Tor sheaves on schemes

Question

I was trying to understand the definition of "Tor sheaves", but since it is defined in the derived category of sheaves of $\mathcal{O}_X$-modules and since I am not acquainted with derived categories I am quite confused and have many (possibly silly) questions in mind. I would appreciate answers to the following questions:

• Given a scheme $X$ and two $\mathcal{O}_X$-modules $\mathcal{F}$ and $\mathcal{G}$, can one compute in practice $Tor_i(\mathcal{F},\mathcal{G})$ by just choosing any flat resolution of $\mathcal{F}$ (resp. $\mathcal{G}$) and applying the functor $\_\otimes\mathcal{G}$ (resp. $\_\otimes\mathcal{F}$) and taking homologies afterwards?

• How would one prove that if $0\rightarrow \mathcal{F}'\rightarrow \mathcal{F}\rightarrow \mathcal{F}''\rightarrow 0$ is exact then there is a long exact sequence

$\dots \rightarrow Tor_{i+1}(\mathcal{F}'',\mathcal{G})\rightarrow Tor_i(\mathcal{F}',\mathcal{G})\rightarrow\dots\rightarrow Tor_1(\mathcal{F}'',\mathcal{G}) \rightarrow \rightarrow \mathcal{F}'\otimes\mathcal{G}\rightarrow \mathcal{F}\otimes\mathcal{G}\rightarrow \mathcal{F}''\otimes\mathcal{G}\rightarrow 0$

• Is it possible to define Tor sheaves without going to the derived category? It is not possible to define it as a left derived functors of $\_\otimes\mathcal{G}$, because there are not enough projectives in the category of sheaves of $\mathcal{O}_X$-modules, but can't one avoid this problem by considering flat resolutions? (I tried to define it this way, but I could not even show that it is independent of the resolution chosen).

Answer 1

Yes to your first question. Of course, one of the key things is to show that it doesn't matter which variable you use. To see this the easiest thing is to choose resolutions for both and consider the double complex and do an argument with zig-zags. This argument is given for the case of tor groups of modules in Section Tag 00LY. Exactly the same works for the case of sheaves of modules.

For your second question, choose a flat resolution $\mathcal{K}_\bullet$ of $\mathcal{G}$. Then we get a short exact sequence of complexes $$ 0 \to \mathcal{F}' \otimes \mathcal{G}_\bullet \to \mathcal{F} \otimes \mathcal{G}_\bullet \to \mathcal{F}'' \otimes \mathcal G_\bullet \to 0 $$ and you get the long exact sequence of tors from the usual application of the snake lemma.

For the third question. Yes, you do not need to use the derived category. To prove that the tor groups are independent of the chosen flat resolution you just have to show that given two flat resolutions $\mathcal G_\bullet$ and $\mathcal{G'}_\bullet$ you get the same thing. Then you first show that you can find a third resolution which comes equipped with a map to both of these. Next, if you have a map of flat resolutions, then you can take the cone on the map and you get a flat resolution of the $0$ sheaf. Finally, you have to show that, if $\mathcal G_\bullet$ is a flat resolution of the $0$ sheaf, then $\mathcal F \otimes \mathcal{G}_\bullet$ is acyclic. This can be done by a simple induction argument.

The point is however that many of these argument are the same in different situations and so learning more general theory (for example about derived categories) will help you recognize situations where the same thing works. Good luck!