Axioms for sheaf cohomology

Question

Let $R$ be a commutative ring and $X$ a topological space. Define a sheafy cohomology theory (see here) to be a collection of functors $\mathrm{H}^q:\mathrm{Sh}(X;R\mathrm{Mod})\to R\mathrm{Mod}$ such that the following conditions are satisfied:

• If $q<0$, then $\mathrm{H}^q(\mathscr{F})=0$. If $q=0$, then there is an isomorphism $\alpha_\mathscr{F}:\mathrm{H}^0(\mathscr{F})\to\mathscr{F}(X)$.

• If $\mathscr{F}$ is flasque or fine, then $\mathrm{H}^q(\mathscr{F})=0$ for $q>0$.

• If $0\to\mathscr{F}\to\mathscr{G}\to\mathscr{H}\to0$ is a short exact sequence of sheaves, then there is a long exact sequence: $$\cdots\to\mathrm{H}^q(\mathscr{F})\to\mathrm{H}^q(\mathscr{G})\to\mathrm{H}^q(\mathscr{H})\to\mathrm{H}^{q+1}(\mathscr{F})\to\cdots$$

My question is as follows: is ordinary sheaf cohomology $\mathrm{H}^q(X,\mathscr{F})$ the only sheafy cohomology theory (removing the flasque sheaf requirement)? If not, are there any other examples of a sheafy cohomology theory?

Answer 1

With respect to the updated question: yes, there are other cohomology theories. For example, if $H$ is ordinary sheaf cohomology, then we can define a new sheaf cohomology theory $K$ by $K^q({\cal F}) = H^q({\cal F}) \times H^{q-1}({\cal F})$. (This is a special instance of a hypercohomology construction which is genuinely important in some areas.)

With respect to the original question: A. Grothendieck, "Sur quelques points d’algèbre homologique" (the Tohoku paper); Cartan-Eilenberg, "Homological algebra".