DOCSLIB.ORG

Low-Dimensional Algebraic K-Theory of Dedekind Domains

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Low-Dimensional Algebraic K-Theory of Dedekind Domains Low-dimensional algebraic K-theory of Dedekind domains What is K-theory? k0¹OFº Z ⊕ Cl¹OFº Relationship with topological K-theory illen was the first to give appropriate definitions of algebraic K-theory. We will be This remarkable fact relates the K0 group of a Dedekind domain to its class Topological K-theory was developed before its algebraic counterpart by Atiyah and using his Q-construction because it is defined more generally, for any exact category C: group, which measures the failure of unique prime factorization. We follow the Hirzebruch. The K0 group was originally just denoted K, and it was defined for a proof outlined in »2; 1:1¼: compact Hausdor space X as: Kn¹Cº := πn+1¹BQCº 1. It is easy to show that every finitely generated projective OF-module is a K¹Xº := K¹VectF¹Xºº We will be focusing on a special exact category, the category of finitely generated direct sum of O -ideals. ¹ º F ¹ º projective modules over a ring R. We denote this by P R . By abuse of notation, we 2. Conversely, every fractional O -ideal I is a finitely generated projective Where Vectf X is the exact category of isomorphism classes of finite-dimensional F R C define the K groups of a ring R to be: OF-module, since: F-vector bundles on X under Whitney sum. Here F is either or . Kn¹Rº := Kn¹P¹Rºº 2.1 By a clever use of Chinese remainder theorem, any fractional ideal of a Dedekind Then a theorem of Swan gives an explicit relationship between this topological domain can be generated by at most 2 elements. Say I = ¹a; bº. K-theory and the algebraic K-theory of C¹X; Fº, the ring of continuous functions from Additionally, we only draw our aention to the case when R is a Dedekind domain. −1 2.2 We can choose generators ¹x; yº for the inverse ideal I in such a way that ax + by = 1 X to F. The statement is: Specifically, we are interested in the case where R is the ring of integers of a number 2.3 The map: ψ : O2 ! I field F. The Q-construction is described in »1; IV:6¼. F K¹VectF¹Xºº K¹P¹C¹X; Fººº ¹m; nº 7! am + bn In particular, finite-dimensional F-vector bundles over X are in 1-to-1 correspondence Admits a section: Generic properties of K groups 2 with finitely-generated projective modules over C¹X; Fº. A full, self-contained proof of ω : I !OF g 7! ¹gx; gyº Swan’s theorem can be found in »3¼. Since we defined K theory as a homotopy theory of a certain space, we get a lot of free Hence I is projective. properties. Note the following product-preserving functors: 3. There is a nice isomorphism: I P is a functor from (commutative rings, ring homomorphisms) to (exact categories, Grothendieck’s K-theory exact functors). I ⊕ K OF ⊕ IK There is also a version of K0 for schemes due to Grothendieck. I Q is a functor from (exact categories, exact functors) to (categories, functors). −1 −1 3.1 By first noting that there exist elements a; b 2 OF such that aI and bK are both A vector bundle over a scheme X is defined to be a locally free coherent sheaf over X. I B is a functor from (categories, functors) to (CW complexes, Cellular maps). integral ideals which are coprime to each other. A coherent sheaf is just a sheaf of OX -modules of finite type. This definition matches 3.2 Hence we can find elements c 2 I−1; d 2 K−1 such that ac + bd = 1. This yields the I πn is a functor from (CW complexes, cellular maps) to (groups, group well with the definition of vector bundles over topological spaces. We then define: homomorphisms). invertible isomorphism: ∗ ∼ K¹Xº := K¹Coh ¹Xºº We conclude that the K-groups are product-preserving functors from the category of I ⊕ K −!O ⊕ IK F ∗ commutative rings to the category of groups. More can be found in »1; pp:350 − 351¼. c −b Where Coh ¹Xº is the exact category of locally free coherent sheaves over X. For aine ¹x; yº 7! »x; y¼ = ¹cx + dy; −bx + ayº d a schemes, there is a clear relationship with the K-theory of the