Show that \(\otimes\) and Tor commute with direct limits: \((\varinjlim_\alpha A_\alpha) \otimes B = \varinjlim(A_\alpha \otimes B)\) and \(\operatorname{Tor}(\varinjlim A_\alpha, B) = \varinjlim \operatorname{Tor}(A_\alpha, B)\).
Solution
Let \((A_\alpha)\) be a directed system of abelian groups.
For tensor products, the universal property immediately gives \[ \left(\varinjlim_\alpha A_\alpha\right)\otimes B \cong \varinjlim_\alpha(A_\alpha\otimes B), \] since \(-\otimes B\) is a left adjoint and therefore preserves colimits.
For \(\operatorname{Tor}\), use a functorial free presentation. Let \(F(A)\) be the free abelian group on the underlying set of \(A\), with its natural surjection \(F(A)\to A\), and let \(K(A)\) be the kernel. Thus \[ 0\to K(A)\to F(A)\to A\to0 \] is functorial in \(A\). Applying this to the directed system and taking direct limits gives an exact sequence \[ 0\to \varinjlim K(A_\alpha) \to \varinjlim F(A_\alpha) \to \varinjlim A_\alpha\to0, \] because filtered colimits are exact in the category of abelian groups. The first two terms are free-resolution terms for the colimit for purposes of computing the kernel after tensoring.
Tensor the presentations with \(B\). Since tensor product commutes with direct limits and filtered colimits preserve kernels of maps occurring in these exact sequences, \[ \begin{aligned} \operatorname{Tor}(\varinjlim A_\alpha,B) &=\ker\!\left((\varinjlim K(A_\alpha))\otimes B \to(\varinjlim F(A_\alpha))\otimes B\right)\\ &\cong \varinjlim\ker\!\left(K(A_\alpha)\otimes B \to F(A_\alpha)\otimes B\right)\\ &=\varinjlim\operatorname{Tor}(A_\alpha,B). \end{aligned} \] Thus both \(\otimes\) and \(\operatorname{Tor}\) commute with directed limits.