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Erratum in Coherence of absolute integral closures.

(communicated by Shimomoto) [Has11, Theorem 3.58] is wrong as stated. That is completions of coherent rings w.r.t finitely generated ideals need not be flat.

Consequently Shimomoto's proof of Asgharzadeh's theorem (non-coherence of R^+ in mixed characteristic in dimension 4) is incorrect. Here is the example: (Shimomoto) consider a valuation ring ** V ** of rank 2.
Choose an element **t** in the maximal ideal but not in the height **1** prime. **V** is not **t**-adically separated and hence the **t**-adic completion can't be flat (since it has to be faithfully flat and hence injective).
This does not affect any of the main or new results. Moreover we believe assuming heavy tools (such as the direct summand theorem and Cohen-Macaulayness of R^+) there are other ways to conclude the non-coherence of R^+ in mixed characteristic in dimension **4**. The novelty of our proof is that we are able to improve on this and prove non-coherence in dimension **3** (where other techniques fail) and give an elementary and (more or less) self contained proof.
In fact just a small modification of the above argument works avoiding completions. Bha20 says that a system of parameters starting with **p** is a regular sequence on R^+. If R^+ was coherent then any permutation of it would also be a regular sequence on R^+ by Asgharzadeh-Shimomoto Corollary 3.10 (since R^+ is local if R is complete local). We know this is not true. As another note, if R^+ was coherent the main theorems of Heitmann-Ma imply the main theorem of Bha20. That is if R^+ was coherent, extended plus closure would be trivial (by standard arguments) and hence would imply a system of parameters starting with **p** is a regular sequence on R^+ which is somewhat absurd and not possible by the previous remarks.