Computer stalks: do route restrictions act like restrictions?

Intend that $X$ is a topological room with a sheaf of rings $\mathcal{O}_X$. As a whole, the stalk at a factor $p \in X$ is the straight restriction of the rings $\mathcal{O}_X(U)$ for all open collections $U$ having $p$.

Below are 2 inquiries on calculating stalks - I assume both need to hold true, given that a straight restriction needs to be some type of "restricting procedure", yet that's much from encouraging for me.

Can I calculate the stalk of $\mathcal{O}_X$ at a factor $p \in X$ by just restricting over standard open collections of $X$ having $p$?

Can I calculate the stalk of $\mathcal{O}_X$ at a factor $p \in X$ by leaving out some limited variety of "huge" open collections around $p$, and afterwards restricting over the continuing to be open collections around $p$?

2019-05-05 07:12:24
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Answers: 3

I assume there is a missing out on word in Akhil Mathew's solution : and also it's "filteringed system".

You can do that due to the fact that stalks are filteringed system colimits (also known as "straight restrictions").

For filteringed system colimits, $\varinjlim_i X_i$, you can take reps of components $x \in \varinjlim_i X_i$ for some $i$ coming from the set of indexes $I$ (in our instance, the open collections $U$). That is, you can locate some $i$ and also $x_i \in X_i$ that mosts likely to your $x$ via the global arrowhead $X_i \longrightarrow \varinjlim_i X_i$. As an example, every component of the stalk $O_{X,p}$ can be stood for by an area $f \in O_X(U)$ for some open set $U$.

Yet this is not real for various other sort of colimits.

As an example, take the push - out of 2 arrowheads $f: A \longrightarrow B$ and also $g: A \longrightarrow C$ in the group of, claim, abelian teams. Components of this push out $B \oplus_A C$ are courses of sets $(b,c) \in B\oplus C$, where you quotient out components of the kind $(f(a), 0) - (0, g(a))$, for all $a\in A$. That is, $(f(a),0) = (0,g(a))$ in $B\oplus_A C$.

Components of $B\oplus_A C$ can not be stood for, as a whole, by components originating from simply $B$ or $C$, which are of the kind $(b,0)$ or $(0,c)$, specifically : so, for a basic $(b,c) \in B\oplus_A C$ there is no $b \in B$, neither $c\in C$, that represents it.

2019-05-08 21:49:13

Yes. The basic declaration is the adhering to : restriction over a poset amounts to restrict over its any kind of coinitial part. Official evidence is very easy (tip : construct maps in both instructions) and also informally it's an analogue of "subsequence has the very same restriction as a series" theory.

2019-05-08 10:03:41

Basically, below's a means to consider the stalk that is extra "down - to - planet" than straight restrictions. A component of the stalk $O_x$ is offered by a set $(f, U)$ where $f$ is an area over the open set $U$ and also $U$ has $x$. 2 sets $(f,U), (g,V)$ are taken into consideration equal if $f=g$ on an area $W$ of $x$ (had in $U \cap V$).

With this definition, it's very easy to see that what takes place at $x$ does not rely on what takes place on $F$, where $F$ is any kind of shut set disjoint from $x$. The stalk is a totally neighborhood building and construction.

When it comes to why this amounts the straight restriction : that's a straight effect of just how the building and construction operates in the majority of acquainted groups with which one could specify a sheaf (collections, teams, rings, etc)

2019-05-08 09:56:09