Non-standard tight closure for affine C-algebras

Transcription

1 manuscripta mathematica manuscript No. (will be inserted by the editor) Hans Schoutens Non-standard tight closure for affine C-algebras Abstract. In this paper, non-standard tight closure is proposed as an alternative for classical tight closure on finitely generated algebras over C. It has the advantage that it admits a functional definition, similar to the characteristic p definition of tight closure, where instead of the characteristic p Frobenius, its ultraproduct, the non-standard Frobenius, is used. This new closure operation cl( ) has the same properties as classical tight closure, to wit, (1) if A is regular, then a = cl(a); (2) if A B is an integral extension of domains, then cl(ab) A cl(a); (3) if A is local and (x 1,..., x n) is a system of parameters, then ((x 1,..., x i) : x i+1) cl(x 1,..., x i) (Colon-Capturing); (4) if a is generated by m elements, then cl(a) contains the integral closure of a m and is contained in the integral closure of a (Briançon-Skoda). Key words. Tight closure non-standard Frobenius ultraproducts 1. Introduction For A a ring of characteristic p > 0, HOCHSTER and HUNEKE have defined a closure operation on ideals of A, called tight closure. To be more precise, an element z A belongs to the tight closure a of an ideal a = (f 1,..., f s )A, if there exists some c A not contained in any minimal prime ideal of A, such that for all sufficiently high powers q of p, we have that cz q (f q 1,...,fq s )A. This closure operation admits all four properties listed in the abstract above; see for instance [7] or the excellent survey article [23]. The same authors have also extended the notion of tight closure to many rings containing a field of characteristic zero, including all finitely generated algebras over a field, and the same four properties still hold. However, even the very definition in characteristic zero is more complicated, involving some reduction process to characteristic p (see for instance [7, Appendix 1] or [4]). Moreover, to prove these properties in characteristic zero, Partially supported by a grant from the National Science Foundation. Department of Mathematics Ohio State University Columbus, OH (USA) schoutens@math.ohio-state.edu

2 2 Hans Schoutens one needs some strong form of Artin Approximation. More precisely, the Artin- Rotthaus Theorem [2] is used to descend properties from complete local rings to arbitrary excellent rings. In this paper, I propose an alternative closure operation (and several variants) for finitely generated C-algebras, using the non-standard Frobenius, that is to say, the ultraproduct of the characteristic p Frobenii. To emphasize this, I termed the closure operation non-standard tight closure. The main goal of the present paper is to establish the four fundamental properties above for this new closure operation and its variants. Originally, I had also included in this paper a proof for a fifth fundamental property, to wit, persistence of non-standard tight closure (that is to say, the fact that for every homomorphism A B, the non-standard tight closure of an ideal a of A is contained in the non-standard tight closure of its extension ab in B). However, since the argument requires a substantial amount of non-noetherian or non-standard commutative algebra, I decided to publish these results in a separate paper. In particular, I have made an attempt to prove the four properties using as little as possible, relying chiefly on some basic facts from tight closure theory in positive characteristic, and often, in fact, mimicking the elementary proofs these properties admit in positive characteristic. 1 After establishing the four main properties, I list in the last section a few applications. Further properties and applications can be found in [14,18 22]. There is also some work in progress ([15, 16]) on extending the present ideas to mixed characteristic. In the first part of this paper, 2 4, some material from [12,13] is reviewed and then used to describe exactly how the transfer from positive to zero characteristic is carried out using non-standard methods. Upon request of the referee of an earlier version of this paper, I will do this in considerable detail. Many of the other papers listed above also make extensive use of this method and therefore I have included several results which are not actually needed in the present paper, but only in the later papers. From 5 on, the theory of non-standard tight closure is then developed. In the remainder of this introduction, I will briefly explain the main ideas and definitions. Let C be a finitely generated C-algebra. Let a be an ideal of C and z C. I will now explain what it means for z to belong to the non-standard tight closure of a. We can write C as a quotient of some polynomial ring C[X 1,..., X n ] by an ideal I. Rather than descending these data to a finitely generated Z-algebra and then taking reductions modulo p as in the Hochster-Huneke tight closure case, we will invoke a different method of transfer. Namely, we observe that C can be viewed as a limit of the fields F alg p (the algebraic closure of the p-element field). To make this limit idea precise, we need some non-standard formalism, to wit, the notion of ultraproduct. I will not explain this at this point, but refer the reader to the next section for more details. Suffices to say here that there is a process that attaches, albeit not uniquely, to any complex number c a sequence c p with each c p F alg p. Choose some generators f 1,...,f m of I and apply this to every coefficient of each f i. We obtain polynomials f ip defined over F alg p. Let C p be 1 At present, there is one exception: to prove that non-standard tight closure is contained in integral closure, I need Néron p-desingularization.

