There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying some nice condition. Then $H^i(X, E)=0$ for $i>0$."

What is the best way to formulate this result? I ask because I'm sure I'll wind up teaching this material one day, and I'd like to get this right.

I asked a similar question over at nLab. Anyone who really understands this material might want to write something over there. If I come to be such a person, I'll do the writing!

Two versions I know:

(1) Suppose that, for any open cover $U_i$ of $X$, there are functions $f_i$ and open sets $V_i$ such that $\sum f_i=1$ and $\mathrm{Supp}(f_i) \subseteq U_i$. Then, for $E$ any sheaf of $\mathcal{O}_X$ modules, $H^i(X,E)=0$. Unravelling the definition of support, $\mathrm{Supp}(f_i) \subseteq U_i$ means that there exist open sets $V_i$ such that $X = U_i \cup V_i$ and $f_i|_{V_i}=0$.

Notice that the existence of partitions of unity is sometimes stated as the weaker condition that $f_i$ is zero on the closed set $X \setminus U_i$. If $X$ is regular, I believe the existence of partitions of unity in one sense implies the other. However, I care about algebraic geometry, and affine schemes have partitions of unity in the weak sense but not the strong.

(2) Any quasi-coherent sheaf on an affine scheme has no higher sheaf cohomology. (Hartshorne III.3.5 in the noetherian case; he cites EGA III.1.3.1 for the general case.) There is a similar result for the sheaf of analytic functions: see Cartan's Theorems.

I have some ideas about how this might generalize to locally ringed spaces other than schemes, but I am holding off because someone probably knows a better answer.

Let me elaborate what one of the main issues is. If $X$ is a topological space, $x$ is a point of $X$, $E$ a sheaf of abelian groups on $X$, and $f$ a section of $E$, then there is only one meaning to the sentence "$f$ vanishes at $x$." It must mean "the image $f$ is zero in the stalk $E_x$." Unpacking the definition of a stalk, this means that there is an open set $U$ containing $x$ so that $f|_U =0$.

In this sense, affine schemes do not have partitions of unity. There do not exist two polynomials, one of which is zero in a neighborhood of $0$ and the other of which is $0$ in a neighborhood of $1$, whose sum is $1$.

Yet this seems to be the sense of partitions of unity described in most theorems. For example, in the Wikipedia definition of fine sheaves, one is working with $\mathcal{E}nd(F)$, for $F$ a general scheme of abelian groups, so this must be what they mean when they say "is 0 outside some element of the open cover". The nLab definition seems to agree.

If $X$ is a locally ringed space, and $E$ is a sheaf of $\mathcal{O}_x$ modules, then there is another sense in which $f$ can vanish at $x$. We can ask that the image of $f$ in $E_x \otimes_{\mathcal{O}_x} k_x$ be zero, where $k_x$ is the residue field of the local ring $E_x$. This is the notion that corresponds to the ordinary sense of vanishing of a section of a vector bundle.

In this sense, affine schemes do have partitions of unity. There are two polynomials, one vanishing at $0$ and the other at $1$, which sum to $1$. Because this sense is weaker, not every sheaf of modules on an affine scheme is acyclic -- only the quasi-coherent ones are.

Into what general framework does the quasi-coherence condition fit?