,
$T(R[k])$ holds for all $k$.
Then $T$ holds for all objects of $D(R)$.
Lemma 15.59.12. Let $R$ be a ring. Let $\alpha : P^\bullet \to Q^\bullet $ be a quasi-isomorphism of K-flat complexes of $R$-modules. For every complex $L^\bullet $ of $R$-modules the induced map is a quasi-isomorphism.
\[ \text{Tot}(\text{id}_ L \otimes \alpha ) : \text{Tot}(L^\bullet \otimes _ R P^\bullet ) \longrightarrow \text{Tot}(L^\bullet \otimes _ R Q^\bullet ) \]
Proof. Choose a quasi-isomorphism $K^\bullet \to L^\bullet $ with $K^\bullet $ a K-flat complex, see Lemma 15.59.10. Consider the commutative diagram
\[ \xymatrix{ \text{Tot}(K^\bullet \otimes _ R P^\bullet ) \ar[r] \ar[d] & \text{Tot}(K^\bullet \otimes _ R Q^\bullet ) \ar[d] \\ \text{Tot}(L^\bullet \otimes _ R P^\bullet ) \ar[r] & \text{Tot}(L^\bullet \otimes _ R Q^\bullet ) } \]
The result follows as by Lemma 15.59.2 the vertical arrows and the top horizontal arrow are quasi-isomorphisms. $\square$
Let $R$ be a ring. Let $M^\bullet $ be an object of $D(R)$. Choose a K-flat resolution $K^\bullet \to M^\bullet $, see Lemma 15.59.10. By Lemmas 15.58.2 and 15.58.4 we obtain an exact functor of triangulated categories
\[ K(R) \longrightarrow K(R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes _ R K^\bullet ) \]
By Lemma 15.59.2 this functor induces a functor $D(R) \to D(R)$ simply because $D(R)$ is the localization of $K(R)$ at quasi-isomorphism. By Lemma 15.59.12 the resulting functor (up to isomorphism) does not depend on the choice of the K-flat resolution.
Definition 15.59.13. Let $R$ be a ring. Let $M^\bullet $ be an object of $D(R)$. The derived tensor product is the exact functor of triangulated categories described above.
\[ - \otimes _ R^{\mathbf{L}} M^\bullet : D(R) \longrightarrow D(R) \]
This functor extends the functor (15.57.0.1). It is clear from our explicit constructions that there is an isomorphism (involving a choice of signs, see below)
\[ M^\bullet \otimes _ R^{\mathbf{L}} L^\bullet \cong L^\bullet \otimes _ R^{\mathbf{L}} M^\bullet \]
whenever both $L^\bullet $ and $M^\bullet $ are in $D(R)$. Hence when we write $M^\bullet \otimes _ R^{\mathbf{L}} L^\bullet $ we will usually be agnostic about which variable we are using to define the derived tensor product with.
Lemma 15.59.14. Let $R$ be a ring. Let $K^\bullet , L^\bullet $ be complexes of $R$-modules. There is a canonical isomorphism functorial in both complexes which uses a sign of $(-1)^{pq}$ for the map $K^ p \otimes _ R L^ q \to L^ q \otimes _ R K^ p$ (see proof for explanation).
\[ K^\bullet \otimes _ R^\mathbf {L} L^\bullet \longrightarrow L^\bullet \otimes _ R^\mathbf {L} K^\bullet \]
Proof. We may and do replace the complexes by K-flat complexes $K^\bullet $ and $L^\bullet $ and then we use the commutativity constraint discussed in Section 15.58. $\square$
Lemma 15.59.15. Let $R$ be a ring. Let $K^\bullet , L^\bullet , M^\bullet $ be complexes of $R$-modules. There is a canonical isomorphism functorial in all three complexes.
\[ (K^\bullet \otimes _ R^\mathbf {L} L^\bullet ) \otimes _ R^\mathbf {L} M^\bullet = K^\bullet \otimes _ R^\mathbf {L} (L^\bullet \otimes _ R^\mathbf {L} M^\bullet ) \]
Proof. Replace the complexes by K-flat complexes and use the associativity constraint in Section 15.58. $\square$
Lemma 15.59.16. Let $R$ be a ring. Let $a : K^\bullet \to L^\bullet $ be a map of complexes of $R$-modules. If $K^\bullet $ is K-flat, then there exist a complex $N^\bullet $ and maps of complexes $b : K^\bullet \to N^\bullet $ and $c : N^\bullet \to L^\bullet $ such that
$N^\bullet $ is K-flat,
$c$ is a quasi-isomorphism,
$a$ is homotopic to $c \circ b$.
If the terms of $K^\bullet $ are flat, then we may choose $N^\bullet $, $b$, and $c$ such that the same is true for $N^\bullet $.
Proof. We will use that the homotopy category $K(R)$ is a triangulated category, see Derived Categories, Proposition 13.10.3. Choose a distinguished triangle $K^\bullet \to L^\bullet \to C^\bullet \to K^\bullet [1]$. Choose a quasi-isomorphism $M^\bullet \to C^\bullet $ with $M^\bullet $ K-flat with flat terms, see Lemma 15.59.10. By the axioms of triangulated categories, we may fit the composition $M^\bullet \to C^\bullet \to K^\bullet [1]$ into a distinguished triangle $K^\bullet \to N^\bullet \to M^\bullet \to K^\bullet [1]$. By Lemma 15.59.5 we see that $N^\bullet $ is K-flat. Again using the axioms of triangulated categories, we can choose a map $N^\bullet \to L^\bullet $ fitting into the following morphism of distinghuised triangles
\[ \xymatrix{ K^\bullet \ar[r] \ar[d] & N^\bullet \ar[r] \ar[d] & M^\bullet \ar[r] \ar[d] & K^\bullet [1] \ar[d] \\ K^\bullet \ar[r] & L^\bullet \ar[r] & C^\bullet \ar[r] & K^\bullet [1] } \]
Since two out of three of the arrows are quasi-isomorphisms, so is the third arrow $N^\bullet \to L^\bullet $ by the long exact sequences of cohomology associated to these distinguished triangles (or you can look at the image of this diagram in $D(R)$ and use Derived Categories, Lemma 13.4.3 if you like). This finishes the proof of (1), (2), and (3). To prove the final assertion, we may choose $N^\bullet $ such that $N^ n \cong M^ n \oplus K^ n$, see Derived Categories, Lemma 13.10.7. Hence we get the desired flatness if the terms of $K^\bullet $ are flat. $\square$
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