3 Non-standard tight closure for affine C-algebras 3 the quotient of F alg p [X 1,..., X n ] modulo the ideal generated by the f ip. These characteristic p rings C p will play a role similar to the reductions modulo p in classical characteristic zero tight closure. Namely, on each C p we have an action of Frobenius. As we could take the limit (in the sense of ultraproducts) of the fields F alg p, so can we similarly take the limit of the C p. This ring, denoted C, will be called the non-standard hull of C, and as such, it is uniquely defined up to C-algebra isomorphism (whereas the C p were not). It contains C as a subring and C C is faithfully flat. Moreover, the action of the Frobenius on each C p (being entirely algebraic) extends to an action of an endomorphism F on C, the limit or non-standard Frobenius on C. Nonetheless, we have to face the following serious obstructions: the non-standard hull is far from Noetherian and the action of F on it does not preserve the subring C. Notwithstanding, we say, in accordance with the characteristic p definition, that z lies in the non-standard tight closure of a, if there exists c C, not contained in any minimal prime ideal of C, such that for all m, we have cf m (z) Fm (a)c (1.1) wheref m (a)c simply means the ideal in C generated by all F m (g) for g a. A variant of this definition is obtained as follows. Apply the downwards process to the data given by z and a = (g 1,..., g s )C and thus obtain elements z p, g ip C p whose coefficients have limits corresponding to the coefficients of z and g i respectively. Then require that z p lies in the (classical) tight closure of (g 1p,...,g sp )C p. I will call this generic tight closure. It contains equational characteristic zero tight closure and is contained in non-standard tight closure; see Theorems 8.5 and The main advantage of non-standard tight closure is its functional definition in terms of the initial data. This is reflected by the ease with which the four main properties can be derived. That in practice non-standard tight closure is impossible to be calculated explicitly (as it depends on the knowledge of an ultrafilter) should not be held against it, for its cousin, classical tight closure, itself is hard to compute, even in positive characteristic. The final and weakest variant, non-standard closure, is obtained by only requiring (1.1) for m = 1. Surprising as it might be, the more since its characteristic p counterpart would be utterly useless, this still gives a closure operation for which the four fundamental properties hold. For an application of this weaker version, see [14]. 2. Ultraproducts In this section, W always denotes an infinite set Non-principal Ultrafilters With a non-principal ultrafilter U on W, we mean a collection of infinite subsets of W closed under intersection, with the property that for any subset F of W,

4 4 Hans Schoutens either F or its complement F belongs to U. In particular, the empty-set does not belong to U and if D U and E is an arbitrary set containing D, then also E U, for otherwise E U, whence = D E U, contradiction. Since every set in U must be infinite, it follows that any cofinite set belongs to U. The existence of non-principal ultrafilters is equivalent with the Axiom of Choice, and we make this set-theoretic assumption henceforth. It follows that for any infinite subset of W, we can find a non-principal ultrafilter containing this set. In the remainder of this section, we also fix a non-principal ultrafilter on W, and omit reference to this fixed ultrafilter from our notation. Non-principal ultrafilters play the role of a decision procedure on the collection of subsets of W by declaring some subsets large (those belonging to U) and declaring the remaining ones small. More precisely, let o w be elements indexed by w W, and let P be a property. We will use the expressions almost all o w satisfy property P or o w satisfies property P for almost all w as an abbreviation of the statement that there exists a set D in the ultrafilter U, such that property P holds for the element o w, whenever w D. Note that this is also equivalent with the statement that the set of all w W for which o w has property P, lies in the ultrafilter Ultraproducts Let O w be sets, for w W. We define an equivalence relation on the Cartesian product O w, by calling two sequences (a w w W) and (b w w W) equivalent, if a w and b w are equal for almost all w. In other words, if the set of indices w W for which a w = b w belongs to the ultrafilter. We will denote the equivalence class of a sequence (a w w W) by ulim a w or a. w The set of all equivalence classes on O w is called the ultraproduct of the O w and is denoted ulim O w or O. w Note that the element-wise and set-wise notations are reconciled by the fact that ulim {o w} = {ulim o w}. w w 2.3. Ultraproducts of Sets For the following properties, the easy proofs of which are omitted, let O w be sets with ultraproduct O If Q w is a subset of O w for each w, then ulim w Q w is a subset of ulim w O w. In fact, ulim w Q w consists of all elements of the form ulim w o w, with almost all o w in Q w.

5 Non-standard tight closure for affine C-algebras If each O w is the graph of a function f w : A w B w, then O is the graph of a function A B, where A and B are the respective ultraproducts of A w and B w. We will denote this function by ulim w f w or f. Moreover, we have an equality for a w A w. ulim (f w(a w )) = (ulim f w)(ulim a w), w w w If each O w comes with an operation w : O w O w O w, then := ulim w w is an operation on O. If all (or, almost all) O w are groups with multiplication w and unit element 1 w, then O is a group with multiplication and unit element 1 := ulim w 1 w. If almost all O w are Abelian groups, then so is O If each O w is a (commutative) ring with addition + w and multiplication w, then O is a (commutative) ring with addition + and multiplication. In fact, in that case, O is just the quotient of the product O w modulo the ideal consisting of all sequences (o w w W) for which almost all o w are zero. From now on, we will drop subscripts on the operations and just write + and for the ring operations on the O w and on O If almost all O w are fields, then so is O. Just to give an example of how to work with ultraproducts, let me give the proof: if a O is non-zero, then by the previous description of the ring structure on O, almost all a w will be non-zero, for some (any) choice of a w O w for which ulim w a w = a. Therefore, letting b w be the inverse of a w whenever this makes sense, and zero otherwise, one verifies that ulim w b w is the inverse of a If C w are rings and O w is an ideal in C w, then O is an ideal in C := ulim w C w. In fact, O is equal to the subset of all elements of the form ulim w o w with almost all o w O w. Moreover, the ultraproduct of the C w /O w is isomorphic to C /O.