underlying ring. A locally free coherent sheaf over X is simply a finitely generated projective O -module. K and K for rings in general 4. Iterating the above isomorphism, and taking ¹1º and ¹2º into consideration, X 0 1 Therefore: we get that finitely generated projective OF-modules are equivalent to Before illen, the groups K and K were already described classically. ∗ 0 1 modules of the form: K¹Coh ¹Xºº K¹P¹OX ¹Xººº K0¹Rº for a ring R is defined to be the group completion (or Grothendieck group) of the n OF ⊕ I A good treatment of this approach is found in »1; I:5¼. monoid ¹P¹Rº; ⊕º. This follows from the more general result that for any exact category Where I is some fractional ideal. Two fractional ideals are isomorphic as O C with biproduct ⊕: F K0¹Cº = Gr¹C; ⊕º modules if and only if they agree in Cl¹OFº. Also: References » ¼ n m For an introduction to K0, see 1; II:2 . O ⊕ I O ⊕ K () m = n ^ I K K ¹Rº is defined to be the abelianisation of the general linear group GL¹Rº. This group F F [1] Weibel, C. (2013). The K-book: An Introduction to Algebraic K-theory (Graduate 1 Hence the semi-group isomorphism: is defined to be the direct limit of the sequence: studies in mathematics ; volume 145). Providence: American Mathematical Society. ∼ [2] Beshenov, A. (2014). Algebraic K-theory of Number Fields. [pdf] master thesis, ¹P¹OFº; ⊕º −! N ⊕ Cl¹OFº Università degli Studi di Milano, Université de Bordeaux. Available at: ::: ! GLn¹Rº ! GLn+1¹Rº ! ::: On ⊕ I 7! ¹n + 1; »I¼º hp://algant.eu/documents/theses/beshenov.pdf [Accessed 06 Jun. 2018]. M 0 F ::: 7! M 7! 7! ::: 0 1 Which upon group completion becomes: [3] Hudson, D. A Bridge between Algebra and Topology: Swan’s Theorem. [pdf] Available at: hps://web.uvic.ca/ drhh/extendedabstract.pdf [Accessed 02 Jun. 2018]. It turns out that the commutator of GL¹Rº is the group E¹Rº generated by elementary Gr¹P¹OFº; ⊕º K0¹OFº Z ⊕ Cl¹OFº matrices with entries in R. We have the equivalence: GL¹R)/E¹Rº R× ⊕ ¹SL¹R)/E¹Rºº R× ⊕ SK ¹Rº ¹O º O× 1 k1 F F Where SK1¹Rº is defined to be the quotient SL¹R)/E¹Rº, measuring which special linear matrices are not elementary. More on K1 can be found in »1; III:1¼. This follows from a theorem of Serre, Milnor and Bass, which they worked out as part of their work on the congruence subgroup problem: SK1¹OFº = 0 This is a highly non-trivial result, and this margin is too small to contain the proof. A proof can be found in »2; 1:2¼. Tudor Ciurca Tudor Ciurca Imperial College London.
Load more

Recommended publications
  • MAXIMAL and NON-MAXIMAL ORDERS 1. Introduction Let K Be A
    MAXIMAL AND NON-MAXIMAL ORDERS LUKE WOLCOTT Abstract. In this paper we compare and contrast various prop- erties of maximal and non-maximal orders in the ring of integers of a number field. 1. Introduction Let K be a number field, and let OK be its ring of integers. An order in K is a subring R ⊆ OK such that OK /R is finite as a quotient of abelian groups. From this definition, it’s clear that there is a unique maximal order, namely OK . There are many properties that all orders in K share, but there are also differences. The main difference between a non-maximal order and the maximal order is that OK is integrally closed, but every non-maximal order is not. The result is that many of the nice properties of Dedekind domains do not hold in arbitrary orders. First we describe properties common to both maximal and non-maximal orders. In doing so, some useful results and constructions given in class for OK are generalized to arbitrary orders. Then we de- scribe a few important differences between maximal and non-maximal orders, and give proofs or counterexamples. Several proofs covered in lecture or the text will not be reproduced here. Throughout, all rings are commutative with unity. 2. Common Properties Proposition 2.1. The ring of integers OK of a number field K is a Noetherian ring, a finitely generated Z-algebra, and a free abelian group of rank n = [K : Q]. Proof: In class we showed that OK is a free abelian group of rank n = [K : Q], and is a ring.
