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The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$.

Let's take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ algebraically closed. If $I,J$ are radical, the standard dictionary tells us $R(I\cap J)$ is the coordinate ring of the variety $\mathbf V(I)\cup \mathbf V(J)$. Furthermore, if $I+J$ is radical then $R/(I+J)$ is the coordinate ring of the intersection $\mathbf V(I)\cap\mathbf V(J)$. Now the elements in the pullback are just pairs of functions which are consistent on the intersection, and the isomorphism tells us we can glue them to get a function defined on the union.

This has a very sheafy feel to it, yet I find the need for $I,J$ to be radical somewhat disconcerting. I don't know any scheme theory, and I'm not sure exactly how to phrase my question except:

What's the underlying sheaf here and in what context is it most natural?

I guess what I'm hoping for is a setting in which every ideal of a ring has some geometric analog, not just radicals.

Arrow
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  • Well, be careful: you might have to take the radical of $I + J$ to get a variety. If you want every ideal to have some meaning then I'm afraid that schemes are the answer. – Hoot Dec 08 '15 at 15:38
  • @Hoot I don't mind schemes at all, I'm just saying I'm not familiar with them. I forgot to assume the sum $I+J$ is radical. I'll fix that now. I would love an answer directing me through the shortest route to "every ideal has some meaning". – Arrow Dec 08 '15 at 15:45
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    Even ignoring the radical bit, I think the awkward thing here is that there's a gluing taking place over a closed subset, and that's something that's not typically covered even in a course on schemes. This paper of Schwede's is the only good reference I know. – Hoot Dec 08 '15 at 16:01

2 Answers2

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I don't know if this is what you want, but here are some related facts :

Let $F$ be any sheaf (of sets) on a topological space. If $Z$ is a closed subset, denote by $F_Z$ the sheaf $F_Z=i_*i^{-1}F$ where $i:Z\rightarrow X$ is the inclusion. Note that if $Y\subset Z$, there is an restriction mapping $F_Z\rightarrow F_Y$. Now let $Y,Z$ be two subset of $X$, then there is a cartesian square of sheaves on $X$ : $$\require{AMScd}\begin{CD}F_{Y\cup Z}@>>>F_Y\\ @VVV@VVV\\ F_Z@>>> F_{Y\cap Z} \end{CD}$$ (To prove this is indeed cartesian, just check stalks). In an abelian context, this yields a short exact sequence (giving rise to a useful long exact sequence in cohomology) : $$0\longrightarrow F_{Y\cup Z}\longrightarrow F_Y\oplus F_Z\longrightarrow F_{Y\cap Z}\longrightarrow 0.$$

The Chinese Remainder Theorem says that the same is true for schemes. More precisely, let $X$ be any scheme, $Y,Z$ be two closed subschemes defined by ideals $\mathcal{I,J}$. Let $Y\cup Z$ be the closed subscheme defined by $\mathcal{I\cap J}$ and $Y\cap Z$ be the closed subscheme defined by $\mathcal{I+J}$. Then the square $$\require{AMScd}\begin{CD}\mathcal{O}_{Y\cup Z}@>>>\mathcal{O}_Y\\ @VVV@VVV\\ \mathcal{O}_Z@>>> \mathcal{O}_{Y\cap Z} \end{CD}$$ is cartesian.

Roland
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  • I don't know much about these things. Firstly, how should I think of $i_*i^{-1}F$? Second, these squares don't look very intuitive. Is something of the form $Y\cup Z\cong Y\sqcup _{Y\cap Z}Z$ true? Where could one learn these things without much background in algebraic geometry? – Arrow Dec 08 '15 at 20:28
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    I don't know how to answer your first question. I just think of $i_*i^{-1}F$ as the restriction of $F$ to $Z$, but seen as a sheaf on $X$... The squares just mean your gluing condition : a section on $Y\cup Z$ is just a pair of sections on $Y$ and $Z$ which coincide on $Y\cap Z$. – Roland Dec 08 '15 at 20:55
  • Oh, right, you're just pushing $i^{-1}F$ back into $X$. I was thinking too formally about adjunctions. – Arrow Dec 08 '15 at 21:04
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    It is true that $Y\cup Z\simeq Y\sqcup_{Y\cap Z} Z$ if $Y$ and $Z$ are closed subschemes of $X$ (because in that case $Y\cap Z$ is a closed subscheme of $Y$ and $Z$ and the pushout exists. This is in the paper linked by Hoot). What things do you want to learn ? Because all I said was just elementary sheaf theory. The paper linked by Hoot is about gluing schemes and contains much more advanced stuff. – Roland Dec 08 '15 at 21:58
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I haven't read all of your questions (and answers) on this topic, so I hope I'm not being redundant, but you might take a look at this article by Ernst Kleinert.

Sig TM
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  • Thanks for the interesting article. For me the real question now is what is the equalizer of the canonical maps $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$. I asked this here. – Arrow Jan 13 '16 at 09:57