6 6 Hans Schoutens If f w : A w B w are ring homomorphisms, then the ultraproduct f is again a ring homomorphism. In particular, if σ w is an endomorphism on A w, then the ultraproduct σ is a ring endomorphism on A := ulim w A w. These examples are just instances of the general principle that algebraic structure carries over to the ultraproduct. The precise formulation of this principle is called Łos Theorem and requires some terminology from model-theory. Since we will only apply the ultraproduct construction to rings, we do not need to introduce the formal language of model-theory and we can get by with the following ad hoc definition Formulae With a quantifier free formula without parameters in the free variables X = (X 1,..., X n ), we will mean in this paper an expression of the form ϕ(x) := m f 1j = 0... f sj = 0 g 1j 0... g tj 0, (2.1) j=1 where each f ij and g ij is a polynomial with integer coefficients in the variables X, and where and are the logical connectives and and or. If instead we allow the f ij and g ij to have coefficients in a ring C, then we call ϕ(x) a quantifier free formula with parameters in C. We allow all possible degenerate cases as well: there might be no variables at all (so that the formula simply declares that certain elements in Z or in C are zero and others are non-zero) or there might be no equations or no negations or perhaps no conditions at all. Put succinctly, a quantifier free formula is a Boolean combination of polynomial equations using the connectives, and (negation), with the understanding that we use distributivity and De Morgan s Laws to rewrite this Boolean expression in the (disjunctive normal) form (2.1). With a formula without parameters in the free variables X, we mean an expression of the form ϕ(x) := (Q 1 Y 1 ) (Q p Y p )ψ(x, Y ), where ψ(x, Y ) is a quantifier free formula without parameters in the free variables X and Y = (Y 1,..., Y p ) and where Q i is either the universal quantifier or the existential quantifier. If instead ψ(x, Y ) has parameters from C, then we call ϕ(x) a formula with parameters in C. A formula with no free variables is called a sentence Satisfaction Let ϕ(x) be a formula in the free variables X = (X 1,...,X n ) with parameters from C (this includes the case that there are no parameters by taking C = Z and

7 Non-standard tight closure for affine C-algebras 7 the case that there are no free variables by taking n = 0). Let A be a C-algebra and let a = (a 1,..., a n ) be a tuple with entries from A. We will give meaning to the expression a satisfies the formula ϕ(x) in A (sometimes abbreviated to ϕ(a) holds in A or is true in A) by induction on the number of quantifiers. Suppose first that ϕ(x) is quantifier free, given by the Boolean expression (2.1). Then ϕ(a) holds in A, if for some j 0, we have that all f ij0 (a) = 0 and all g ij0 (a) 0. For the general case, suppose ϕ(x) is of the form ( Y )ψ(x, Y ) (respectively, ( Y )ψ(x, Y )), where the satisfaction relation is already defined for the formula ψ(x, Y ). Then ϕ(a) holds in A, if there is some b A such that ψ(a, b) holds in A (respectively, if ψ(a, b) holds in A, for all b A). The subset of A n consisting of all tuples satisfying ϕ(x) will be called the subset defined by ϕ, and any subset that arises in such way will be called a definable subset of A n. Note that if n = 0, then there is no mention of tuples in A. In other words, a sentence is either true or false in A. By convention, we set A 0 equal to the singleton { } (that is to say, A 0 consists of the empty tuple ). If ϕ is a sentence, then the set defined by it is either { } or, according to whether ϕ is true or false in A. There is a connection between definable sets and Zariski-constructible sets, where the relationship is the most transparent over algebraically closed fields, as we will explain below. In general, we can make the following observations Constructible Sets Let C be a ring. Let ϕ(x) a quantifier free formula with parameters from C, given as in (2.1). Let Σ ϕ(x) denote the C-constructible subset of A n C consisting of all prime ideals p of Spec C[X] which for some j 0 contain all f ij0 and do not contain any g ij0. In particular, if n = 0, so that A 0 C is by definition Spec C, then the C-constructible subset Σ ϕ associated to ϕ is a subset of Spec C. Let A be a C-algebra and assume moreover that A is a domain (we will never use constructible sets associated to formulae if A is not a domain). For an n-tuple a over A, we have that ϕ(a) holds in A if, and only if, the prime ideal p a of A[X] generated by the X i a i lies in the constructible set Σ ϕ(x) (strictly speaking, we should say that it lies in the base change Σ ϕ(x) SpecC Spec A, but for notational clarity, we will omit any reference to base changes). We will express the latter fact by saying that p a is an A-rational point of Σ ϕ(x). If ϕ is a sentence, then Σ ϕ is a constructible subset of Spec C and hence its base change to Spec A is a constructible subset of Spec A. Since A is a domain, Spec A has a unique A- rational point (corresponding to the zero-ideal) and hence ϕ holds in A if, and only if, this point belongs to Σ ϕ. Conversely, if Σ is a C-constructible subset of A n C, then we can associate to it a quantifier free formula ϕ Σ (X) with parameters from C as follows. However, here there is some ambiguity, as a constructible set is more intrinsically defined than a formula. Suppose first that Σ is the Zariski closed subset V(I), where I is an ideal in C[X]. Choose a system of generators, so that I = (f 1,..., f s )C[X] and set ϕ Σ (X) equal to the quantifier free formula f 1 (X) = = f s (X) = 0. Let A be a C-algebra without zero-divisors. It follows that an n-tuple a is an A-rational