    [Show full text]
  • 6 Ideal Norms and the Dedekind-Kummer Theorem
    18.785 Number theory I Fall 2017 Lecture #6 09/25/2017 6 Ideal norms and the Dedekind-Kummer theorem In order to better understand how ideals split in Dedekind extensions we want to extend our definition of the norm map to ideals. Recall that for a ring extension B=A in which B is a free A-module of finite rank, we defined the norm map NB=A : B ! A as ×b NB=A(b) := det(B −! B); the determinant of the multiplication-by-b map with respect to an A-basis for B. If B is a free A-module we could define the norm of a B-ideal to be the A-ideal generated by the norms of its elements, but in the case we are most interested in (our \AKLB" setup) B is typically not a free A-module (even though it is finitely generated as an A-module). To get around this limitation, we introduce the notion of the module index, which we will use to define the norm of an ideal. In the special case where B is a free A-module, the norm of a B-ideal will be equal to the A-ideal generated by the norms of elements. 6.1 The module index Our strategy is to define the norm of a B-ideal as the intersection of the norms of its localizations at maximal ideals of A (note that B is an A-module, so we can view any ideal of B as an A-module). Recall that by Proposition 2.6 any A-module M in a K-vector space is equal to the intersection of its localizations at primes of A; this applies, in particular, to ideals (and fractional ideals) of A and B.
    [Show full text]
  • Multiplication Semimodules
    Acta Univ. Sapientiae, Mathematica, 11, 1 (2019) 172{185 DOI: 10.2478/ausm-2019-0014 Multiplication semimodules Rafieh Razavi Nazari Shaban Ghalandarzadeh Faculty of Mathematics, Faculty of Mathematics, K. N. Toosi University of Technology, K. N. Toosi University of Technology, Tehran, Iran Tehran, Iran email: [email protected] email: [email protected] Abstract. Let S be a semiring. An S-semimodule M is called a mul- tiplication semimodule if for each subsemimodule N of M there exists an ideal I of S such that N = IM. In this paper we investigate some properties of multiplication semimodules and generalize some results on multiplication modules to semimodules. We show that every multiplica- tively cancellative multiplication semimodule is finitely generated and projective. Moreover, we characterize finitely generated cancellative mul- tiplication S-semimodules when S is a yoked semiring such that every maximal ideal of S is subtractive. 1 Introduction In this paper, we study multiplication semimodules and extend some results of [7] and [17] to semimodules over semirings. A semiring is a nonempty set S together with two binary operations addition (+) and multiplication (·) such that (S; +) is a commutative monoid with identity element 0; (S; :) is a monoid with identity element 1 6= 0; 0a = 0 = a0 for all a 2 S; a(b + c) = ab + ac and (b + c)a = ba + ca for every a; b; c 2 S. We say that S is a commutative semiring if the monoid (S; :) is commutative. In this paper we assume that all semirings are commutative. A nonempty subset I of a semiring S is called an ideal of S if a + b 2 I and sa 2 I for all a; b 2 I and s 2 S.
    [Show full text]
  • Mension One in Noetherian Rings R
    LECTURE 5-12 Continuing in Chapter 11 of Eisenbud, we prove further results about ideals of codi- mension one in Noetherian rings R. Say that an ideal I has pure codimension one if every associated prime ideal of I (that is, of the quotient ring R=I as an R-module) has codimen- sion one; the term unmixed is often used instead of pure. The trivial case I = R is included in the definition. Then if R is a Noetherian domain such that for every maximal ideal P the ring RP is a UFD and if I ⊂ R is an ideal, then I is invertible (as a module or an ideal) if and only if it has pure codimension one. If I ⊂ K(R) is an invertible fractional ideal, then I is uniquely expressible as a finite product of powers of prime ideals of codimension one. Thus C(R) is a free abelian group generated by the codimension one primes of R. To prove this suppose first that I is invertible. If we localize at any maximal ideal then I becomes principal, generated by a non-zero-divisor. Since a UFD is integrally closed, an earlier theorem shows that I is pure of codimension one. Conversely, if P is a prime ideal of codimension one and M is maximal, then either P ⊂ M, in which case PM ⊂ RM is ∼ principal since RM is a UFD, or P 6⊂ M, in which case PM = RM ; in both case PM = RM , so P is invertible. Now we show that any ideal I of pure codimension one is a finite product of codi- mension one primes.
