On the Nullstellensätze for Stein spaces and C-analytic sets

• Published in: Transactions of the American Mathematical Society Vol. 368; no. 6; pp. 3899 - 3929
• Main Authors: Acquistapace, Francesca, Broglia, Fabrizio, Fernando, José F.
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-06-2016
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/6436

In this work we prove the real Nullstellensatz for the ring 𝒪(X) of analytic functions on a C-analytic set X
⊂ ℝ
n
in terms of the saturation of Łojasiewicz's radical in 𝒪(X): The
ideal
ℐ
(
Z
(
a
)
)
of the zero-set
Z
(
a
)
of an ideal 𝔞 of 𝒪(X) coincides with the saturation
a
L
˜
of Łojasiewicz's radical
a
L
. If
Z
(
a
)
has ‘good properties’ concerning Hilbert's 17th Problem, then
ℐ
(
Z
(
a
)
)
=
a
r
˜
where
a
r
stands for the real radical of 𝔞. The same holds if we replace
a
r
with the real-analytic radical
a
ra
of 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 𝔞 be a saturated ideal of 𝒪(ℝ
n
) and Y
ℝ
n
the germ of the
support of the coherent sheaf that extends 𝔞𝒪ℝ
n
to a suitable complex open neighborhood of
ℝ
n
. We study the relationship between a normal primary decomposition of 𝔞 and the decomposition of
Y
ℝ
n
as the union of its irreducible components. If 𝔞 ≔ 𝔭 is prime, then
ℐ
(
Z
(
p
)
)
=
p
if and only if the (complex) dimension of Y
ℝ
n
coincides with the (real) dimension
of
Z
(
p
)
.
2010 Mathematics Subject Classification. Primary 32C15, 32C25, 32C05, 32C07; Secondary 11E25, 26E05.
Key words and phrases. Nullstellensatz, Stein space, closed ideal, radical, real Nullstellensatz, C-analytic set, saturated ideal,
Łojasiewicz's radical, convex ideal, H-sets, H
a-set, real ideal, real radical, real-analytic ideal, real-analytic
radical, quasi-real ideal.