CONTINUOUS CLOSURE, AXES CLOSURE, AND NATURAL CLOSURE NEIL EPSTEIN AND MELVIN HOCHSTER Abstract. Let R be a reduced affine C-algebra, with corresponding affine algebraic set X.Let (X)betheringofcontinuous(Euclideantopology)C- C valued functions on X.Brennerdefinedthecontinuous closure Icont of an ideal I as I (X) R.Healsointroducedanalgebraicnotionofaxes closure Iax that C \ always contains Icont,andaskedwhethertheycoincide.Weextendthenotion of axes closure to general Noetherian rings, defining f Iax if its image is in 2 IS for every homomorphism R S,whereS is a one-dimensional complete ! seminormal local ring. We also introduce the natural closure I\ of I.Oneof many characterizations is I\ = I+ f R : n>0withf n In+1 .Weshow { 2 9 2 } that I\ Iax,andthatwhencontinuousclosureisdefined,I\ Icont Iax. ✓ ✓ ✓ Under mild hypotheses on the ring, we show that I\ = Iax when I is primary to a maximal ideal, and that if I has no embedded primes, then I = I\ if and only if I = Iax,sothatIcont agrees as well. We deduce that in the @f polynomial ring [x1,...,xn], if f =0atallpointswhereallofthe are C @xi 0, then f ( @f ,..., @f )R.WecharacterizeIcont for monomial ideals in 2 @x1 @xn polynomial rings over C,butweshowthattheinequalitiesI\ Icont and ✓ Icont Iax can be strict for monomial ideals even in dimension 3. Thus, Icont ax✓ and I need not agree, although we prove they are equal in C[x1,x2]. Contents 1. Introduction 2 2. Properties of continuous closure 4 3. Seminormal rings 8 4. Axes closure and one-dimensional seminormal rings 13 5. Special and inner integral closure, and natural closure 18 6. The ideal generated by the partial derivatives 25 7. Naturally closed primary ideals are axes closed 26 8. Multiplying by invertible ideals and rings of dimension 2 33 9. A negative example and a fiber criterion for exclusion from the continuous closure 35 10. Mixed natural closure and continuous closure for monomial ideals in polynomial rings 37 11. A bigger axes closure 44 Acknowledgments 47 Date:July27,2016. 2010 Mathematics Subject Classification. Primary 13B22, 13F45; Secondary 13A18, 46E25, 13B40, 13A15. Key words and phrases. continuous closure, axes closure, natural closure, seminormal ring. The second named author is grateful for support from the National Science Foundation, grants DMS-0901145 and DMS-1401384. 1 2 NEILEPSTEINANDMELVINHOCHSTER References 47 1. Introduction Holger Brenner [Bre06] introduced a new closure operation on ideals in finitely generated C-algebras called continuous closure, and asks whether it is the same as an algebraic notion called axes closure that he introduces. He proves this for ideals in a polynomial ring that are primary to a maximal ideal and generated by monomials. We shall relate this closure to some variant notions of integral closure, special part of the integral closure over a local ring, introduced in [Eps10], and inner integral closure, a notion explored here that exists without an explicit name in the literature, and also to a notion we introduce called natural closure. We shall prove that if I is an unmixed ideal in any affine C-algebra, then I is continuously closed if and only if it is axes closed. See Theorem 7.8, Corollary 7.14, and Corollary 7.15. We also provide further conditions under which continuous closure equals axes closure or natural closure. In consequence we can prove, for example, that if f is a polynomial in C[x1,...,xn] that vanishes wherever its partial derivatives all vanish, then there are continuous n functions gj from C C such that ! n @f f = g . j @x j=1 j X See Theorem 6.1. On the other hand, we show that continuous closure is sometimes strictly smaller than axes closure. Indeed, in 9, we give an example (followed by a method of § generating such examples) of a monomial ideal in a polynomial ring over C which is continuously closed but not axes closed. After hearing the second named author give a talk on the results of this paper, Koll´ar[Kol12] studied continuous closure in the context of coherent sheaves on schemes over C and has given an algebraic characterization that permits the notion of continuous closure to be defined in a larger context. In a further paper [FK13], continuous closure is studied over topological fields other than C, particularly for the field of real numbers. Let R be a finitely generated C-algebra. Map a polynomial ring C[x1,...,xn] ⇣ R onto R as C-algebras. Let A C[x1,...,xn] be the kernel ideal, and let X ✓ be the set of points in Cn where all elements of A vanish. X may be identified with the set of maximal ideals of R.ThenX has