Rings of differential operators and ´etale homomorphisms by G´ısli M´asson B.S., University of Iceland (1985) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1991 c Massachusetts Institute of Technology 1991 Signature of Author .................................................. Department of Mathematics May 2, 1991 Certified by .......................................................... Michael Artin Professor of Mathematics Thesis Supervisor Accepted by .......................................................... Sigurdur Helgason Chairman, Departmental Graduate Committee Department of Mathematics Rings of differential operators and ´etale homomorphisms by G´ısli M´asson Submitted to the Department of Mathematics on May 2, 1991, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Rings of differential operators over rings of krull dimension 1 were studied by Musson, Smith and Stafford, and Muhasky. In particular it was proved that if k is a field of characteristic zero, and A is a finitely generated reduced k-algebra, then (i) D(A), the ring of differential operators on A, has a minimal essential twosided ideal J , (ii) A contains a minimal essential left D(A)-submodule, J,and (iii) the algebras C(A):=A/J and H(A):=D(A)/J are finite dimensional vector spaces over k. In this case the algebras C(A)andH(A) were studied by Brown and Smith. We use ´etale homomorphisms to obtain similar results in a somewhat more general setting. While an example of Bernstein, Gelfand and Gelfand shows that the statements above do not hold for all reduced finitely generated k-algebras of higher krull dimension, we present sufficient conditions on commutative, noetherian and reduced rings so that analogues of the ideal J and module J can be constructed, and for which statements similar to (i) and (ii) hold. Besides being applicable to rings of higher krull dimension, our method also reveals that the earlier results are a reflection of the (rather natural) fact that differential operators preserve supports of regular functions. We then use our results to prove the existence of minimal reflexive modules for rings of differential operators over domains which satisfy the condition S2. Thisisrelatedto results of Chamarie and Stafford. We also compute the dimension of C(A) (in terms of intersection multiplicities) when A is a completion of an affine plane curve over an algebraically closed field of characteristic zero, and we give a description of H(A) for some algebras A of higher krull dimension, an analogue of a theorem of Brown. Finally we point out that we give a generalization of the formula for applying differ- ential operators to inverses, extending a result of Nakai and Hart. Thesis Supervisor: Michael Artin Title: Professor of Mathematics 2 Acknowledgements First of all I would like to thank my advisor Michael Artin, who always had the right ideas, and who set the record straight on countless matters. I would also like to thank him for the financial support I received as a research assistant the last two years of my studies. The mathematics department at MIT I thank for its support and virtually unlimited resources, and I would in particular like to thank Phyllis Ruby, our graduate administra- tor, for all the letters and other help which made real life considerably simpler. I would like to thank all the friends I made during my time at MIT, especially all my officemates for providing me with good surroundings and nice time on and off work. Special thanks go to Amnon Yekutieli for all the mathematical discussions and for the good time in Texas, to Wolfram Gerdes for the beer and the intercontinental cultural excursions, to Tom Roby for being my primary reference on the english language, and to Jan Pedersen for the coffee and Danish. You all made my time in the Boston area an unforgettable experience. I thank my teachers at the University of Iceland for providing me with a solid under- graduate education, and I also thank them for their hospitality during my stay in Iceland in the spring and summer of 1990. All my friends in Iceland I thank for making it a place where I want to return to, and Hans Jakob Beck and Snorri Bergmann get special thanks for keeping me in the illusion that every place has the same luxurious computer resources as I got spoiled by at MIT. My parents, Sigr´un G´ıslad´ottir and J´ohann M´ar Mar´ıusson, deserve very special thanks for always being supportive of my decisions, and father, this is probably what our early games with a pencil and graph paper finally lead to. Finally my very best thanks go to my wife, my companion and my very best friend, Freyja Hreinsd´ottir, for taking care of me and our son, N¨okkvi, while I was out working. I thank her for the mathematics, and for pointing out all the mistakes and the places where there was room for improvement in my thesis. Last but not least I do thank her for all the patience and willingness to go along even if it meant that she had to sacrifice the most. Freyja, I hope that now I can start giving something in return for all your help. 3 Contents Introduction 6 1 Definition and first properties of rings of differential operators 9 1.1 Basic properties of differential operators .................. 