## 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)

### Citations:

Zbl 0235.57015; Zbl 0284.12005; Zbl 0216.08001; Zbl 0218.22015