Base change for the functors Tor. (Q1882764)

If \(K\) is a commutative ring and \(\mathcal{B}\) a small category then the category of \(\mathcal{B}\)-modules is the category of all functors \(\mathcal{B}\to K\)-\(\mathcal{M}\)\textit{od} with the natural transformations as homomorphisms. If \(F\) is a \(\mathcal{B}^{op}\)-module and \(G\) is a \(\mathcal{B}\)-module, the tensor product \(F\otimes_{\mathcal{B}} G\) is defined as \((\bigoplus_{X\in Ob(\mathcal{B})}F(X)\otimes G(X))/\sim\), where \(\sim\) identifies all elements \(x\phi\otimes_{K}y\) and \(x\otimes_{K}\phi y\) with \(\phi:X\to Y\), \(x\in F(Y)\) and \(y\in G(X)\). Using projective resolutions for two \(\mathcal{B}\)-modules we can construct the derived functors Tor\(_{n}^{\mathcal{B}}(F,G)\). If \(\mathcal{B}\) and \(\mathcal{C}\) are small categories, every \(\mathcal{B}\times \mathcal{C}^{op}\)-module is called a \(\mathcal{B}\)-\(\mathcal{C}\)-module. If \(M\) is a \(\mathcal{B}\)-\(\mathcal{C}\)-module then it induces two functors \(L_{M}:\mathcal{B}^{op}\)-\({\mathcal M}\)\textit{od}\(\to \mathcal{C}^{op}\)-\({\mathcal M}\)\textit{od}, \(L_{M}(F)=F\otimes_{\mathcal{B}} M\) and \(R_{M}:\mathcal{C}\)-\({\mathcal M}\)\textit{od}\(\to\mathcal{B}\)-\({\mathcal M}\)\textit{od}, \(R_{M}(G)=M\otimes_{\mathcal{C}}G\). The main theorem of the paper proves that if \(R_{M}\) or \(L_{M}\) is exact and preserves the projective modules then there exists a natural isomorphism Tor\(_{*}^{\mathcal{B}}(F, R_{M}(G))\cong \text{Tor}_{*}^{\mathcal{C}}(L_{M}(F), G)\). Using this theorem, the author gives a simple proof for a theorem of \textit{T. Pirashvili} and \textit{B. Richter} [K-Theory 25, 39--49 (2002; Zbl 1013.16004)].