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Weak normality and seminormality in the mixed characteristic case. (English) Zbl 1514.13016

The class of weakly normal and seminormal singularities include normal singularities as well as some other singularities that, although non-normal, have been under special attention in algebraic geometry and commutative algebra. For example, Du Bois singularities in equal characteristic zero and \(F\)-finite \(F\)-injective rings with a dualizing complex in prime characteristic are weakly normal (Remark 1.11 of [J. Kollár and S. J. Kovács, J. Am. Math. Soc. 23, No. 3, 791–813 (2010; Zbl 1202.14003)] and Theorem 4.7 of [K. Schwede, Am. J. Math. 131, No. 2, 445–473 (2009; Zbl 1164.14001)]).
Weak normality implies seminormality and the two concepts coincide in equal characteristic zero, but they differ in prime characteristic.
In the second section of the paper under review, some equivalent definitions of any of these two concepts are described. For example, for reduced rings there exists an alternative definition which agrees with the original – different looking – definition while the latter (original) definition does not imply/need reducedness.
In the third section, the authors provide an example of a mixed characteristic local domain which is seminormal but it is not weakly normal. This demonstrates that, also in mixed characteristic, weak normality is strictly stronger than seminormality.
In the forth (last) section of their paper, the authors prove the following local Bertini Theorem for weak normality in mixed characteristic:
{Corollary 4.7 of the paper under review}. Let \((R,\mathfrak{m}=(x_0,\ldots,x_n),k)\) be a weakly normal complete local domain of mixed characteristic with coefficient ring \(V\) and infinite residue field. Assume that the canonical map to the integral closure, \[ R\rightarrow \overline{R}\subset \mathrm{Frac}(R) \] is unramified in codimension \(1\). Then there exists a dense Zariski open subset \(\mathcal{U}\subseteq \mathbb{P}^d(k)\) such that for every \(\alpha:=(\alpha_0:\cdots:\alpha_n)\in \mathrm{Sp}^{-1}_V(\mathcal{U})\subseteq \mathbb{P}^n(V)\) and every lift \((\tilde{\alpha}_0,\ldots,\tilde{\alpha}_n)\in V^{n+1}\backslash \{0,\ldots,0\}\) of \(\alpha\) through the quotient map \(V^{n+1}\backslash \{0\}\rightarrow \mathbb{P}^n(V)\), the followings hold:
(1) \((R/(\sum_{i=0}^n \tilde{\alpha}_ix_i)R)_\mathfrak{p}\) is weakly normal for every \(\mathfrak{p}\in V((\sum_{i=0}^n \tilde{\alpha}_ix_i)R)\cap \mathrm{Spec}^\circ(R)\).
(2) If \(\mathrm{depth}(R)\ge 3\), then \(R/(\sum_{i=0}^n \tilde{\alpha}_ix_i)R\) is weakly normal.
The assumption on the integral closure map \(R\rightarrow \overline{R}\) in the above corollary is satisfied when \(R\) is \((R_1)\). Then the authors provide an example of a weakly normal non-normal complete local domain which is \((R_1)\) (Example 4.8 of the paper).
Then the authors end the paper by proposing some questions. For example they ask about the necessity of the infiniteness of the residue field \(k\) in the above corollary, or the necessity of the condition on the integral closure map. Referring to some results in the literature, they support the idea that the assumption on the integral closure map can be really necessary.

MSC:

13E05 Commutative Noetherian rings and modules
13F45 Seminormal rings
13J10 Complete rings, completion
13N05 Modules of differentials
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References:

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