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Finite generation of the groups \(K_i\) of rings of algebraic integers. (English) Zbl 0355.18018

Algebr. \(K\)-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 179-198 (1973).
Let \(A\) be the ring of algebraic integers in a number field \(F\). In 1972 A. Borel [C. R. Acad. Sci., Paris, Sér. A 274, 1700–1702 (1972; Zbl 0235.57015)] computed the ranks of the higher \(K\)-groups \(K_i(A)\). S. Lichtenbaum [Algebr. \(K\)-Theory II, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 342, 489–501 (1973; Zbl 0284.12005)] has made some conjectures about the torsion part of \(K_i(A)\). In this essential paper the author proves that \(K_i(A)\) is a finitely generated abelian group for all \(i\geq 0\). The statement is also proved for \(A\) a maximal order in a semisimple \(Q\)-algebra.
The proof uses a result of Solomon-Tits (see L. Solomon, Theory Finite Groups, Symp. Harvard Univ. 1968, 213–221 (1969; Zbl 0216.08001)): Let \(V\) be an \(n\)-dimensional vector space over \(F\) and let \(\mathcal V\) denote the building of \(V\) (e.g. if \(n=2\) then \(\mathcal V\) is the projective space \(\mathbb P(V)\)).Then \(H_i(\mathcal V)\), the reduced integral homology of \(\mathcal V\), is zero except \(H_{n-2}(\mathcal V)=\text{st}(V)=\) the Steinberg module of \(V\). It is a free \(\mathbb Z\)-module on which \(\text{GL}(V)\) acts. Let \(Q_n\) be the subcategory of \(Q=Q\mathbb P(A)\) consisting of projective \(A\)-modules of rank \(\leq n\).
Theorem 3: The inclusion \(Q_{n-1}\to Q_n\) induces a long exact sequence \[ \rightarrow H_iQ_{n-1}\rightarrow H_iQ_n\rightarrow \oplus_\alpha H_{i-n}(\text{GL}(P_\alpha), \text{st}(V_\alpha))\rightarrow H_{i-1}Q_{n-1}\rightarrow \] where the \(P_\alpha\) represent the isomorphism classes of projective \(A\)-modules of rank \(n\) and \(V_\alpha=P_\alpha\otimes_A F\). ( \(H_iQ_n\) denotes the homology of the classifying space of the category \(Q_n\).)
The proof requires a clever use of a spectral sequence and a generalization of the Jordan–Zassenhaus theorem. The result of the title is then proved by using duality theorems of Borel–Serre and a suitable generalization of M. S. Raghunathan’s finiteness theorem [Invent. Math. 4, 318–335 (1968; Zbl 0218.22015)]. By induction from the long exact sequence in Theorem 3 all \(H_iQ_n\) are finitely generated and taking the direct limit also \(H_iQ\) is finitely generated for all \(i\). Since \(BQ\) is a homotopy associative and commutative \(H\)-space it follows that \(K_i(A)=\pi_{i+1}(BQ)\) is finitely generated.
Reviewer: G. Almkvist (Lund)

MSC:

19F99 \(K\)-theory in number theory
11R70 \(K\)-theory of global fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)