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Bachelorarbeit Zur Erlangung des akademischen Grades Bachelor of Science Dimension theory for commutative rings Eingereicht von: Daniel Heiß Eingereicht bei: Prof. Dr. Niko Naumann Universität Regensburg Fakultät für Mathematik Ausgabetermin: 22.08.2013 (Semester 5) Abgabetermin: 25.09.2013 Contents Contents I Introduction 1 II Primary Decomposition 2 2.1 Primary ideals . .2 2.2 The Lasker-Noether decomposition theorem . .4 III Krull’s principal ideal theorem 6 3.1 The n-th symbolic powers . .6 3.2 Krull’s principal ideal theorem . .7 IV A first boundary for dim(R[X]) 11 V Dimension theory of K[X1;:::;Xn] 14 VI Dimension theory of Noetherian rings 16 6.1 Dimension under polynomial extensions . 16 6.2 Non-transcendental extensions . 17 Summary of non-transcendental extensions . 21 VII Prüfer domains and a note on Seidenberg’s F -rings 21 7.1 Dimension theory of Prüfer domains . 21 7.2 A note on F -rings . 22 Summary of polynomial extensions . 27 VIII A journey to the power series ring 27 Summary and comparison with polynomial extensions . 29 References iii ii I Introduction I Introduction In mathematics there are several concepts of dimension. Some of them are quite intuitive, like the Hamel dimension: The Hamel dimension of a vector space is the minimal number of n vectors that are needed to generate the vector space. For the euclidean space R this coin- cides with the “natural” term of dimension which is the minimal number of coordinates that are necessary to specify each point of the space. On the other hand there are less intuitive concepts of dimension: Studying fractals one usually comes across the Hausdorff dimension which associates to any metric space a certain non-negative real number, which (for fractals) can be non-integer. For example the Sierpinski triangle (a famous fractal) has the Hausdorff log(3) dimension log(2) 2= Q. In algebraic geometry one of the basic objects of study are affine algebraic varieties. The dimension of an affine algebraic variety Y is defined as the supremum of all natural numbers n 2 N such that there exists a chain Z0 ( Z1 ( ::: ( Zn of irreducible closed subsets of Y . This idea of dimension is clearly the generalization of the intuitive Hamel dimension; and it can be translated into notion of commutative algebra: For this let k be an algebraically closed field and m ≥ 1 an integer. For an affine algebraic subvariety Y of km the chain of irre- ducible closed subsets of Y correspond by Hilbert’s Nullstellensatz to a chain of prime ideals p0 ( p1 ( ::: ( pn of k[X1;:::;Xm] containing the vanishing ideal I(Y ) of Y . This chain in turn corresponds to a chain of prime ideals p¯0 ( ::: ( p¯n in the ring k[X1;:::;Xm]=I(Y ). So the dimension of a variety Y can be interpreted as the supremum of the lengths of chains of prime ideals in the ring k[X1;:::;Xm]=I(Y ). So we translated the concept of geometric dimension into notion of pure algebra. There this concept of dimension is called the Krull dimension of a ring and throughout this thesis we will call it just “the dimension”. Clearly the Krull dimension of any field k is zero and any principal ideal domain that is not a field is one-dimensional. The main interest of this thesis will be the behavior of the dimension of a commutative unitary ring R under certain extensions R −! S. Mainly we study polynomial extensions and see that in several cases the dimension rises by the number of adjoint indeterminates. This thesis is organized as follows: We start with a brief introduction to primary de- composition in order to proof Krull’s principal ideal theorem and similar results in section III. Equipped with this we are able to give a lower and an upper bound for the dimension dim(R[X]) of the polynomial ring R[X] as it is provided by [Sei53]. Then we follow princi- pally [BMRH73] and we will give an elementary proof that dim k[X1;:::;Xn] = n for any field k or more generally any Artinian ring. Next we pass from Artinian rings to Noethe- rian rings and we will see – still following [BMRH73] – that we have a similar behavior: For any Noetherian ring we have dim(R[X]) = dim(R) + 1 fitting the previous result for zero-dimensional Noetherian rings. Still in the