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.<\/p>\n
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<\/p>\n
b^n + \\sum_{i = 1, …, n} (-1)^i (n choose i) a_i b^{n – i} = 0<\/p>\n
This is a result of Reid, Roberts, Singh, see [1, equation 5.1]. These authors use weakly subintegral extension<\/em> 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<\/a> 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.<\/p>\n [1] Reid, Les; Roberts, Leslie G., Singh, Balwant, On weak subintegrality,<\/em> J. Pure Appl. Algebra 114<\/strong> (1996), no. 1, 93\u2013109.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading