Abstract
We characterize when an ideal of the algebra \({A(\mathbb{R}^d)}\) of real analytic functions on \({\mathbb{R}^d}\) which is determined by the germ at \({\mathbb {R}^d}\) of a complex analytic set V is complemented under the assumption that either V is homogeneous or \({V\cap \mathbb{R}^d}\) is compact. The characterization is given in terms of properties of the real singularities of V. In particular, for an arbitrary complex analytic variety V complementedness of the corresponding ideal in \({A(\mathbb{R}^d)}\) implies that the real part of V is coherent. We also describe the closed ideals of \({A(\mathbb{R}^d)}\) as sections of coherent sheaves.
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Domański, P., Vogt, D. The coherence of complemented ideals in the space of real analytic functions. Math. Ann. 347, 395–409 (2010). https://doi.org/10.1007/s00208-009-0438-1
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DOI: https://doi.org/10.1007/s00208-009-0438-1