8 8 Hans Schoutens point of Σ if, and only if, a satisfies the formula ϕ Σ. Therefore, if we make a different choice of generators I = (f 1,...,f s)c[x], although we get a different formula ϕ, it defines in any C-algebra A without zero-divisors the same definable set, to wit, the collection of A-rational points of Σ. To associate a formula to an arbitrary constructible set, we do this recursively by letting ϕ Σ ϕ Ψ, ϕ Σ ϕ Ψ and ϕ Σ correspond to the constructible sets Σ Ψ, Σ Ψ and Σ respectively. We say that two formulae ϕ(x) and ψ(x) in the same free variables X = (X 1,..., X n ) are equivalent over a ring A, if they hold on exactly the same tuples from A (that is to say, if they define the same subsets in A n ). In particular, if ϕ and ψ are sentences, then they are equivalent in A if they are both true or both false in A. If ϕ(x) and ψ(x) are equivalent for all rings A in a certain class K, then we say that ϕ(x) and ψ(x) are equivalent modulo the class K. In particular, if Σ is a constructible set in A n C, then any two formulae associated to it are equivalent modulo the class of all C-algebras without zero-divisors. In this sense, there is a one-one correspondence between constructible subsets of A n C and quantifier free formulae with parameters from C modulo the above equivalence relation Quantifier Elimination For certain rings (or classes of rings), every formula is equivalent to a quantifier free formula; this phenomenon is known under the name Quantifier Elimination. We will only encounter it for the following class. Theorem 2.1 (Quantifier Elimination for algebraically closed fields). If K is the class of all algebraically closed fields, then any formula without parameters is equivalent modulo K to a quantifier free formula without parameters. More generally, if F is a field and K(F) the class of all algebraically closed fields containing F, then any formula with parameters from F is equivalent modulo K(F) to a quantifier free formula with parameters from F. Proof (Sketch of proof). These statements can be seen as translations in modeltheoretic terms of Chevalley s Theorem which says that the projection of a constructible set is again constructible. I will only explain this for the first assertion. As already observed, a quantifier free formula ϕ(x) (without parameters) corresponds to a constructible set Σ ϕ(x) in A n Z and the tuples in Kn satisfying ϕ(x) are precisely the K-rational points Σ ϕ(x) (K) of Σ ϕ(x). The key observation is now the following. Let ψ(x, Y ) be a quantifier free formula and put ξ(x) := ( Y )ψ(x, Y ), where X = (X 1,...,X n ) and Y = (Y 1,..., Y m ). Let Ψ be the subset of K n+m defined by ψ(x, Y ) and let Ξ be the subset of K n defined by ξ(x). Therefore, if we identify K n+m with the collection of K-rational points of A n+m K, then Ψ = Σ ψ(x,y ) (K). Moreover, if p: A n+m K A n K is the projection on the first n coordinates then p(ψ) = Ξ. By Chevalley s Theorem, p(σ ψ(x,y ) ) (as a subset in A n Z ) is again constructible, say of the form Σ χ(x), for some quantifier free formula χ(x). Hence

9 Non-standard tight closure for affine C-algebras 9 Ξ = Σ χ(x) (K), showing that ξ(x) is equivalent modulo K to χ(x). Since χ(x) does not depend on K, we have in fact an equivalence of formulae modulo the class K. To get rid of an arbitrary chain of quantifiers, we use induction on the number of quantifiers, noting that the complement of a set defined by ( Y )ψ(x, Y ) is the set defined by ( Y ) ψ(x, Y ), where ( ) denotes negation. For some alternative proofs, see [5, Corollary A.5.2] or [9, Theorem 1.6]. Therefore, when dealing with algebraically closed fields, we may forget altogether about formulae and use constructible sets instead. However, we will not always be able to work just in algebraically closed fields and so we need to formulate a general transfer principle for ultraproducts. Recall that a sentence is a formula without free variables. Theorem 2.2 (Łos Theorem). Let C be a ring and let A w be C-algebras. If ϕ is a sentence with parameters from C, then ϕ holds in almost all A w if, and only if, ϕ holds in the ultraproduct A. More generally, let ϕ(x 1,...,X n ) be a formula with parameters from C and let a w be an n-tuple in A w with ultraproduct a. Then ϕ(a w ) holds in almost all A w if, and only if, ϕ(a ) holds in A. The proof is tedious but not hard; one simply has to unwind the definition of formula (see [5, Theorem 9.5.1] for a more general treatment). Note that A is naturally a C-algebra, so that it makes sense to assert that ϕ is true or false in A. Applying Łos Theorem to a quantifier free formula, we get the following equational version. Theorem 2.3 (Equational Łos Theorem). Suppose each A w is a C-algebra and let A denote their ultraproduct. Let X be an n-tuple of variables, let f C[X] and let a w be an n-tuple in A w. Then f(a ) = 0 in A if, and only if, f(a w ) = 0 in A w for almost all w. Moreover, instead of a single equation f = 0, we may take in the above statement any system of equations and negations of equations over C. Let us list some applications of Łos Theorem that are relevant for the present paper Ultraproducts of Fields Let K w be a collection of fields and K their ultraproduct, which is again a field by Assertion (or by an application of Łos Theorem). Note that the converse also holds: if an ultraproduct of rings is a field, then almost all of these rings are fields themselves If for each prime number p, only finitely many K w have characteristic p, then K has characteristic zero. Indeed, for every prime number p, the equation px 1 = 0 has a solution in all but finitely many of the K w and hence it has a solution in K, by Theorem 2.3.

10 10 Hans Schoutens If almost all K w are algebraically closed fields, then so is K, and conversely. Indeed, for each n 2, consider the sentence σ n := ( Y 0,...,Y n )( X)Y n = 0 Y n X n + + Y 1 X + Y 0 = 0. This sentence is true in any algebraically closed field, whence in almost all K w, and therefore, by Łos Theorem, in K. However, a field in which every σ n holds is algebraically closed. The converse is proven in the same way. We have the following important corollary which can be thought of as a model theoretic Lefschetz Principle (here F alg p denotes the algebraic closure of the p- element field). Theorem 2.4 (Lefschetz Principle). Let W be the set of prime numbers, endowed with some non-principal ultrafilter. The ultraproduct of the fields F alg p is isomorphic with the field C of complex numbers, that is to say, we have an isomorphism ulim p Falg p = C. Proof. Let F denote the ultraproduct of the fields F alg p. By Assertion 2.8.2, the field F is algebraically closed, and by Assertion 2.8.1, its characteristic is zero. Using elementary set theory, one calculates that the cardinality of F is equal to that of the continuum. The theorem now follows since any two algebraically closed fields of the same uncountable cardinality are (non-canonically) isomorphic by Steinitz s Theorem (see [5]). Remark 2.5. We can extend the above result as follows: every uncountable algebraically closed field K of characteristic zero is the ultraproduct of algebraically closed fields of prime characteristic. More precisely, any ultraproduct of algebraically closed fields of different prime characteristic, but of the same uncountable cardinality as K, is isomorphic to K (in particular, we may do this for any choice of non-principal ultrafilter). So, if we would want to, we may view C also as an ultraproduct of algebraically closed fields of prime characteristic of cardinality the continuum, instead of the (countable) algebraically closed fields F alg p. In the sequel, we will fix once and for all one such isomorphism between C and F. Although some of the constructions will depend on the choice of this isomorphism (as well as on the choice of the non-principal ultrafilter), we will always obtain isomorphic objects (in the appropriate category), regardless of the particular choices made Ultraproducts of Rings Let A w be a collection of rings and let A be their ultraproduct.