    [Show full text]
  • Unique Factorization of Ideals in OK
    IDEAL FACTORIZATION KEITH CONRAD 1. Introduction We will prove here the fundamental theorem of ideal theory in number fields: every nonzero proper ideal in the integers of a number field admits unique factorization into a product of nonzero prime ideals. Then we will explore how far the techniques can be generalized to other domains. Definition 1.1. For ideals a and b in a commutative ring, write a j b if b = ac for an ideal c. Theorem 1.2. For elements α and β in a commutative ring, α j β as elements if and only if (α) j (β) as ideals. Proof. If α j β then β = αγ for some γ in the ring, so (β) = (αγ) = (α)(γ). Thus (α) j (β) as ideals. Conversely, if (α) j (β), write (β) = (α)c for an ideal c. Since (α)c = αc = fαc : c 2 cg and β 2 (β), β = αc for some c 2 c. Thus α j β in the ring. Theorem 1.2 says that passing from elements to the principal ideals they generate does not change divisibility relations. However, irreducibility can change. p Example 1.3. In Z[ −5],p 2 is irreduciblep as an element but the principal ideal (2) factors nontrivially: (2) = (2; 1 + −5)(2; 1 − −p5). p To see that neither of the idealsp (2; 1 + −5) and (2; 1 − −5) is the unit ideal, we give two arguments. Suppose (2; 1 + −5) = (1). Then we can write p p p 1 = 2(a + b −5) + (1 + −5)(c + d −5) for some integers a; b; c, and d.
    [Show full text]
  • Ideals and Class Groups of Number Fields
    Ideals and class groups of number fields A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science by Minjiao Yang August, 2018 ○C Copyright All rights reserved Except for previously published materials Thesis written by Minjiao Yang B.S., Kent State University, 2015 M.S., Kent State University, 2018 Approved by Gang Yu , Advisor Andrew Tonge , Chair, Department of Mathematics Science James L. Blank , Dean, College of Arts and Science TABLE OF CONTENTS…………………………………………………………...….…...iii ACKNOWLEDGEMENTS…………………………………………………………....…...iv CHAPTER I. Introduction…………………………………………………………………......1 II. Algebraic Numbers and Integers……………………………………………......3 III. Rings of Integers…………………………………………………………….......9 Some basic properties…………………………………………………………...9 Factorization of algebraic integers and the unit group………………….……....13 Quadratic integers……………………………………………………………….17 IV. Ideals……..………………………………………………………………….......21 A review of ideals of commutative……………………………………………...21 Ideal theory of integer ring 풪푘…………………………………………………..22 V. Ideal class group and class number………………………………………….......28 Finiteness of 퐶푙푘………………………………………………………………....29 The Minkowski bound…………………………………………………………...32 Further remarks…………………………………………………………………..34 BIBLIOGRAPHY…………………………………………………………….………….......36 iii ACKNOWLEDGEMENTS I want to thank my advisor Dr. Gang Yu who has been very supportive, patient and encouraging throughout this tremendous and enchanting experience. Also, I want to thank my thesis committee members Dr. Ulrike Vorhauer and Dr. Stephen Gagola who help me correct mistakes I made in my thesis and provided many helpful advices. iv Chapter 1. Introduction Algebraic number theory is a branch of number theory which leads the way in the world of mathematics. It uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Concepts and results in algebraic number theory are very important in learning mathematics.
