# Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following.

There are two main advantages of this approach. Firstly that we have managed to make the complex $RF(A)$, rather than its less powerful cohomology $R^iF(A)$, into an invariant of A, unique up to quasi-isomorphism. Secondly, the derived functor has simply become the original functor applied to complexes (though not arbitrary ones, they have to be in $R$). This gives easier and more conceptual proofs for results about derived functors that usually require complicated double complex, spectral sequence type arguments. We give some examples [...]

Let's work in the category of $R$-modules for a fixed commutative ring $R$. The only argument I'm aware of that Tor is balanced uses spectral sequences (which can be disguised and done with only double complexes, but that's really the same thing). But Thomas says derived categories give a more conceptual proof, arguing as follows (my paraphrase): Tensor product is symmetric, thus the corresponding derived functor and its homology (the usual Tor groups) are symmetric. This follows from tensoring complexes of flat modules.

This works, assuming one has the derived category machinery built up. But to build up that machinery, it seems one needs to use spectral sequences. For example, Weibel treats the derived category in chapter 10 of his book on homological algebra, and develops Tor as a derived functor using a spectral sequence arguments (Lemma 10.6.2). So in fact, contrary to what Thomas wrote, it seems we do not avoid spectral sequences after all.

So, can the theory be developed without spectral sequences or complicated double complex arguments, giving the kind of conceptual proofs Thomas suggests exist? Or is he wrong?

I should note I haven't actually read chapter 10 of Weibel and am a complete novice when it comes to derived categories, so maybe that lemma isn't crucial and this is a stupid question. But it sure looks important. I also looked at Gelfand and Manin's book, but could not find a proof Tor is balanced. I should note that Thomas's original example is actually about sheaves, but I consider the case of modules for simplicity.

• You don't need spectral sequences, but the argument presented in [Weibel, §2.7] is basically what you get when you unfold the spectral sequence argument. – Zhen Lin Dec 25 '15 at 9:13
• @ZhenLin That was what I was alluding to when I said you could "disguise" the argument. I guess what i want to say is that Thomas is being a bit misleading because using the derived category language doesn't actually save you any trouble in proving Tor is symmetric. It just hides the trouble under the hood, so to speak. Do you agree? – user4571 Dec 25 '15 at 9:15
• Well, there is such a thing as conservation of work, yes. I suppose the difficulty is in showing that the "obvious" thing is in fact a derived functor. – Zhen Lin Dec 25 '15 at 9:22
• @ZhenLin So I think the key question, can one show the "obvious" thing is a derived functor in a way that doesn't boil down to the proof in [Weibel, §2.7]? – user4571 Dec 25 '15 at 9:25