11 Non-standard tight closure for affine C-algebras If each A w is local with maximal ideal m w and residue field k w := A w /m w, then A is local with maximal ideal m := ulim w m w and residue field k := ulim w k w. Indeed, a ring is local if, and only if, the sum of any two non-units is again a non-unit. This statement is clearly expressible by means of a sentence, so that by Łos Theorem, A is local. The remaining assertions now follow easily from Assertion In fact, the same argument shows that the converse is also true: if A is local, then so are almost all A w If A w are local rings of embedding dimension e, then so is A. Indeed, by assumption almost all m w are generated by e elements x iw. It follows that m is generated by the e ultraproducts x i Almost all A w are domains (respectively, reduced) if, and only if, A is a domain (respectively, reduced). Indeed, being a domain is captured by the formula ( X, Y )X = 0 Y = 0 XY 0 and being reduced by the formula ( X)X = 0 X 2 0. In particular, using Assertion 2.3.6, we see that an ultraproduct of ideals is a prime (respectively, radical, maximal) ideal if, and only if, almost all ideals are prime (respectively, reduced, maximal) If almost all A w are Artinian local rings of length l, then so is A. This follows from [11]. If we do not restrict the length of the Artinian rings A w, then A will no longer be Artinian (nor even Noetherian). In fact, apart from the above example, Noetherianity is never preserved in ultraproducts (for it cannot be expressed in general by means of formulae). This also shows that in general the ultraproduct of primary ideals need not be primary. Nonetheless, we still have the following unidirectional version: if A is Artinian, then so are almost all A w. This yields: If I w are ideals in the local rings (A w, m w ), such that in (A, m ), their ultraproduct I is m -primary, then almost all I w are m w - primary Suppose A w and B w are rings with respective ultraproducts A and B. Then A w = Bw for almost all w if, and only if, A = B. Moreover, if S w are rings such that A w and B w are S w -algebras, then almost all A w and B w are isomorphic as S w -algebras if, and only if, A and B are isomorphic as S -algebras, where S is the ultraproduct of the S w.

12 12 Hans Schoutens This follows from Assertion and Łos Theorem, since we can express by means of formulae that a homomorphism is injective and surjective For elements x 1w,..., x dw in A w with ultraproduct x i, the sequence (x 1,..., x d ) is A -regular if, and only if, the sequence (x 1w,..., x dw ) is A w -regular for almost all w. Let me illustrate the argument by taking d = 2. We want to write down a formula in two free variables X and Y expressing that X is not a zero-divisor and Y is not a zero-divisor modulo X. The following formula ϕ(x, Y ) does exactly this ( Z 1, Z 2 )( Z 3 )(Z 1 = 0 Z 1 X 0) (Z 1 = Z 3 X Z 1 Y Z 2 X). By Łos Theorem, ϕ(x 1, x 2 ) holds in A if, and only if, ϕ(x 1w, x 2w ) holds in almost all A w. I conclude this section, with discussing an example of an ultraproduct of rings that will play a crucial role in the sequel. For each w, let K w be a field and put A w := K w [X], where X is a finite fixed tuple of variables. Let A denote the ultraproduct ulim w A w. It follows from Assertion that A contains the ultraproduct K := ulim w K w. Since each A w is a Z[X]-algebra, so is A by Łos Theorem. Moreover, by that same theorem, the image of the variables X i in A are algebraically independent over K. Therefore, we have a canonical embedding of K [X] into A. The main property of this embedding was discovered by VAN DEN DRIES in [24]. Theorem 2.6 (VAN DEN DRIES). If all K w are fields with ultraproduct K, then the canonical embedding K [X] ulim w K w[x] is faithfully flat, for every finite tuple of variables X. Proof. See [10, Theorem 1.7]; for an alternative proof, see [17, A.2]. Set A := K [X] and A immediate corollary. := ulim w K w [X]. We have the following Corollary 2.7. For every ideal I of A, we have that IA A = I. Another useful property of this embedding is that an ideal I is prime in A if, and only if, IA is prime in A ; see Corollary 4.2 below.

13 Non-standard tight closure for affine C-algebras Approximations and Non-standard Hulls As observed in Assertion 2.9.4, in most cases, an ultraproduct is not Noetherian. In particular, no positive dimensional finitely generated C-algebra arises as an ultraproduct. Nonetheless, these are the algebras on which we want to define a tight closure operation. The idea therefore is to embed such an algebra in an ultraproduct of rings of prime characteristic, where tight closure is already defined. Of course, this embedding should be canonical and there should be enough transfer between the algebra and the ultraproduct Notations To treat a more general situation, we fix some field K of characteristic zero which arises as an ultraproduct of fields of prime characteristic. By Remark 2.5, this includes the case of an uncountable algebraically closed field of characteristic zero, and in particular, the case that K = C. The underlying index set W will always be the set of prime numbers and we will no longer write it. Instead, we express this notationally by using for index the letter p, which always stands for an arbitrary prime number (there will never be need to specialize the particular prime number). Moreover, we fix some (unnamed) non-principal ultrafilter on the set of prime numbers. For each p, fix a field K p of characteristic p, so that the ultraproduct ulim p K p is isomorphic to K. Fix also an isomorphism between K and ulim p K p. Finally, fix a tuple of variables X, set A p := K p [X] and let A be their ultraproduct. By the discussion at the end of the previous section, we may view K[X] as a subring of A (after identification via the fixed isomorphism between K and ulim p K p ), and we will denote this subring by A Approximations and Codes By the isomorphism we just fixed, every element c K can be written as an ultraproduct ulim p c p, where c p K p. If c p is a second choice of elements in K p for which c = ulim p c p, then almost all c p = c p. We refer to any such choice of c p as an approximation of c. If c is a tuple (c 1,...,c n ), then taking an approximation c ip for every c i, yields a tuple c p := (c 1p,...,c np ), called an approximation of c, and any two choices of approximations will be the same for almost all p. We want to extend the notion of approximation to other algebraic objects as well. At the same time, we will define the notion of family and fiber, which will be used in carrying out the transfer in the next section Polynomials. Let f K[X]. Write f = ν a νx ν with a ν K. Choose for each a ν an approximation a νp and put f p := ν a νp X ν.

