0
$\begingroup$

On Demailly book (Complex Analytic and Differential Geometry), on page 102 there is a result (3.22 - Strong Noetherian Property) that states that :

Let $\mathcal{F}$ be a coherent analytic sheaf on a complex manifold $M$ and let $\mathcal{F}_{1}\subset ... \subset\mathcal{F}_n\subset ...$, be an increasing sequence of coherent subsheaves of $\mathcal{F}$. Then, the sequence $(\mathcal{F}_k)$ is stationary on every compact subset of $M$.

My questions are :

(a) When he writes that $(\mathcal{F}_n)$ is an increasing sequence of subsheaves does he mean that for each open set $U$, we have that $\mathcal{F}_{n}(U)\subset\mathcal{F}_{n+1}(U)$ ? (as a sequence of submodules of $\mathcal{F}(U))$

(b) If so, does he mean that the same happens for any sequence $\mathcal{F}_{n}(K)$, with $K$ compact ? But how is this defined if $K$ is not an open set. Are we talking about restriction of sheaves ? If so, what is exactly the meaning of this last sentence ?

Thanks a lot.

1 Answers 1

1

For question (a), he means it's an increasing sequence of submodules on each stalk $\mathcal{F}_{n, x}$ (the condition you made only forms an increasing sequence of sub-presheaves of $\mathcal{F}_{n}$ ). (b) means there is a uniform integer on each compact set that stablizes all stalk sequences on that compact set.