0
$\begingroup$

I'm working to understand the Grothendieck topology version of the Zariski topology of schemes. Explained simply, it replaces the notion of "open subschemes" with "open immersions". So instead of $U\subseteq X$, we have $U\hookrightarrow X$.

The intersection $U\cap V$ between two open subschemes is replaced with the canonical immersion of the fiber product $(U\times_X V)\hookrightarrow X$. Is there a correspondingly simple analogue of the union, or do I have to construct it explicitly?

  • 1
    I'm not sure if this is what you want, but usually, in place of the union considerations, you just consider a covering as a surjective map from the coproduct.2012-11-03

2 Answers 2

2

A covering of $U$ is replaced by a family of morphisms $U_i\to U$ the union of whose images is $U$.

  • 0
    There's something bugging me about this answer. It doesn't feel... "intrinsic" enough. I feel the analogue of union of $U_i$'s should feel independent of $X$ in some way. But then again, the fiber product isn't really intrinsic either, being the pullback of maps to the base scheme, so I guess you can't do much better.2012-11-03
  • 0
    @Arthur: you are talking about some topology on $X$, I can't see how to make it independ of $X$. Or I misunderstand your remark.2012-11-03
  • 0
    No, you understand it perfectly. I've just been going around with a sense that the fiber product was less independent of the base space than it really is, and thought whatever analogue the union had would be the same. The confusion in my earlier comment is really me going through the process of reorganizing the (practically non-existant) intuition of "fiber product" in my head.2012-11-03
0

Maybe the following can help motivate this definition. In the case of $\mathbb{R}$-valued functions on a topological space $X$, if $\mathcal{U}$ is a covering sieve on $X$, so that the union of its open sets is $X$, then defining a function $\mathcal{U}$-locally on X is equivalent to defining a continuous function on $X$ (SGA 4 1/2, I, 2.3). One also defines a notion of vector bundles given $\mathcal{U}$-locally, and in this case the corresponding statement is that if $\mathcal{U}$ is a covering sieve, then there is an equivalence between the categories of vector bundles on $X$ and vector bundles given $\mathcal{U}$-locally (SGA 4 1/2, I, 3.3). And finally in the case of (flat) schemes over $X$, the corresponding statement is that if $\mathcal{U}$ is the sieve generated by a family $\{U_i\}$ of flat schemes over $X$, and the union of the images of $U_i$ is $X$, then there is an equivalence between the categories of quasi-coherent modules on $X$ and quasi-coherent modules given $\mathcal{U}$-locally (SGA 4 1/2, I, 4.5).