9 1.1.1 Endomorphisms, derivations and the derivation ring ........ 9 1.1.2 Definition of rings of differential operators ............. 11 1.2 Elementary results .............................. 14 1.2.1 A collection of basic lemmas ..................... 14 1.2.2 Modules of differentials ........................ 17 2 Etale´ homomorphisms and differential operators 20 2.1 Basic properties of ´etale homomorphisms .................. 21 2.1.1 Formally ´etale homomorphisms ................... 21 2.1.2 Etale´ homomorphisms, essentially of finite type .......... 24 2.2 Behavior of differential operators under ´etale homomorphisms ...... 26 2.2.1 Differentials and formally ´etale homomorphisms .......... 26 2.2.2 Differential operators and formally ´etale morphisms ........ 30 2.2.3 Finitely presented modules of differentials ............. 36 2.2.4 Etale´ descent for differential operators ............... 43 2.3 Differential operators and algebraic relations ................ 47 2.3.1 Elements satisfying polynomial relations .............. 47 4 2.3.2 Proof of theorem 2.3.2 ........................ 52 2.3.3 Right ideals and ´etale homomorphisms ............... 63 3 Minimal twosided ideals in rings of differential operators 68 3.1 Supports in reduced commutative rings ................... 69 3.1.1 Associated prime ideals ........................ 69 3.1.2 Geometrical unibranches ....................... 76 3.2 Minimal twosided ideals and minimal left submodules ........... 84 3.2.1 Differential operators and supports ................. 84 3.2.2 Essential submodules ......................... 88 3.2.3 Minimal nonzero ideals in rings of differential operators ...... 91 3.2.4 Applications of theorem 3.2.11 .................... 96 4 Reflexive modules and rings of differential operators 101 4.1 Some noncommutative ring theory ...................... 102 4.1.1 Rings of quotients ........................... 102 4.1.2 Reflexive modules ........................... 105 4.2 Rings satisfying the condition S2 ...................... 111 4.2.1 Minimal reflexive modules ...................... 111 5 Algebras related to rings of differential operators 117 5.1 The algebra C(A) ............................... 118 5.1.1 A dimension formula related to affine plane curves ......... 118 5.2 The algebra H(A) ............................... 123 5.2.1 A description when the irreducible components of A are regular . 123 Bibliography 130 5 Introduction In [25] it was proved that if k is an algebraically closed field of characteristic zero, and A is a finitely generated k-algebra, and a domain of krull dimension 1, then (i) D(A), the ring of differential operators on A, has a minimal essential twosided ideal J , (ii) A contains a minimal essential left D(A)-submodule, J,and (iii) the algebras C(A):=A/J and H(A):=D(A)/J are finite dimensional vector spaces over k. A similar result was independently obtained in [17], in the slightly more general case that k is (any) field of characteristic zero and A is a finitely generated reduced k-algebra of krull dimension 1. In [25] it was moreover shown that the result above also holds if A is replaced by its localization at a maximal ideal. In both papers, one of the main techniques was to use the fact that if A is as above, and the map Spec Ae → Spec A is set theoretically injective (where Ae denotes the nor- malization of A), then the rings D(A)andD(Ae) are Morita equivalent (a result motivated by an example of [18]). Since Ae is regular in this case, its ring of differential operators is well known, and using the Morita equivalence it is hence possible to derive many of the properties of D(A) from the same properties of D(Ae). There are two reasons why this method fails for rings of higher krull dimension: First of all it may happen that D(A)andD(Ae) are no longer Morita equivalent if A is not of krull dimension 1; and secondly, rings of differential operators over normal domains may be rather unpleasant. A well known example is the ring of differential operators on the 6 cubic cone, originally given in [1]. The goal of this thesis is to generalize the results stated at the beginning of the intro- duction to some rings of higher krull dimension, and to show how ´etale homomorphisms may be used to do that. In particular we show that if A is a commutative, noetherian and reduced k-algebra such that A has an ´etale covering B with irreducible components that are geometrical unibranches, then an analogue of the module J, as above, may be constructed.