Noetherian case and now following [Cla65] we will further study the Krull dimension under ring extensions R −! R[x] where x is not an indeterminate. It will turn out that the dimension either decreases by one or does not 1 II Primary Decomposition change at all and we will give some necessary and sufficient conditions for the dimension to decrease. In section VII we will find out that the beautiful property of the dimension under polynomial extensions in the case of Noetherian rings is also true for semi-hereditary rings, especially it is true for Prüfer domains. Further we will discuss rings that do not behave like Noetherian rings or Prüfer domains under polynomial extentions. Especially we will have a look at F -rings provided by Seidenberg in [Sei53, Sei54]. Eventually we will make a short visit to the formal power series rings in one and several indeterminates observing that in the Noetherian case the behavior is just like in the polynomial case while in the non-Noetherian case various weird things can happen. In this last section we only want to give a rough overview of Arnold’s work in [Arn73a,Arn73b,Arn82] concerning the large theory about the Krull dimension under power series extensions. Notation. Throughout the whole thesis R denotes a commutative unitary non-zero ring of finite Krull dimension. Further the set of natural numbers N is supposed to contain zero. If a ring does not contain any proper zero-divisors we call it a domain. Moreover if not otherwise specified X; X1;:::;Xn denote indeterminates and p ⊆ R is assumed to be a prime ideal. The set of all prime ideals of R is denoted by Spec(R). When passing to a residue class ring R=a for some ideal a we denote with x¯ the residue class of x for any element x 2 R and similar by b¯ we mean the image of the ideal b. II Primary Decomposition In this section we will briefly develop the theory of primary decomposition which is a basic tool studying ideals. Mainly we follow [ZS75], but also [AM69]. 2.1 Primary ideals We start with some basic facts about the radical of an ideal. p n o n Definition 2.1. Let a ⊆ R be an ideal. Then we call a := x 2 R 9n 2 N: x 2 a the radical of a. p p Lemma 2.2. (i) If a ⊆ b for some ideals a; b ⊆ R, then a ⊆ b. p (ii) For any prime ideal p 2 Spec(R) the equation pn = p holds for all n ≥ 1. (iii) Let I be a finite set and for all i 2 I let qi ⊆ R be an ideal. Then we have pT T p i2I qi = i2I qi. p (iv) Let a ⊆ R be an ideal. Then a is the intersection of all prime ideals p 2 Spec(R) containing a. Proof. Those are well-known properties. See [AM69, 1.13f.] among others. 2 2 II Primary Decomposition p T Example 2.3. (i) For the nilradical of R we have N(R) := (0) = p2Spec(R) p. (ii) For a prime number p 2 we see pT (pn) = p(0) = (0) 6= (p) = T p(pn). Z n2N n2N This shows that we cannot omit the finiteness of I in Lemma 2.2 (iii). In some sence the concept of prime ideals corresponds to those of prime elements. Now we introduce ideals that – in a similar way – correspond to powers of prime elements. Definition 2.4. Let q ( R be an ideal such that p for x; y 2 R with xy 2 q and x2 = q we have y 2 q: p Then q is called primary ideal of R. In this case one says that q is p-primary if p = q. p Remark: In Proposition 2.8 we will see that for a primary ideal q the radical q ⊆ R is a prime ideal. Remark 2.5. An ideal q ⊆ R is primary if and only if every zero-divisor of R=q is nilpotent. Example 2.6. (i) Every prime ideal p is p-primary. n (ii) The primary ideals of Z are (0) and (p ) where p is a prime number and n ≥ 1. (iii) Consider the ideal q := (X; Y 2) ⊆ R := K[X; Y ] where K denotes a field. Since we have R=q =∼ K[Y ]=(Y 2) all zero-divisors of R=q are multiples of Y hence nilpotent. p 2 So q ⊆ R is primary. But we have p := q = (X; Y ) and p ( q ( p. This shows that a primary ideal need not be a power of a prime ideal. (iv) Conversely not every power of a prime ideal is primary: Let R := K[X; Y; Z]=(XY −Z2) and p := (X;¯ Z¯). Then p is prime since R=p =∼ K[Y ], but p 2.2 X¯ · Y¯ = Z¯2 2 p2 and X¯ 2 = p2; Y¯ 2 = p2 = p: So p2 is not primary although it is a power of a prime ideal. We have seen that in general the powers of prime ideals need not be primary.
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