To set the stage, I first state a well known result. Namely, suppose that A ⊂ B is a ring extension such that \Spec(B) —> \Spec(A) is universally closed. Then A —> B is integral, i.e., every element b of B satisfies a monic polynomial over A.

Now suppose that A ⊂ B is a ring extension such that Spec(B) —> Spec(A) is a universal homeomorphism. Then what kind of equation does every element b of B satisfy? The answer seems to be: there exist p > 0 and elements a_1, a_2, … in A such that for each n > p we have

b^n + \sum_{i = 1, …, n} (-1)^i (n choose i) a_i b^{n – i} = 0

This is a result of Reid, Roberts, Singh, see [1, equation 5.1]. These authors use *weakly subintegral extension* to indicate a A ⊂ B which is (a) integral, (b) induces a bijection on spectra, and (c) purely inseparable extensions of residue fields. By the characterization of universal homeomorphisms of Lemma Tag 04DF this means that \Spec(B) —> \Spec(A) is a universal homeomorphism. By the same token, if φ : A —> B is a ring map inducing a universal homeomorphism on spectra, then φ(A) ⊂ B is weakly subintegral.

[1] Reid, Les; Roberts, Leslie G., Singh, Balwant, *On weak subintegrality,* J. Pure Appl. Algebra **114** (1996), no. 1, 93–109.

A much more elementary result (due to Traverso in char 0 and Yanagihara in arbitrary char) is that if A->B is a weakly subintegral extension (universal homeomorphism on Spec) then there exists b in B such that either:

(1) both b^2, b^3 are in A, or

(2) both b^p, pb are in A for some prime number p.

It is then easily seen that any weakly subintegral extension is the filtered union of all finite extensions A[b_1,b_2,b_3,…,b_n] such that b_k satisfies (1) or (2) with respect to A[b_1,b_2,…,b_{k-1}] for all 1<=k B => (B\otimes_A B)_red

is exact if and only if A->B is weakly subintegral ( Spec(B)->Spec(A) univ. homeo.). This is due to Manaresi. If we allow A->B to be non-injective, then the condition is of course that im(A->B)=ker(B=>B\otimes_A B)_red) and that the kernel of A->B is a nilideal.

Some of these results and related stuff can be found in Appendix B to my paper “Submersions and effective descent of étale morphisms”.

OK, thanks. What is the exact reference for this result? I tried to find this result but I couldn’t get access to Yanagihara’s papers. It also isn’t stated in your paper.

The first result is (an easy consequence of) Thm 1 in Yanagihara’s “Some results on weakly normal ring extensions” (J. Math Soc. Japan 35, 1983). This is also mentioned in the introduction of [1]. I find this result and Reid-Roberts-Singh’s result very beautiful and satisfying but it seems to me that their range of applications is rather limited.

The second result (which got mangled into the first in my previous comment—I don’t know why) is that

A -> B => (B\otimes_A B)_red

is exact iff A->B is *weakly normal* (I wrote weakly subintegral by mistake before), that is, if Spec(B)->Spec(A) does not factor through any non-trivial universal homeomorphism. This is due to Manaresi (1980) and this is also found in the appendix of my paper cited above.

Both these results (especially the second) are quite elementary whereas the result of RRS looks more complicated. Haven’t read their proof though.

I also think these results you mention are much easier to work with! Thank you for the details.