On the Nullstellensätze for Stein spaces and 𝐶-analytic sets

Abstract

In this work we prove the real Nullstellensatz for the ring O ( X ) \mathcal {O}(X) of analytic functions on a C C -analytic set X ⊂ R n X\subset \mathbb {R}^n in terms of the saturation of Łojasiewicz’s radical in O ( X ) \mathcal {O}(X) : The ideal I ( Z ( a ) ) \mathcal {I}(\mathcal {Z}(\mathfrak {a})) of the zero-set Z ( a ) \mathcal {Z}(\mathfrak {a}) of an ideal a \mathfrak {a} of O ( X ) \mathcal {O}(X) coincides with the saturation a L ~ \widetilde {\sqrt [L]{\mathfrak {a}}} of Łojasiewicz’s radical a L \sqrt [L]{\mathfrak {a}} . If Z ( a ) \mathcal {Z}(\mathfrak {a}) has ‘good properties’ concerning Hilbert’s 17th Problem, then I ( Z ( a ) ) = a r ~ \mathcal {I}(\mathcal {Z}(\mathfrak {a}))=\widetilde {\sqrt [\mathsf {r}]{\mathfrak {a}}} where a r \sqrt [\mathsf {r}]{\mathfrak {a}} stands for the real radical of a \mathfrak {a} . The same holds if we replace a r \sqrt [\mathsf {r}]{\mathfrak {a}} with the real-analytic radical a r a \sqrt [\mathsf {ra}]{\mathfrak {a}} of a \mathfrak {a} , which is a natural generalization of the real radical ideal in the C C -analytic setting. We revisit the classical results concerning (Hilbert’s) Nullstellensatz in the framework of (complex) Stein spaces. Let a \mathfrak {a} be a saturated ideal of O ( R n ) \mathcal {O}(\mathbb {R}^n) and Y R n Y_{\mathbb {R}^n} the germ of the support of the coherent sheaf that extends a O R n \mathfrak {a}\mathcal {O}_{\mathbb {R}^n} to a suitable complex open neighborhood of R n \mathbb {R}^n . We study the relationship between a normal primary decomposition of a \mathfrak {a} and the decomposition of Y R n Y_{\mathbb {R}^n} as the union of its irreducible components. If a := p \mathfrak {a}:=\mathfrak {p} is prime, then I ( Z ( p ) ) = p \mathcal {I}(\mathcal {Z}(\mathfrak {p}))=\mathfrak {p} if and only if the (complex) dimension of Y R n Y_{\mathbb {R}^n} coincides with the (real) dimension of Z ( p ) \mathcal {Z}(\mathfrak {p}) .