What is the definition for a fine sheaf/ a partition of the unity on a sheaf?

Question

From Griffiths and Harris, p. 42: A sheaf $\mathcal F$ on $M$ is called fine, if for $U=\bigcup_i U_i\subseteq M$ there is a partition of the unity subordinate to the cover $(U_i\mid i\in I)$. By this we mean a collection of homomorphisms $\eta_i:\mathcal F(U_i)\to\mathcal F(U)$ such that for $\sigma\in\mathcal F(U)$ we have (i) $\operatorname{supp}(\eta_i\sigma)\subseteq U_i$ and (ii) $\sum\eta_i(\sigma|_{U_i})=\sigma$.

I have three problems with this definition:

1. Is the sum in (ii) well defined? Even if ($U_i\mid i\in I)$ is locally finite I am not sure about this.
2. Should not there be some relation between the $\eta_i$ for different choices of $U$?
3. How is $\operatorname{supp}(\sigma)$ defined? My idea was: $p\notin \operatorname{supp}(\sigma)$ iff there is a neighborhood $V$ of $p$ such that $\sigma|_V=0$. Is this right?

While thinking about these problems I came up with the following definition:

A sheaf $\mathcal F$ on $M$ is called fine, if for every locally finite open cover $(U_i\mid i\in I)$ of $M$ there are homomorphisms $$ \eta^U_i:\mathcal F(U\cap U_i)\to\mathcal F(U) $$ for every open set $U\subseteq M$ such that

1. For $V\subset U$ the following diagram commutes: $$\require{AMScd} \begin{CD} \mathcal F(U\cap U_i) @>{\eta_i^U}>> \mathcal F(U)\\ @V{r^{U\cap U_i}_{V\cap U_i}}VV @VV{r^U_V}V \\ \mathcal F(V\cap U_i) @>{\eta_i^V}>> \mathcal F(V) \end{CD}$$
2. It is $$\sum_{i\in I} \eta_i^U\circ r^U_{U\cap U_i}=\operatorname{id}:\mathcal F(U)\to\mathcal F(U),$$ where $r^U_V:\mathcal F(U)\to\mathcal F(V)$ are the restriction homomorphisms.
3. It is $$\operatorname{supp}(\eta^U_i\sigma)\subseteq U_i$$ for every $\sigma\in\mathcal F(U\cap U_i)$ and $U$ open.

Is this a reasonable definition?

Answer 1

1. An explicit assumption of the existence of the partition of unity is that this sum should be locally finite, i.e. that for any $x\in U_i$ that there's a neighborhood on which all but finitely many of the $\sigma$ vanish. If Griffiths and Harris didn't state this, it may have been an oversight, or perhaps they assumed it was implicit in "partition of unity". A more conceptual way to think about it is that we need this condition to give us the ability to construct a sheaf endomorphism which is the identity on the open set, which we can do as long as we have well-defined sums locally. See for example the nLab definition.

   Certainly if the open cover is locally finite, then the sum itself is finite and so trivially is well-defined.

2. It's not really necessary. The partition of unity is by no means unique, even for fixed $U$, so there does not have to be any particular relation between (arbitrarily chosen) partitions of unity of different open sets. There is no need for the "agreement" diagram in the first condition of your definition, because every partition of unity restricts in an obvious way to smaller open sets, and we never really need to "glue" partitions of unity because we can take $U$ to be arbitrarily large - so if we make it the entire space, by restriction we get "globally agreeing" partitions of unity, which we can make arbitrarily fine.

   So in fact there is no point in having the cover be only on $U$ rather than $M$ in the first place (and indeed most definitions I can find online only talk about $M$) - are you sure that $U$ doesn't refer to the set $\{U_i\}$ in the definition rather than their union?

3. The support of $\sigma$ is just all $p$ for which $\sigma(p)\ne 0$.