Topology, intersections and flat modules
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- by Carmelo A. Finocchiaro and Dario Spirito PDF
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Abstract:
It is well known that, in general, multiplication by an ideal $I$ does not commute with the intersection of a family of ideals, but that this fact holds if $I$ is flat and the family is finite. We generalize this result by showing that finite families of ideals can be replaced by compact subspaces of a natural topological space, and that ideals can be replaced by submodules of an epimorphic extension of a base ring. As a particular case, we give a new proof of a conjecture by Glaz and Vasconcelos.References
- D. D. Anderson, On the ideal equation $I(B\cap C)=IB\cap IC$, Canad. Math. Bull. 26 (1983), no. 3, 331–332. MR 703406, DOI 10.4153/CMB-1983-054-3
- Pham Ngoc Ánh, Dolors Herbera, and Claudia Menini, $\textrm {AB}$-$5^\ast$ and linear compactness, J. Algebra 200 (1998), no. 1, 99–117. MR 1603265, DOI 10.1006/jabr.1997.7216
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
- David E. Dobbs, Richard Fedder, and Marco Fontana, Abstract Riemann surfaces of integral domains and spectral spaces, Ann. Mat. Pura Appl. (4) 148 (1987), 101–115 (English, with Italian summary). MR 932760, DOI 10.1007/BF01774285
- Carmelo Antonio Finocchiaro, Spectral spaces and ultrafilters, Comm. Algebra 42 (2014), no. 4, 1496–1508. MR 3169645, DOI 10.1080/00927872.2012.741875
- Carmelo A. Finocchiaro, Marco Fontana, and K. Alan Loper, The constructible topology on spaces of valuation domains, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6199–6216. MR 3105748, DOI 10.1090/S0002-9947-2013-05741-8
- Carmelo Antonio Finocchiaro, Marco Fontana, and Dario Spirito, New distinguished classes of spectral spaces: A survey, Proceedings of the Graz conference, to appear.
- Carmelo Antonio Finocchiaro and Dario Spirito, Some topological considerations on semistar operations, J. Algebra 409 (2014), 199–218. MR 3198840, DOI 10.1016/j.jalgebra.2014.04.002
- Marco Fontana and James A. Huckaba, Localizing systems and semistar operations, Non-Noetherian commutative ring theory, Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 169–197. MR 1858162
- Marco Fontana, James A. Huckaba, and Ira J. Papick, Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203, Marcel Dekker, Inc., New York, 1997. MR 1413297
- Robert Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, vol. 90, Queen’s University, Kingston, ON, 1992. Corrected reprint of the 1972 edition. MR 1204267
- Sarah Glaz and Wolmer V. Vasconcelos, Flat ideals. II, Manuscripta Math. 22 (1977), no. 4, 325–341. MR 472797, DOI 10.1007/BF01168220
- William Heinzer and Jack Ohm, Noetherian intersections of integral domains, Trans. Amer. Math. Soc. 167 (1972), 291–308. MR 296095, DOI 10.1090/S0002-9947-1972-0296095-6
- M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. MR 251026, DOI 10.1090/S0002-9947-1969-0251026-X
- John L. Kelley, General topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. MR 0370454
- Manfred Knebusch and Digen Zhang, Manis valuations and Prüfer extensions. I, Lecture Notes in Mathematics, vol. 1791, Springer-Verlag, Berlin, 2002. A new chapter in commutative algebra. MR 1937245, DOI 10.1007/b84018
- Daniel Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128 (French). MR 254100
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Bruce Olberding, Affine schemes and topological closures in the Zariski-Riemann space of valuation rings, J. Pure Appl. Algebra 219 (2015), no. 5, 1720–1741. MR 3299704, DOI 10.1016/j.jpaa.2014.07.009
- Bruce Olberding, Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings, J. Pure Appl. Algebra 212 (2008), no. 7, 1797–1821. MR 2400744, DOI 10.1016/j.jpaa.2007.11.008
- Bruce Olberding, Noetherian spaces of integrally closed rings with an application to intersections of valuation rings, Comm. Algebra 38 (2010), no. 9, 3318–3332. MR 2724221, DOI 10.1080/00927870903114979
- Bruce Olberding, Topological aspects of irredundant intersections of ideals and valuation rings, Proceedings of the Graz conference, to appear.
- Giampaolo Picozza and Francesca Tartarone, Flat ideals and stability in integral domains, J. Algebra 324 (2010), no. 8, 1790–1802. MR 2678822, DOI 10.1016/j.jalgebra.2010.07.021
- Niels Schwartz and Marcus Tressl, Elementary properties of minimal and maximal points in Zariski spectra, J. Algebra 323 (2010), no. 3, 698–728. MR 2574858, DOI 10.1016/j.jalgebra.2009.11.003
- Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876
Additional Information
- Carmelo A. Finocchiaro
- Affiliation: Dipartimento di Matematica e Fisica, Università degli Studi “Roma Tre”, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy
- Address at time of publication: Institute of Analysis and Number Theory, Graz University of Technology, 8010 Graz, Steyrergasse 31/II, Austria
- Email: carmelo@mat.uniroma3.it, finocchiaro@math.tugraz.at
- Dario Spirito
- Affiliation: Dipartimento di Matematica e Fisica, Università degli Studi “Roma Tre”, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy
- Email: spirito@mat.uniroma3.it
- Received by editor(s): October 1, 2015
- Received by editor(s) in revised form: December 10, 2015
- Published electronically: April 25, 2016
- Communicated by: Irena Peeva
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4125-4133
- MSC (2010): Primary 13A15, 13A18, 13C11
- DOI: https://doi.org/10.1090/proc/13131
- MathSciNet review: 3531166