a Euclidean topology, and the topological space X is independent of the presentation of R.Welet (Y ) denote the ring of complex-valued continuous functions on any space Y . PolynomialC functions n on C ,whenrestrictedtoX,yieldaringC[X] which is isomorphic to Rred,the original ring modulo the ideal N of nilpotents. (Nothing will be lost in the sequel if we restrict attention to reduced rings R (i.e., rings without nonzero nilpotents).) Thus, we have a C-homomorphism R (X)whichisinjectivewhenR is reduced. The continuous closure of I R, denoted! C Icont, is the contraction of I (X)toR. That is, if I =(f ,...,f )R✓,thenf Icont precisely when there areC continuous 1 m 2 functions gi : X C such that ! f = g f + + g f , |X 1 1|X ··· m m|X CONTINUOUS CLOSURE, AXES CLOSURE, AND NATURAL CLOSURE 3 where h indicates the image of h R in (X). Henceforth, we focus on the case |X 2 C where R is reduced, and omit X from the notation. However, we can state many of the results without this hypothesis:| one can typically pass at once in the proofs to the case where the ring is reduced. In this paper we study this closure and several other closures that are related, obtaining satisfying answers to many questions that were open even for polynomial rings. Let L be an algebraically closed field. We are especially concerned with the case where L = C is the complex numbers. A finitely generated L-algebra R is called a ring of axes over L if it is one-dimensional, reduced, and either smooth, with just one irreducible component, or else is such that the corresponding algebraic set is the union of n smooth irreducible curves, and there is a unique singular point, which is the intersection of any two of the components, such that the completion of the local ring at that point is isomorphic with L[[ x1,...,xn]] /(xixj 1 i<j n). We now restrict to the case of the complex numbers. In [Bre06]| Brenner obtains a structure theorem for the ideals of such a completed local ring that enables him to prove that in a ring of axes over C, for every ideal I, I = Icont.Theaxes closure Iax of an ideal I of R is defined to be the set of elements r such that for every C- homomorphism R S,whereS is a ring of axes, one has r IS. (Here as in the sequel, we use the! standard abuse of notation r IS to mean2 that '(r) '(I)S, where ' : R S is the map from R to S.) The2 results of [Bre06] imply2 that Icont Iax in! general, and that they agree for ideals of polynomial rings that are primary✓ to maximal ideals and are generated by monomials. As mentioned above, we prove here that continuous closure coincides with axes closure for all ideals of affine C-algebras that are primary to a maximal ideal, and in many other cases. We also show that an unmixed ideal (one that has no embedded primes) is axes closed if and only if it is continuously closed, and that there exist continuously closed ideals which are not axes closed, which answers a question raised by Brenner. In 3 we prove that an element r of an affine C-algebra is in the axes closure of I § R if and only if x IS for every homomorphism of R to an excellent (respectively,✓ complete) Noetherian2 one-dimensional seminormal ring S.Weuse the latter definition to extend the notion of axes closure to all Noetherian rings. See Corollaries 4.2 and 4.4 and Definition 4.3. Here is a brief sketch of the contents of the paper: In 2, we discuss some important properties of continuous closure that we will need.§ Some of this material is reviewed from [Bre06], but in some cases we need sharper or more general results. 3 is devoted to seminormal rings and their connec- tions to continuous and axes closures.§ In 4, we extend the definition of axes closure to general Noetherian rings, characterizing§ it by maps to excellent one-dimensional seminormal rings, and we show that this agrees with the original definition in Bren- ner’s setting. In 5 we discuss the concepts of special and inner integral closure, and introduce the notion§ of natural closure. We also introduce the notion of I-relevant ideals, which are used to characterize when an ideal is naturally closed, and which play a key role in proving the results of 7. We show that the natural closure is contained in the axes closure and, wherever§ it is defined, the continuous closure. This “traps” continuous closure between two algebraically defined closures. This is the main tool used in 7 to prove results on when axes closure and continuous closure agree. § 4 NEILEPSTEINANDMELVINHOCHSTER One of the main results of 6 has already been stated in the second paragraph of this Introduction.