Ring constructions on spectral spaces A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering by Christopher Francis Tedd School of Mathematics 2016 2 Contents Abstract5 Declaration6 Copyright statement7 Acknowledgements9 Dedication 11 Epigraph 13 Introduction 15 Summary of notation ........................... 21 Summary of references .......................... 23 Literature on the Hochster construction ............. 23 1. Preliminaries 25 The prime spectrum of a ring ...................... 25 Spectral spaces .............................. 28 The specialisation order ...................... 38 The inverse space of a spectral space ............... 41 Spec A as a spectral space ........................ 44 Valuations and valuation rings ...................... 49 2. Hochster’s construction 51 Representing a spectral space X within the spectrum of a ring of functions on X ........................... 51 The example of X = Spec Z .................... 59 3 Extensions and indices .......................... 60 Representing spectral maps via ring homomorphisms ......... 69 3. Ring constructions on finite posets 73 Summary of techniques .......................... 73 Intersecting (discrete) valuation rings ............... 73 Fibre sums of spaces ........................ 74 The method of Lewis ........................... 78 ‘Gluing’ of domains over maximal ideals ............. 79 ‘Joining’ of maximal ideals ..................... 83 The iterative procedure ....................... 86 Application to 1-dimensional Noetherian spectral spaces .... 92 The method of Ershov .......................... 97 4. Comparison of constructions 107 Comparison of finite constructions ................... 107 Gluing of 1-dimensional domains ................. 112 Joining of maximal points ..................... 117 Comparison of constructions on a general spectral space ....... 120 The comparison of valuation rings ................ 122 The comparison of admissibility conditions ........... 126 The example of X = Spec Z inv .................. 132 Bibliography 137 Word count 29,506 4 Abstract Ring constructions on spectral spaces; a thesis submitted to the University of Manchester for the degree of Doctor of Philosophy by Christopher Francis Tedd, September 27, 2016. In the paper [14] Hochster gave a topological characterisation of those spaces X which arise as the prime spectrum of a commutative ring: they are the spectral spaces, defined as those topological spaces which are T0, quasi-compact and sober, whose quasi-compact and open subsets form a basis for the topology and are closed under finite intersections. It is well known that the prime spec- trum of a ring is always spectral; Hochster proved the converse by describing a construction which, starting from such a space X, builds a ring having the desired prime spectrum; however the construction given is (in Hochster’s own words) very intricate, and has not been further exploited in the literature (a passing exception perhaps being the use of [14] Theorem 4 in the example on page 272 of [24]). In the finite setting, alternative constructions of a ring having a given spectrum are provided by Lewis [17] and Ershov [7], which, particularly in light of the work of Fontana in [8], appear to be more tractable, and at least more readily understood. This insight into the methods of constructing rings with a given spectrum is used to prove a result about which spaces may arise as the prime spectrum of a Noetherian ring: it is shown that every 1-dimensional Noetherian spectral space may be realised as the prime spectrum of a Noethe- rian ring. A close analysis of the two finite constructions considered here re- veals considerable similarities between their underlying operation, despite their radically different presentations. Furthermore, we generalise the framework of Ershov’s construction beyond the finite setting, finding that the ring we thus define on a space X contains the ring defined by Hochster’s construction on X as a subring. We find that in certain examples these rings coincide, but that in general the containment is proper, and that the spectrum of the ring provided by our generalised construction is not necessarily homeomorphic to our origi- nal space. We then offer an additional condition on our ring which may (—and indeed in certain examples does) serve to repair this disparity. It is hoped that the analysis of the constructions presented herein, and the demonstration of the heretofore unrecognised connections between the disparate ring constructions proposed by various authors, will facilitate further investigation into the prime ideal structure of commutative rings. 5 Declaration No portion of the work referred to in this thesis has been submitted in sup- port of an application for another degree or qualification of this or any other university or other institute of learning. 6 Copyright statement The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and they have given The University of Manchester certain rights to use such Copyright, including for administrative purposes. Copies of this thesis, either in full or in extracts and whether in hard or elec- tronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appro- priate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. The ownership of certain Copyright, patents, designs, trade marks and other in- tellectual property (the “Intellectual Property”) and any reproductions of copy- right works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permis- sion of the owner(s) of the relevant Intellectual Property and/or Reproductions. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the Uni- versity IP Policy (see http://documents.manchester.ac.uk/display.aspx?DocID= 24420), in any relevant Thesis restriction declarations deposited in the Univer- sity Library, The University Library’s regulations (see http://www.manchester 7 .ac.uk/library/about/regulations/) and in The University’s policy on Presenta- tion of Theses. 8 Acknowledgements Thanks must first of all go to my supervisor Dr. Marcus Tressl, whose knowl- edge, insight, confidence and equanimity throughout have been central to the success of this project. Thanks also to my parents for their unfailing financial and logistical support, without which none of this would have been possible. And thanks for constant friendship to Sophie and to all my fellow Maths stu- dents who have made Manchester such a supportive place: Alex, Adam, Lyn- dsey, Nic, Siân, Rosie, Alex, Jonas, Goran, Jamie, Laura, Joe, Inga, Matthew and David, my office-mates and all past and present attendees of the Pure Post- graduate seminar, Friday pub, Hallowe’en and Christmas parties. I would also like to thank my examiners Dr. Vincent Astier and Dr. Hendrik Süss for their diligence, attention to detail, and constructive commentary on the thesis. 9 10 Dedicated to the memory of Gerard McCavana 01/10/1958—31/12/2013 & Robert Ernest Charles Tedd 29/7/1927—01/04/2014 11 12 “craving peace its slow so slow drop into our laps but as far from it and myself as ever” — Tom Paulin, The Road to Inver (after Pessoa) 13 14 Introduction A topological space X is spectral if it satisfies the following properties: (1) X is T0; (2) X is quasi-compact; (3) X has a basis of sets which are quasi-compact and open; (4) The quasi-compact open sets of X are closed under finite intersections; and (5) X is sober | every non-empty irreducible closed set of X is the closure of a unique point. A map f : X ! Y of spectral spaces is a spectral map if the preimage under f of all quasi-compact open sets of Y are quasi-compact and open in X. In the paper [14], Hochster showed that the spectral spaces are exactly the spaces that arise as the prime spectrum of some (commutative, unital) ring (| throughout the thesis a ring will always be commutative with 1), and that every spectral map of spectral spaces arises from a ring homomorphism under the prime spectrum functor, which takes a ring homomorphism ' : A ! B to the map which sends a prime ideal of B to its preimage under '. Indeed, it is easy to show that the prime spectrum of a ring is spectral, and that homomorphisms of rings induce spectral maps; Hochster demonstrated the converse by describing a construction which, starting from such a space X, builds a ring having the desired prime spectrum, in such a way that a spectral map between any two spectral spaces may be realised by a homomorphism of the constructed rings. This construction is the central object of study of this thesis, which, due to its intricate nature, is relatively unexplored and unexploited in the subsequent 15 16 Introduction literature. It is hoped that the elucidation of this construction, and of the heretofore unrecognised connections between this and other ring constructions proposed by various authors