Low-dimensional algebraic K-theory of Dedekind domains

What is K-theory? k0(OF)  Z ⊕ Cl(OF) 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, 1.1]: compact Hausdor space X as: Kn(C) := πn+1(BQC) 1. It is easy to show that every finitely generated projective OF-module is a K(X) := K(VectF(X)) We will be focusing on a special exact category, the category of finitely generated direct sum of O -ideals. ( ) F ( ) 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 . Kn(R) := Kn(P(R)) 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 = (a, b). K-theory and the algebraic K-theory of C(X, F), 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 (x, y) 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.6]. F K(VectF(X))  K(P(C(X, F))) (m, n) 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 C(X, F). A full, self-contained proof of ω : I → OF g 7→ (gx, gy) Swan’s theorem can be found in [3]. 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 ∈ 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 ∈ I−1, d ∈ 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: ∗ ∼ K(X) := K(Coh (X)) 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 − 351]. c −b Where Coh (X) is the exact category of locally free coherent sheaves over X. For aine (x, y) 7→ [x, y] = (cx + dy, −bx + ay) 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 (1) and (2) 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: K(Coh (X))  K(P(OX (X))) K0(R) 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.5]. monoid (P(R), ⊕). 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 K0(C) = Gr(C, ⊕) modules if and only if they agree in Cl(OF). Also: References [ ] n m For an introduction to K0, see 1, II.2 . O ⊕ I  O ⊕ K ⇐⇒ m = n ∧ I  K K (R) is defined to be the abelianisation of the general linear group GL(R). 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, (P(OF), ⊕) −→ N ⊕ Cl(OF) Università degli Studi di Milano, Université de Bordeaux. Available at: ... → GLn(R) → GLn+1(R) → ... On ⊕ I 7→ (n + 1, [I]) 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 GL(R) is the group E(R) generated by elementary Gr(P(OF), ⊕)  K0(OF)  Z ⊕ Cl(OF)  matrices with entries in R. We have the equivalence: GL(R)/E(R) R× ⊕ (SL(R)/E(R)) R× ⊕ SK (R) (O ) O×   1 k1 F  F Where SK1(R) is defined to be the quotient SL(R)/E(R), measuring which special linear matrices are not elementary. More on K1 can be found in [1, III.1]. This follows from a theorem of Serre, Milnor and Bass, which they worked out as part of their work on the congruence subgroup problem:

SK1(OF) = 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, 1.2].

Tudor Ciurca Tudor Ciurca Imperial College London