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).