Let $A$ be a commutative ring with $1$ and $M,N$ be $A$-modules. Can you give a quick proof that $\textrm{Tor}\_i(M,N) \cong \textrm{Tor}\_i(N, M)$ using derived categories?


In his Homological algebra book, Weibel proves this with an argument via a double complexes: the so-called "acyclic assembly lemma", and from what I understand this argument can be essentially reworded into the language of spectral sequences. Hartshorne's discussion (in "Residues and Duality") of derivatives of functors in two variables is quite short, but it's not clear to me if this result (commutativity of Tor) immediately follows from the relevant derived category formalism.



Commutativity of Tor
Yes, the derived category of a commutative ring is symmetric monoidal $M\otimes\_A^{\mathbb L}N=N\otimes\_A^{\mathbb L}M$ and Tor is the homology of this tensor product.