    [Show full text]
  • The Different Ideal
    THE DIFFERENT IDEAL KEITH CONRAD 1. Introduction The discriminant of a number field K tells us which primes p in Z ramify in OK : the prime factors of the discriminant. However, the way we have seen how to compute the discriminant doesn't address the following themes: (a) determine which prime ideals in OK ramify (that is, which p in OK have e(pjp) > 1 rather than which p have e(pjp) > 1 for some p), (b) determine the multiplicity of a prime in the discriminant. (We only know the mul- tiplicity is positive for the ramified primes.) Example 1.1. Let K = Q(α), where α3 − α − 1 = 0. The polynomial T 3 − T − 1 has discriminant −23, which is squarefree, so OK = Z[α] and we can detect how a prime p 3 factors in OK by seeing how T − T − 1 factors in Fp[T ]. Since disc(OK ) = −23, only the prime 23 ramifies. Since T 3−T −1 ≡ (T −3)(T −10)2 mod 23, (23) = pq2. One prime over 23 has multiplicity 1 and the other has multiplicity 2. The discriminant tells us some prime over 23 ramifies, but not which ones ramify. Only q does. The discriminant of K is, by definition, the determinant of the matrix (TrK=Q(eiej)), where e1; : : : ; en is an arbitrary Z-basis of OK . By a finer analysis of the trace, we will construct an ideal in OK which is divisible precisely by the ramified primes in OK . This ideal is called the different ideal. (It is related to differentiation, hence the name I think.) In the case of Example 1.1, for instance, we will see that the different ideal is q, so the different singles out the particular prime over 23 that ramifies.
    [Show full text]
  • Algebraic Number Theory Course Notes
    Math 80220 Spring 2018 Notre Dame Introduction to Algebraic Number Theory Course Notes Andrei Jorza Contents 1 Euclidean domains 2 2 Fields and rings of integers 6 2.1 Number fields . 6 2.2 Number Rings . 7 2.3 Trace and Norm . 8 3 Dedekind domains 11 3.1 Noetherian rings . 11 3.2 Unique factorization in Dedekind domains . 11 4 Ideals in number fields 13 4.1 Norms of ideals . 13 4.2 The class group . 14 4.3 Units . 18 4.4 S-integers and S-units . 20 5 Adeles and p-adics 21 5.1 p-adic completions . 21 5.2 Adeles and ideles . 25 6 Ramification and Galois theory 29 6.1 Prime ideals under extensions . 29 6.2 Ramification and inertia indices . 30 6.3 Factoring prime ideals in extensions . 32 6.4 Ramification in Galois extensions . 33 6.5 Applications of ramification . 35 7 Higher ramification 37 7.1 The different ideal . 38 8 Dirichlet series, ζ-functions and L-functions 39 8.1 Formal Dirichlet series . 39 8.2 Analytic properties of Dirichlet series . 40 8.3 Counting ideals and the Dedekind zeta function . 41 8.4 The analytic class number formula . 44 8.5 Functional equations . 45 8.6 L-functions of characters . 47 1 8.7 Euler products and factorizations of Dedekind zeta functions . 48 8.8 Analytic density of primes . 50 8.9 Primes in arithmetic progression . 51 8.10 The Chebotarëv density theorem . 52 8.11 Fourier transforms . 53 8.12 Gauss sums of characters . 55 8.13 Special values of L-functions at non-positive integers .
    [Show full text]
  • (1), 1984 Characterizations of *-Multiplication Domains
    Canad. Math. Bull. Vol. 27 (1), 1984 CHARACTERIZATIONS OF *-MULTIPLICATION DOMAINS BY EVAN G. HOUSTON, SAROJ B. MALIK AND JOE L. MOTT ABSTRACT. Let '* be a finite-type star-operation on an integral domain D. If D is integrally closed, then D is a *-multiplication domain (the *-finite *-ideals form a group) if and only if each upper to 0 in D[x] contains an element / with c(f)* = D. A finite-type star operation on D[x] naturally induces a finite-type star operation on D, and, if each *-prime ideal P of D[x] satisfies PHD = 0 or P = (P H D)D[x], then D[x] is a *-multiplication domain if and only if D is. Introduction. Throughout this paper D will denote an integral domain, and K will denote its quotient field. We shall be concerned with finite-type star operations. A star operation on D is a mapping I-+I* from the set of non-zero fractional ideals of D to itself which satisfies the following axioms for each element aGK and each pair of non-zero fractional ideals I, J of D: (i) (a)* = (a) and (al)* = ai*, (h) Içzl* and IÇ:J^I*Œ j*, (hi) /** = /*. The star operation * is said to be of finite type if each non-zero ideal I of D satisfies I* = U {J* | J is a non-zero finitely generated ideal contained in I}. For the pertinent facts about star operations, we refer the reader to [2, Sections 32 and 34] or to [6]. If * is a star operation on D, then we call D a *-multiplication domain if for each non-zero finitely generated ideal I of D there is a finitely generated fractional ideal J of D with (I/)* = D.
