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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the Nullstellensätze for Stein spaces and $C$-analytic sets
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by Francesca Acquistapace, Fabrizio Broglia and José F. Fernando PDF
Trans. Amer. Math. Soc. 368 (2016), 3899-3929 Request permission

Abstract:

In this work we prove the real Nullstellensatz for the ring $\mathcal {O}(X)$ of analytic functions on a $C$-analytic set $X\subset \mathbb {R}^n$ in terms of the saturation of Łojasiewicz’s radical in $\mathcal {O}(X)$: The ideal $\mathcal {I}(\mathcal {Z}(\mathfrak {a}))$ of the zero-set $\mathcal {Z}(\mathfrak {a})$ of an ideal $\mathfrak {a}$ of $\mathcal {O}(X)$ coincides with the saturation $\widetilde {\sqrt [L]{\mathfrak {a}}}$ of Łojasiewicz’s radical $\sqrt [L]{\mathfrak {a}}$. 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 $\mathcal {O}(\mathbb {R}^n)$ and $Y_{\mathbb {R}^n}$ the germ of the support of the coherent sheaf that extends $\mathfrak {a}\mathcal {O}_{\mathbb {R}^n}$ 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
  • MR Author ID: 41870
  • 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
  • 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.
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 3899-3929
  • MSC (2010): Primary 32C15, 32C25, 32C05, 32C07; Secondary 11E25, 26E05
  • DOI: https://doi.org/10.1090/tran/6436
  • MathSciNet review: 3453361