Lemma 15.61.1. Let $R$ be a ring. Let $A, B, C$ be $R$-algebras and let $B \to C$ be an $R$-algebra map. Then the induced map

is an $A$-algebra homomorphism.

The simplest example of the product maps comes from the following situation. Suppose that $K^\bullet , L^\bullet \in D(R)$. Then there are maps

15.61.0.1

\begin{equation} \label{more-algebra-equation-simple-tor-product} H^ i(K^\bullet ) \otimes _ R H^ j(L^\bullet ) \longrightarrow H^{i + j}(K^\bullet \otimes _ R^{\mathbf{L}} L^\bullet ) \end{equation}

Namely, to define these maps we may assume that one of $K^\bullet , L^\bullet $ is a K-flat complex of $R$-modules (for example a bounded above complex of free or projective $R$-modules). In that case $K^\bullet \otimes _ R^{\mathbf{L}} L^\bullet $ is represented by the complex $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$, see Section 15.57 (or Section 15.56). Next, suppose that $\xi \in H^ i(K^\bullet )$ and $\zeta \in H^ j(L^\bullet )$. Choose $k \in \mathop{\mathrm{Ker}}(K^ i \to K^{i + 1})$ and $l \in \mathop{\mathrm{Ker}}(L^ j \to L^{j + 1})$ representing $\xi $ and $\zeta $. Then we set

\[ \xi \cup \zeta = \text{class of }k \otimes l\text{ in } H^{i + j}(\text{Tot}(K^\bullet \otimes _ R L^\bullet )). \]

This make sense because the formula (see Homology, Definition 12.22.3) for the differential $\text{d}$ on the total complex shows that $k \otimes l$ is a cocycle. Moreover, if $k' = d_ K(k'')$ for some $k'' \in K^{i - 1}$, then $k' \otimes l = \text{d}(k'' \otimes l)$ because $l$ is a cocycle. Similarly, altering the choice of $l$ representing $\zeta $ does not change the class of $k \otimes l$. It is equally clear that $\cup $ is bilinear, and hence to a general element of $H^ i(K^\bullet ) \otimes _ R H^ j(L^\bullet )$ we assign

\[ \sum \xi _ i \otimes \zeta _ i \longmapsto \sum \xi _ i \cup \zeta _ i \]

in $H^{i + j}(\text{Tot}(K^\bullet \otimes _ R L^\bullet ))$.

Let $R \to A$ be a ring map. Let $K^\bullet , L^\bullet \in D(R)$. Then we have a canonical identification

15.61.0.2

\begin{equation} \label{more-algebra-equation-pullback-derived-tensor-product} (K^\bullet \otimes _ R^{\mathbf{L}} A) \otimes _ A^{\mathbf{L}} (L^\bullet \otimes _ R^{\mathbf{L}} A) = (K^\bullet \otimes _ R^{\mathbf{L}} L^\bullet ) \otimes _ R^{\mathbf{L}} A \end{equation}

in $D(A)$. It is constructed as follows. First, choose K-flat resolutions $P^\bullet \to K^\bullet $ and $Q^\bullet \to L^\bullet $ over $R$. Then the left hand side is represented by the complex $\text{Tot}((P^\bullet \otimes _ R A) \otimes _ A (Q^\bullet \otimes _ R A))$ and the right hand side by the complex $\text{Tot}(P^\bullet \otimes _ R Q^\bullet ) \otimes _ R A$. These complexes are canonically isomorphic. Thus the construction above induces products

\[ \text{Tor}^ R_ n(K^\bullet , A) \otimes _ A \text{Tor}^ R_ m(L^\bullet , A) \longrightarrow \text{Tor}_{n + m}^ R(K^\bullet \otimes _ R^\mathbf {L} L^\bullet , A) \]

which are occasionally useful.

Let $M$, $N$ be $R$-modules. Using the general construction above, the canonical map $M \otimes _ R^\mathbf {L} N \to M \otimes _ R N$ and functoriality of $\text{Tor}$ we obtain canonical maps

15.61.0.3

\begin{equation} \label{more-algebra-equation-tor-product} \text{Tor}^ R_ n(M, A) \otimes _ A \text{Tor}^ R_ m(N, A) \longrightarrow \text{Tor}_{n + m}^ R(M \otimes _ R N, A) \end{equation}

Here is a direct construction using projective resolutions. First, choose projective resolutions

\[ P_\bullet \to M, \quad Q_\bullet \to N, \quad T_\bullet \to M \otimes _ R N \]

over $R$. We have $H_0(\text{Tot}(P_\bullet \otimes _ R Q_\bullet )) = M \otimes _ R N$ by right exactness of $\otimes _ R$. Hence Derived Categories, Lemmas 13.19.6 and 13.19.7 guarantee the existence and uniqueness of a map of complexes $\mu : \text{Tot}(P_\bullet \otimes _ R Q_\bullet ) \to T_\bullet $ such that $H_0(\mu ) = \text{id}_{M \otimes _ R N}$. This induces a canonical map

\begin{align*} (M \otimes _ R^{\mathbf{L}} A) \otimes _ A^{\mathbf{L}} (N \otimes _ R^{\mathbf{L}} A) & = \text{Tot}((P_\bullet \otimes _ R A) \otimes _ A (Q_\bullet \otimes _ R A)) \\ & = \text{Tot}(P_\bullet \otimes _ R Q_\bullet ) \otimes _ R A \\ & \to T_\bullet \otimes _ R A \\ & = (M \otimes _ R N) \otimes _ R^{\mathbf{L}} A \end{align*}

in $D(A)$. Hence the products (15.61.0.3) above are constructed using (15.61.0.1) over $A$ to construct

\[ \text{Tor}^ R_ n(M, A) \otimes _ A \text{Tor}^ R_ m(N, A) \to H^{-n-m}((M \otimes _ R^{\mathbf{L}} A) \otimes _ A^{\mathbf{L}} (N \otimes _ R^{\mathbf{L}} A)) \]

and then composing by the displayed map above to end up in $\text{Tor}_{n + m}^ R(M \otimes _ R N, A)$.

An interesting special case of the above occurs when $M = N = B$ where $B$ is an $R$-algebra. In this case we obtain maps

\[ \text{Tor}_ n^ R(B, A) \otimes _ A \text{Tor}_ m^ R(B, A) \longrightarrow \text{Tor}_ n^ R(B \otimes _ R B, A) \longrightarrow \text{Tor}_ n^ R(B, A) \]

the second arrow being induced by the multiplication map $B \otimes _ R B \to B$ via functoriality for $\text{Tor}$. In other words we obtain an $A$-algebra structure on $\text{Tor}^ R_{\star }(B, A)$. This algebra structure has many intriguing properties (associativity, graded commutative, $B$-algebra structure, divided powers in some case, etc) which we will discuss elsewhere (insert future reference here).

Lemma 15.61.1. Let $R$ be a ring. Let $A, B, C$ be $R$-algebras and let $B \to C$ be an $R$-algebra map. Then the induced map

\[ \text{Tor}^ R_{\star }(B, A) \longrightarrow \text{Tor}^ R_{\star }(C, A) \]

is an $A$-algebra homomorphism.

**Proof.**
Omitted. Hint: You can prove this by working through the definitions, writing all the complexes explicitly.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)