different definitions of fine sheaves / partitions of unity

Question

I have the feeling that in the literature there are two different definitions of what a partition of unity for a sheaf is supposed to be. A partition of unity for a sheaf of abelian groups $\newcommand{\c}{\mathcal} \c{F}$ on a topological space $X$ subordinate to an open cover $(U_i)_{i \in I}$ of an open subset $V$ of $X$ is a family $(f_i)_{i \in I}$ of morphisms $\DeclareMathOperator{Hom}{Hom} f_i \in \Hom(\c{F}|_V, \c{F}|_V)$ satisfying some properties.

For the "strong" definition we require this familiy to satisfy: $\DeclareMathOperator{supp}{supp}$

• $\supp f_i \subset U_i$ for all $i$
• The family $(\supp f_i)_i$ of subsets of $V$ is locally finite
• $\sum_{i \in I} f_i = \mathrm{id}_\c{F}|_V$ (well-defined because the supports are locally finite)

The support of a section $s$ of a sheaf $\c{G}$ is defined as $$ \supp s = \{x \in U: s_x \neq 0\} $$ and we view the $f_i$ as global sections of the sheaf $\c{Hom}(\c{F}|_V, \c{F}|_V)$.

However, we can also use a weaker definition where we require that for any section $s \in \Gamma \left(W, \c{F}|_V \right)$ over some open $W \subset V$:

• $\supp f_i(s) \subset U_i$ for all $i$
• The family $(\supp f_i(s))_i$ is locally finite
• $\sum_{i \in I} f_i(s) = s$ (well-defined because the supports are locally finite)

As far as I can see, these are not equivalent. As far as I know, the weak definition is sufficient to prove that fine sheaves on a paracompact space are soft, and hence acyclic (although we could weaken this even further by only requiring partitions of unity for finite covers).

It seems to me that the weaker definition also is genuinely more useful. For example it is very easy to show that the sheaf of divisors on a manifolds is fine using the second definition but not using the first.

Is one of these definitions nonstandard? If the weaker one also is sufficient for the sheaf to be acyclic, why would one use the first one?

EDIT: To illustrate the difference between the two definitions consider the sheaf $\c{D}$ of divisors on a Riemann surface $X$. A divisor is a function $X \to \mathbb{Z}$ which is nonzero only on a closed discrete subset of $X$. For a cover $(U_i)_{i \in I}$ of $X$ it is very easy to find a "weak" partition of unity for $\c{D}$. Choose a total ordering on $I$ and put $$ f_i(s) = s \cdot \chi_{U_i \setminus \bigcup_{j < i} U_j} $$ where $\chi_{U_i \setminus \bigcup_{j < i} U_j}$ is an indicator function. Since any subset of a discrete space is closed and discrete this is again a divisor and we quickly check that this satisfies all three axioms. However if $i \in I$ is a minimal element we have $\supp f_i = \overline{U_i}$, so $(f_i)_i$ does not satisfy the conditions for being a "strong" partition of unity.

Answer 1

I just noticed that if $X$ is paracompact (which seems to be the only setting where one should care about fine sheaves) this is not that hard. Evidently, a "strong" partition of unity also is a "weak" one and we will show that the existence of "weak" partitions of unity for any open cover implies the existence of "strong" partitions for any cover (i. e. "weakly" fine and "strongly" fine are equivalent).

For a given open cover $(U_i)_i$ of $V \subset X$ we can choose a locally finite shrinking $(W_i)_i$ (Munkres, Lemma 41.6) i. e. a locally finite open cover of $V$ such that $\overline{W_i} \subset U_i$. Then there exists a "weak" partition of unity $(f_i)_i$ subordinate to $(W_i)_i$. In fact, this is also a "strong" partition of unity for $(U_i)_i$:

• $\DeclareMathOperator{supp}{supp}\supp f_i \subset \overline{W_i} \subset U_i$ since $f_i$ is equal to $0$ on $X \setminus \overline{W_i}$
• $(W_i)_i$ locally finite implies that $(\overline{W_i})_i$ is locally finite (Munkres, Lemma 39.1) and hence $(\supp f_i)_i$ is locally finite.
• By construction $\sum_i f_i = \mathrm{id}_{\mathcal{F}|_V}$