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Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor $$ \otimes_R^\mathbf{L}: \mathbf{D^-(R-mod)} \times \mathbf{D^-(R-mod)} \to \mathbf{D^-(Ab Gps)}. $$ I don't love this notation but I'm emulating Weibel for ease of translation. For reference, a nice pdf of chapter 10 of Weibel is available here.

For cochains $A,B \in \mathbf{D^-(R-mod)}$, theorem 10.6.3 reads $$ \mathbf{Tor}_i^R(A, B) \cong H^{-i}(A \otimes_R^\mathbf{L} B). $$ where $\mathbf{Tor}_i^R(A, B)$ is the hypertor functor.

Now, I am confused by the statement of the theorem because the hypertor functor is defined for chain complexes, while the total tensor product functor is defined on cochain complexes. My impression is that one makes sense of it by reindexing the cochain complexes $A,B$ to regard them as chain complexes when computing $\mathbf{Tor}_i^R(A, B)$. The proof of 10.6.3 indeed reindexes at one point, but the proof is sufficiently terse that I am struggling to understand how it plays into the statement of the theorem.

  1. How should I regard $A,B$ as cochain complexes for the total tensor product, but as chain complexes for hypertor?
  2. If the answer to 1 truly is "reindex", how can reindexing be canonical here?
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