14 14 Hans Schoutens If we make different choices of approximations for the a ν, the resulting polynomials will still be equal for almost all p (since only finitely many coefficients are involved). We will call any such choice of polynomials f p an approximation of f. It follows that the element ulim p f p in A lies in the subring A and is equal to f. More generally, let f Z[U, X], where U is another tuple of (parametric) variables. With a fiber of f over a field F, we mean a polynomial f(u, X) in F[X], obtained by substituting for U some tuple u over F. For instance, if f = U ν X ν and a is the tuple over K of the coefficients (a ν ) of f (as above), then f = f(a, X). For this reason, we call f a family of polynomials and we call a a code of f with respect to the family f. It follows that if a p is an approximation of a, then f(a p, X) is an approximation of f Ideals. Let I be an ideal in A, say of the form (f 1,...,f s )A. Choose for each f i an approximation f ip and let I p be the ideal in A p generated by these f ip. Again, one checks that if we make a different choice of approximations of the f i, then for almost all p, the ideal obtained in this way is equal to I p. In fact, by Łos Theorem more is true: if the g i are a different set of generators of I and if g ip is an approximation for each g i, then the ideals I p and (g 1p,..., g tp )A p are equal for almost all p. We call any choice of I p an approximation of I. This time, the ultraproduct I := ulim p I p of the I p is no longer the original ideal I. Instead, we have that I = IA. In particular, we can retrieve I from the approximation I p, since I = I A by Corollary 2.7. To rephrase this in terms of families and fibers, let I be an ideal in Z[U, X], where U is another tuple of (parametric) variables; we call any such an ideal a family of ideals. With a fiber of the family I over a field F, we mean the image I(u) of I under the homomorphism Z[U, X] F[X] given by the substitution U = u. In other words, if I is generated by polynomials f i, then I(u) is generated by the fibers f i (u, X). As for polynomials, we can find an approximation of an ideal I in A as follows. Choose some ideal I in some polynomial ring Z[U, X] and choose a tuple a over K, such that I = I(a). It is clear that we can always find such data. We call any such choice of tuple a a code of I with respect to the family I. If we take an approximation a p of the code a, then I(a p ) is an approximation of I Algebras. Let B be a finitely generated K-algebra of the form A/I, for some ideal I. Let I p be an approximation of I and put B p := A p /I p. I claim that taking a different presentation B = K[Y ]/J and a choice of approximation J p yields algebras K p [Y ]/J p which are isomorphic as K p -algebras to the B p, for almost all p. It is therefore justified to call any choice of B p an approximation of B. To prove the claim, let the assignment X i P i (Y ), with P i K[Y ], induce a K-algebra isomorphism between K[X]/I and K[Y ]/J. Let P ip be an approximation of P i. It follows that for almost all p, the K p -algebra homomorphism φ p : K p [X] K p [Y ] given by X i P ip maps the ideal I p into the ideal J p. Moreover, by applying the same reasoning to an inverse of the given isomorphism,

15 Non-standard tight closure for affine C-algebras 15 we see that φ p induces an isomorphism between K p [X]/I p and K p [Y ]/J p, for almost all p (see also below). The ultraproduct B of the B p is in general bigger than B, and will be called the non-standard hull of B. See 3.4 below for more details. A finitely generated Z-algebra A of the form Z[U, X]/I with U some variables and I an ideal in Z[U, X] (that is to say, a family of ideals), is called a family of algebras. A fiber of this family A over a field F, is the algebra A(u) := F[X]/I(u) for some fiber I(u) over F of I (that is to say, for some tuple u over F ). If B is a finitely generated K-algebra of the form A/I and if a is a code for I with respect to some family I, then we call a also a code for B with respect to the family A := Z[U, X]/I. If a p is an approximation of a, then A(a p ) is an approximation of B = A(a) Homomorphisms. Let B and C be finitely generated K-algebras of the form B = A/I and C = A/J, for some ideals I and J, and let φ: B C be a K-algebra homomorphism given by the rule X i F i, for some F i A. Let I p, J p and F ip be approximations of I, J and F i respectively. It is an easy exercise to show, using Łos Theorem, that with B p := A p /I p and C p := A p /J p (so that they are approximations of B and C respectively), the K p -algebra endomorphism of A p given by X i F ip induces a homomorphism φ p : B p C p, for almost all p. We call φ p an approximation of φ. Its ultraproductφ then gives a homomorphism B C giving rise to a commutative diagram φ B C B φ C. (3.1) We could similarly define the notions of family, fiber and code for homomorphisms; we leave the details to the reader Summary In all of the above cases, we have the following underlying principle at work. Let O be an algebraic object defined over K (such as a polynomial, an ideal in K[X], a finitely generated K-algebra or a K-algebra homomorphism). We can find a family O (defined over Z[U, X]) and a tuple u over K, such that the fiber O(u) is precisely the original object O. Moreover, to obtain an approximation O p of O, we then simply have to take an approximation u p of u and take for O p the fiber O(u p ). Put succinctly, an object is encoded by a tuple over K, that is to say, a code; an approximation is then encoded by an approximation of the code. Moreover, all the families discussed so far are made up from families of ideals.

