(Tor)
For an \(A\)-module \(M\), the functors \(\Tor^A_i(M,\wait)\), \(i \geq 0\), are the left derived functors of the right exact functor \(M \tensor_A \wait\). A short exact sequence \(0 \to N' \to N \to N'' \to 0\) induces a long exact sequence \[ \cdots \to \Tor^A_{i+1}(M, N'') \to \Tor^A_i(M,N') \to \Tor^A_i(M,N) \to \Tor^A_i(M,N'') \to \Tor^A_{i-1}(M,N') \to \cdots \] terminating in \[ \Tor^A_1(M,N'') \to M\tensor_A N' \to M \tensor_A N \to M \tensor_A N'' \to 0 , \] so that \(\Tor^A_0(M,N) \cong M \tensor_A N\).
\(\Tor_1^A(M,N)\) vanishing for all \(N\) is equivalent to \(\Tor_i^A(M,N)\) vanishing for all \(i>0\) and all \(N\), and both are equivalent to \(M\) being flat. Moreover, \(M\) is flat if and only if \(\Tor_1^A(M,A/I)=0\) for every finitely generated ideal \(I\subseteq A\).
\(\Tor\) is computed from a projective resolution of either argument, and \(\Tor_i^A(M,N)\cong\Tor_i^A(N,M)\). Over \(\ZZ\), resolving \(\ZZ/m\) by \(0\to\ZZ\mapsvia{m}\ZZ\to\ZZ/m\to 0\) and tensoring with an abelian group \(B\) gives \(\Tor^\ZZ_1(\ZZ/m,B)\cong\ts{b\in B\st mb=0}\), the \(m\)-torsion of \(B\); in particular \(\Tor^\ZZ_1(\ZZ/m,\ZZ/n) \cong \ZZ/{\gcd(m,n)}\).
Dependencies
D-DEFDERIVDerived functorsusesD-DEFFLATFlat modulesuses
Backlinks
T-FLATCRITThe local and infinitesimal criteria for flatnessuses