Functional analysis/Algebraic geometry
Functional calculus on Nœtherian schemes
Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 57-61.

The present note is devoted to the functional calculus problem for sections of a quasi-coherent sheaf on a Nœtherian scheme. We prove scheme-theoretic analogs of the known results on the multivariable holomorphic functional calculus over Fréchet modules which are mainly due to of J. Taylor and M. Putinar. The generalization of the Taylor joint spectrum considered in the paper leads to subvarieties of an algebraic variety over an algebraically closed field. In particular, every algebraic variety is represented as the joint spectrum of related coordinate multiplication operators.

La présente Note est consacrée à un problème de calcul fonctionnel sur les sections d'un faisceau quasi cohérent d'un schéma nœthérien. Nous démontrons des analogues des résultats connus du calcul fonctionnel holomorphe en plusieurs variables sur les modules de Fréchet, essentiellement dus à J. Taylor et M. Putinar. Nous considérons un analogue du spectre joint de Taylor dans un cadre très général, conduisant à des sous-variétés d'une variété algébrique sur un corps algébriquement clos. En particulier, toute variété algébrique est réalisée comme le spectre joint des opérateurs de multiplication par les coordonnées correspondantes.

Published online:
DOI: 10.1016/j.crma.2014.10.007
Dosi, Anar 1

1 Middle East Technical University Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey
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     title = {Functional calculus on {N{\oe}therian} schemes},
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Dosi, Anar. Functional calculus on Nœtherian schemes. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 57-61. doi : 10.1016/j.crma.2014.10.007.

[1] Beltiţă, D. Spectrum for a solvable Lie algebra of operators, Stud. Math., Volume 135 (1999) no. 2, pp. 163-178

[2] Bourbaki, N. Commutative Algebra, Mir, Moscow, 1971 (Ch. 1–7)

[3] Cade, P.; Yang, R. Projective spectrum and cyclic cohomology, J. Funct. Anal., Volume 265 (2013), pp. 1916-1933

[4] Cho, M.; Takaguchi, M. Joint spectra of matrices, Sci. Rep. Hirosaki Univ., Volume 26 (1979), pp. 15-19

[5] Colojoařa, I.; Foiaş, C. Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968

[6] Dosi, A.A. Formally-radical functions in elements of a nilpotent Lie algebra and noncommutative localizations, Algebra Colloq., Volume 17 (2010) no. 1, pp. 749-788

[7] Dosi, A.A. Taylor functional calculus for supernilpotent Lie algebra of operators, J. Oper. Theory, Volume 63 (2010) no. 1, pp. 101-126

[8] Dosiev, A.A. Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems, J. Funct. Anal., Volume 230 (2006) no. 2, pp. 446-493

[9] Eschmeier, J.; Putinar, M. Spectral theory and sheaf theory, III, J. Reine Angew. Math., Volume 354 (1984), pp. 150-163

[10] Eschmeier, J.; Putinar, M. Spectral Decompositions and Analytic Sheaves, London Math. Soc., Clarendon Press, Oxford, 1996

[11] Hartshorne, R. (Grad. Texts Math.), Volume vol. 52, Springer-Verlag (1977), p. 496

[12] Helemskii, A.Ya. Homology of Banach and Topological Algebras, MGU, 1986

[13] Putinar, M. Functional calculus with sections of an analytic space, J. Oper. Theory, Volume 4 (1980), pp. 297-306

[14] Putinar, M. Spectral theory and sheaf theory. II, Math. Z., Volume 192 (1986), pp. 473-490

[15] Putinar, M. Spectral theory and sheaf theory. IV, Proc. Symp. Pure Math., Volume 51 (1990), pp. 273-293

[16] Quillen, D. Projective modules over polynomial rings, Invent. Math., Volume 36 (1976), pp. 167-171

[17] Taylor, J.L. A joint spectrum for several commuting operators, J. Funct. Anal., Volume 6 (1970), pp. 172-191

[18] Taylor, J.L. A general framework for a multi-operator functional calculus, Adv. Math., Volume 9 (1972), pp. 183-252

[19] Vasilescu, F.-H. Analytic Functional Calculus and Spectral Decompositions, Reidel, Dordrecht, The Netherlands, 1982

[20] Yang, R. Projective spectrum in Banach algebras, J. Topol. Anal., Volume 1 (2009) no. 3, pp. 289-306

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