16 16 Hans Schoutens 3.4. Non-standard Hulls Let B be a finitely generated K-algebra and let B p be an approximation of B. We call the ultraproduct B := ulim p B p the non-standard hull of B. Since any two approximations of B are isomorphic as K p -algebras by the argument in 3.2.3, it follows from Assertion that B is uniquely defined up to K-algebra isomorphism. As we are only interested in objects up to isomorphism, it does not matter which approximation we take to calculate a non-standard hull. Corollary 3.1. The canonical homomorphism B B is faithfully flat. Proof. Suppose B is of the form A/I, for some ideal I of A. We showed that IA = I and from Assertion 2.3.6, after taking some approximation, we get that B is equal to A /I. Therefore, the homomorphism B B is just the base change of the embedding A A and hence is faithfully flat by Theorem 2.6. The next result explains somehow the terminology non-standard. Proposition 3.2. For each p, take some f p in A p. Then the f p are an approximation of an element f A if, and only if, there is a uniform bound N on the degree of each f p. The proof is left as an exercise to the reader. As a corollary, one might say that A consist of all elements of finite degree in its non-standard hull A, or, formulated differently, that A consists of the standard elements of A. 4. Transfer In this section, we keep the notations introduced in 3.1. In particular, K is the ultraproduct of characteristic p fields K p, X is a fixed tuple of variables, A := K[X] and A p := K p [X], and A is the ultraproduct of the A p (whence the nonstandard hull of A). Recall that for a finitely generated K-algebra B = A/I, we called the characteristic p rings B p := A p /I p an approximation of B, where I p was obtained from I by replacing each coefficient c K in each member of a generating set of I, by some c p K p with the property that ulim p c p = c (after identification of K with ulim p K p ). Although neither I p nor B p is uniquely defined, any different choice of this data will be isomorphic for almost all p. The objective of this section is to show how many algebraic properties pass from B to any of its approximations and vice versa. We will also deal with localizations of finitely generated K-algebras. Therefore, let us call a ring B a K-affine algebra if it is either a finitely generated K-algebra or a localization at a prime ideal of a finitely generated K-algebra (we express the latter fact by calling B a local K-affine algebra).

17 Non-standard tight closure for affine C-algebras Definability in families Let P be a property of t polynomials in a polynomial ring over a field in the variables X. We say that P is definable in families 2, if for each choice of t families of polynomials f 1,..., f t (in some parametric variables U; see 3.2), we can find a formula without parameters ϕ(u), such that for each field F, a tuple u over F satisfies ϕ(u) if, and only if, the t fibers f i (u) have property P. If we would allow only algebraically closed fields F in the above definition, then we could replace the formula ϕ by a Z-constructible subset Σ by Theorem 2.1. This would suffice for treating the case of main interest, K = C. However, in the current generality, where K is allowed to be non-algebraically closed, we need the formalism of formulae. Since properties of ideals can be translated into properties of polynomials by considering a generating set of the ideal, we may include in the above definition properties about s ideals and t polynomials in a polynomial ring over a field. More precisely, if we are given s families of ideals I 1,...,I s and t families of polynomials f 1,..., f t, then a property of the s fibers I i (u) and the t fibers f i (u), can be viewed as a property of the sm + t fibers f i (u), where I i is generated by the m polynomials f t+(i 1)m+1,...,f t+im. Since this allows for less cumbersome notation, we will in the sequel consider properties of polynomials and ideals. A similar convention can be made for families of algebras and elements and ideals in these algebras. We will leave it up to the reader to translate everything back to properties of polynomials (see also 4.4 below) Properties which are definable in families In the following, each given property (involving ideals I, J, J 1,... and/or elements f, g, g 1,... ), is definable in families The ideal membership property expressing that f I. This is definable in families for the following reason. There exists for each pair of natural numbers (d, n), a bound d such that if f 0,..., f s are polynomials in n variables of degree at most d over a field and if f 0 lies in the ideal generated by the remaining f i, then there exist polynomials g i of degree at most d, such that f 0 = g 1 f g s f s (see for instance [10], where this is shown to follow from Theorem 2.6). Now, let f 0 be a family of polynomials and let I be a family of ideals, say generated by f 1,..., f s. Let n be the number of X-variables and d be the maximum of the X-degree of all f i. For a tuple u over a field F, we have that f 0 (u, X) belongs to the fiber I(u), if there exist polynomials g i over F of degree at most d, such that f 0 (u, X) = g 1 f 1 (u, X) + + g s f s (u, X). (4.1) 2 Note that in [12,13], to be definable in families was called there to be asymptotically definable.

18 18 Hans Schoutens Let c i be the tuple of all coefficients of g i (so that it is the code of g i with respect to some appropriate choice of family of ideals of degree d ). The existence of the g i satisfying (4.1) is then equivalent with the existence of tuples c i over F satisfying the equations obtained by expanding the right hand side of (4.1) and comparing coefficients with the left hand side. Note that the latter set of equations involve the u. Therefore, f 0 (u, X) belongs to I(u) if, and only if, this system has a solution in F for the c i. This latter statement, as a statement about the tuple u, can be expressed by a formula (with only equations and existential quantifiers), showing that ideal membership is definable in families The ideal containment property expressing that I J. Also, the properties expressing that one ideal is the sum, the product or the intersection of two (or some other fixed number of) ideals and the property that I = (J 1 : J 2 ) The primality property expressing that I is prime. As for the previous cases and for most cases that we will encounter, this property is definable in families due to the existence of certain bounds. More precisely, for each pair (d, n), we can find a bound d, such that if I is an ideal generated by polynomials in n variables over a field of degree at most d and if for all polynomials f and g of degree at most d, we have that fg I implies that either f or g belongs to I, then in fact I is prime (see [10]). From this and the fact that the ideal membership problem is definable in families, it follows that primality is definable in families The property expressing that J is a prime ideal containing I. This is immediate from Assertions and The properties expressing that an ideal is radical, primary or maximal. Also, the properties expressing that one ideal is the radical of another, or that one ideal is an associated prime ideal of another, or that one ideal has a primary decomposition given by some other fixed number of ideals. Again this follows from the corresponding existence of bounds, proven in [10]. Note that there is also a bound on the possible number of associated primes of an ideal in terms of the degrees of its generators For any fixed l, the property expressing that J is a prime ideal containing I such that A J /IA J has length l. Immediate from Assertion and the bounds proven in [12, Theorem 2.4].