    [Show full text]
  • Ideal Class of an Algebraic Number Field of Unit Rank One
    MATHEMATICS OF COMPUTATION VOLUME 50, NUMBER 182 APRIL 1988, PAGES 569-579 On the Infrastructure of the Principal Ideal Class of an Algebraic Number Field of Unit Rank One By Johannes Buchmann* and H. C. Williams** Abstract. Let R be the regulator and let D be the absolute value of the discriminant of an order 0 of an algebraic number field of unit rank 1. It is shown how the infra- structure idea of Shanks can be used to decrease the number of binary operations needed to compute R from the best known 0(RD£) for most continued fraction methods to 0(Rll2De). These ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of 0 is principal. 1. Introduction. In [16] Shanks introduced an idea which has since been modified and extended by Lenstra [13], Schoof [15] and Williams [17]. This idea can be used to decrease the number of binary operations needed to compute the regulator of a real quadratic order of discriminant D from 0(Dll2+e) to 0(D1/4+£) for every s > 0. In [21] and [17] Williams et al. showed that Shanks' idea could be extended to complex cubic fields. In this note we show that it can be further extended to any order 0 of an algebraic number field 7 of unit rank one, i.e., to orders of real quadratic, complex cubic, and totally complex quartic fields. We present an algorithm which computes the regulator R of 0 in 0(Rll2De) binary operations.
    [Show full text]
  • Dedekind Domains
    NOTES ON DEDEKIND RINGS J. P. MAY Contents 1. Fractional ideals 1 2. The definition of Dedekind rings and DVR’s 2 3. Characterizations and properties of DVR’s 3 4. Completions of discrete valuation rings 6 5. Characterizations and properties of Dedekind rings 7 6. Ideals and fractional ideals in Dedekind rings 10 7. The structure of modules over a Dedekind ring 11 These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least three years to cover all of the basic algebra that every educated mathematician should know. Throughout these notes, R is an integral domain with field of fractions K. 1. Fractional ideals Definition 1.1. A fractional ideal A of R is a sub R-module of K for which there is a non-zero element d of R such that dA ⊂ R. Define A−1 to be the set of all k ∈ K, including zero, such that kA ⊂ R. For fractional ideals A and B define AB to be the set of finite linear combinations of elements ab with a ∈ A and b ∈ B. Observe that AB and A−1 are fractional ideals. The set of isomorphism classes of non-zero fractional ideals is a commutative monoid with unit R under this product.
    [Show full text]
  • Math 221 : Midterm
    Math 221 : Midterm Due online by 10am Oct. 28, or as hard copy in class by 10:10am Oct. 28 (This midterm is out of 50 points, with a 5 point extra credit problem.) 1. (6pts) In this problem we use primary ideal decomposition to prove Theorem (Krull’s intersection theorem). If A is a local noetherian ring with maximal ideal T i m, then i≥1 m = 0. (In class we did this with the additional assumptions that A is an integral domain and has Krull dimension 1. There are a number of different ways to prove this theorem in full generality, and you can find a different proof given in Eisenbud.) T i T Let m¥ = i≥1 m . Let mm¥ = k qk be a primary decomposition of the ideal mm¥. Prove that each qk contains m¥. Use this to show that mm¥ = m¥, and then deduce Krull’s intersection theorem. 2. Let A = C[x, y]/(y2 − x3). It follows from Problem Set 6 that A is noetherian of Krull dimension 1, but not a Dedekind domain; so it must be the case that A is not integrally closed in Frac(A) a) (2pts) Show this directly by showing that the element y/x 2 Frac(A) is integral over A but does not lie in A. b) (5pts) Show that there is an isomorphism C(t) =∼ Frac(A) which sends t 7! y/x. Show that this isomorphism identifies A with a proper subring of C[t], and that the integral closure of A in C(t) is precisely C[t].
    [Show full text]