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This is a crosspost of this MSE question.

I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will trivially imply the usual version if they are assumed comaximal.

The answer to this question prompts me to ask what is the equalizer of the natural arrows below, sending $(r_j+I_j)_j$ to $(r_j+I_j+I_i)_{i,j}$ and $(r_i+I_i+I_j)_{i,j}$: $$\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$$ The elegant example given there shows the equalizer is not generally $R/\bigcap_j I_j$, which was my naive geometric intuition. This can be rephrased as saying the closed pasting lemma does not hold for regular functions. I'm having trouble figuring out what is the equalizer. Unfortunately, I just don't have enough feel/knowledge/experience to figure it out.

Update. The comment by Steven Landsburg directs me to this MO question and the answer to it mentioning a paper by Kleinert. Although it is certainly related, I dont think it answers my question (apologies if I'm wrong). Maybe my initial phrasing was vague, so I'll try and clarify below.

I am not looking for conditions which guarantee $R/\bigcap_jI_i$ is the equalizer of the above arrows. Instead, I would like to understand geometrically what the general equalizer is, so that I can understand the geometric issues that arise when one looks at the sheaf condition for finite closed coverings. Julian Rosen's answer hints tangent spaces may be involved. This is exactly along the lines of what I'm looking for - something like "the equalizer is comrpised of elements of $R/\bigcap _j I_j$ that satisfy some geometric conditions".

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    $\begingroup$ See the first answer here:mathoverflow.net/questions/21782/…. $\endgroup$
    – Steven Landsburg
    Commented Jan 14, 2016 at 23:49
  • $\begingroup$ @StevenLandsburg I edited the question to clarify what exactly I'm looking for. (I think) It's not quite the same as the answer you linked to. $\endgroup$
    – Arrow
    Commented Jan 15, 2016 at 15:05
  • $\begingroup$ As Julian already mentioned, the equalizer is the ring of families of sections on $V(I_i)$ which are compatible on the intersections $V(I_i) \cap V(I_j)$. I don't think that in general there is a different geometric description. (It is more interesting to find out when the sheaf condition is satisfied, i.e. the sections glue to a section on the union of the $V(I_i)$.) $\endgroup$
    – HeinrichD
    Commented Sep 23, 2016 at 12:52

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