19 Non-standard tight closure for affine C-algebras For any fixed l, the property expressing that J is a prime ideal with I J 1 J such that J 1 (A J /IA J ) has height (respectively, depth) l. Also, the property expressing that (x 1,...,x d ) is (part of) a system of parameters of (A J /IA J ). Follows from the bounds proven in [12, Proposition 5.1] The property expressing that J is a prime ideal containing I such that the sequence (g 1,..., g d ) is A J /IA J -regular. Follows from the bound proven in [12, Corollary 5.2] The property expressing that the K-algebra endomorphism ϕ on A given by X i g i maps the ideal I inside the ideal J. Also the property expressing that the induced homomorphism A/I A/J is injective. If I = (f 1,...,f s )A, then the statement ϕ(i) J amounts to the ideal membership in J of the compositions f i (g 1,..., g n ) and, therefore, is definable in families by Assertion It follows from [12, Theorem 2.7] that the property expressing that the kernel of the induced homomorphism φ: A/I A/J is equal to an ideal J 1 (A/I) is definable in families. From this the definability in families of the injectivity of φ is clear. Shortly, we will discuss properties of affine algebras and their ideals, but let us first indicate the main application of definability in families to the transfer problem. We keep the notations and definitions from 3.1. Proposition 4.1. Let P be a property about s ideals and t polynomials in a polynomial ring over a field. Assume that P is definable in families. Let a 1,...,a s and g 1,...,g t be ideals and polynomials in A. Choose approximations a ip and g jp of a i and g j respectively. The ideals a 1,..., a s and the elements g 1,..., g t satisfy property P if, and only if, the ideals a 1p,..., a sp and the polynomials g 1p,..., g tp satisfy property P for almost all p. Proof. Choose families of ideals I i and families of polynomials g i and choose a tupleaover K such that I i (a) = a i and g j (a, X) = g j (X) (this is always possible by the observations made in 3.3). Let a p be an approximation of a. As explained in 3.2, we can use I i (a p ) and g j (a p, X) as approximations of a i and g j (X) respectively, so that we may assume that a ip = I i (a p ) and g jp = g j (a p, X). Let ϕ(u) be the formula witnessing that P is definable in families for the families I i and g j. Assume that the ideals a ip and the polynomials g jp satisfy property P for almost all p. Therefore, ϕ(a p ) is true in K p for almost all p. By Łos Theorem, ϕ(a) holds in K. This in turn means that the ideals a i and the polynomials g j have property P. The converse holds by reversing the arguments.

20 20 Hans Schoutens Although we will not need this, the converse of Proposition 4.1 also holds: if a property P as above has the property that it holds on ideals a i and polynomials g i if, and only if, it holds on almost all approximations a ip and g ip, then P is definable in families. Applying Proposition 4.1 to Assertions and 4.2.3, we get the following corollary. Corollary 4.2. Let I and J be ideals in A with approximations I p and J p. Let I and J be their respective ultraproducts, viewed as ideals in A. Then the following are true. 1. Almost all I p J p if, and only if, I J if, and only if, I J. 2. Almost all I p are prime if, and only if, I is prime if, and only if, I is. We will shortly obtain more transfer results of this kind, but first we extend our definition of approximation to incorporate local affine algebras Approximations and non-standard hulls local case Let R be a local K-affine algebra given as A J /IA J with I J and J prime. By Corollary 4.2, almost all J p are prime ideals containing I p. In particular, for almost all p, the ring R p := (A p ) Jp /I p (A p ) Jp is well-defined (in the remaining case, we can put R p := 0). It follows from Corollary 4.2 that ulim R p = (A ) J /I (A ) J, (4.2) p We call the ultraproduct of the R p the non-standard hull of R and denote it R. This is well-defined up to K-algebra isomorphism. Indeed, if we take a different presentation of R as a localization of a finitely generated K-algebra, then the resulting local K p -algebras will almost all be isomorphic to R p, since both choices have the same ultraproduct equal to the right hand side of (4.2). Any such choice of R p will therefore be called an approximation of R. Corollary 4.3. For R a C-affine algebra R, the canonical homomorphism R R is faithfully flat. Proof. Follows immediately from Theorem 2.6 (or from Corollary 3.1) by base change. Terminology. Let R be a local K-affine algebra with approximation R p. Suppose R = A J /IA J and let I p and J p be approximations of I and J respectively. For x an element of R, choose f, g A with g / J, such that x is equal to the image in R of the fraction f/g. Let f p and g p be approximations of f and g respectively. It follows that almost all g p / J p. The collection of elements x p := f p /g p R p is called an approximation of x. Similarly, for a an ideal in R, choose an ideal J 1 in A such that I J 1 J and a = J 1 R, and let J 1p be an approximation of J 1. The collection of ideals a p := J 1p R p is called an approximation of a. A similar convention is in place if R is just of the form A/I. In other words, we extend the notion of an approximation of an element or an ideal to arbitrary affine algebras.