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On the Nullstellensätze for Stein spaces and $ C$-analytic sets


Authors: Francesca Acquistapace, Fabrizio Broglia and José F. Fernando
Journal: Trans. Amer. Math. Soc. 368 (2016), 3899-3929
MSC (2010): Primary 32C15, 32C25, 32C05, 32C07; Secondary 11E25, 26E05
Published electronically: August 20, 2015
MathSciNet review: 3453361
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Abstract: In this work we prove the real Nullstellensatz for the ring $ \EuScript {O}(X)$ of analytic functions on a $ C$-analytic set $ X\subset \mathbb{R}^n$ in terms of the saturation of Łojasiewicz's radical in $ \EuScript {O}(X)$: The ideal $ \mathcal {I}(\mathcal {Z}(\mathfrak{a}))$ of the zero-set $ \mathcal {Z}(\mathfrak{a})$ of an ideal $ \mathfrak{a}$ of $ \EuScript {O}(X)$ coincides with the saturation $ \widetilde {\sqrt [\mathit {L}]{\mathfrak{a}}}$ of Łojasiewicz's radical $ \sqrt [\mathit {L}]{\mathfrak{a}}$. If $ \mathcal {Z}(\mathfrak{a})$ has `good properties' concerning Hilbert's 17th Problem, then $ \mathcal {I}(\mathcal {Z}(\mathfrak{a}))=\widetilde {\sqrt [\mathsf {r}]{\mathfrak{a}}}$ where $ \sqrt [\mathsf {r}]{\mathfrak{a}}$ stands for the real radical of $ \mathfrak{a}$. The same holds if we replace $ \sqrt [\mathsf {r}]{\mathfrak{a}}$ with the real-analytic radical $ \sqrt [\mathsf {ra}]{\mathfrak{a}}$ of $ \mathfrak{a}$, which is a natural generalization of the real radical ideal in the $ C$-analytic setting. We revisit the classical results concerning (Hilbert's) Nullstellensatz in the framework of (complex) Stein spaces.

Let $ \mathfrak{a}$ be a saturated ideal of $ \EuScript {O}(\mathbb{R}^n)$ and $ Y_{\mathbb{R}^n}$ the germ of the support of the coherent sheaf that extends $ \mathfrak{a}\EuScript {O}_{\mathbb{R}^n}$ to a suitable complex open neighborhood of $ \mathbb{R}^n$. We study the relationship between a normal primary decomposition of $ \mathfrak{a}$ and the decomposition of $ Y_{\mathbb{R}^n}$ as the union of its irreducible components. If $ \mathfrak{a}:=\mathfrak{p}$ is prime, then $ \mathcal {I}(\mathcal {Z}(\mathfrak{p}))=\mathfrak{p}$ if and only if the (complex) dimension of $ Y_{\mathbb{R}^n}$ coincides with the (real) dimension of $ \mathcal {Z}(\mathfrak{p})$.


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Additional Information

Francesca Acquistapace
Affiliation: Dipartimento di Matematica, Università degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
Email: acquistf@dm.unipi.it

Fabrizio Broglia
Affiliation: Dipartimento di Matematica, Università degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
Email: broglia@dm.unipi.it

José F. Fernando
Affiliation: Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: josefer@mat.ucm.es

DOI: https://doi.org/10.1090/tran/6436
Keywords: Nullstellensatz, Stein space, closed ideal, radical, real Nullstellensatz, $C$-analytic set, saturated ideal, {\L}ojasiewicz's radical, convex ideal, $H$-sets, $H^{\mathsf a}$-set, real ideal, real radical, real-analytic ideal, real-analytic radical, quasi-real ideal
Received by editor(s): January 27, 2014
Received by editor(s) in revised form: March 24, 2014
Published electronically: August 20, 2015
Additional Notes: The authors were supported by Spanish GAAR MTM2011-22435. The first and second authors were also supported by Italian GNSAGA of INdAM and MIUR. This article is the fruit of the close collaboration of the authors in the last ten years and has been performed in the course of several research stays of the first two authors in the Department of Algebra at the Universidad Complutense de Madrid and of the third author in the Department of Mathematics at the Università di Pisa.
Article copyright: © Copyright 2015 American Mathematical Society