Lemma 15.63.1. 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.

Lemma 15.63.1. 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.
$\square$

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