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Galois descent and class groups of orders. (English) Zbl 0579.16002
Orders and their applications, Proc. Conf., Oberwolfach/Ger. 1984, Lect. Notes Math. 1142, 239-255 (1985).
[For the entire collection see Zbl 0561.00005.]
Let R be a Dedekind ring with quotient field K, let A be a central simple algebra over K and let \(\Lambda\) be a hereditary R-order in A. Let \(K_ 0^{lf}(\Lambda)\) be the Grothendieck group built upon finitely generated locally free \(\Lambda\)-modules. Then the class group Cl(\(\Lambda)\) is defined as the kernel of the rank map from \(K_ 0^{lf}(\Lambda)\) to \({\mathbb{Z}}\) [see I. Reiner, Maximal Orders (1975; Zbl 0305.16001) or R. G. Swan, Lect. Notes Pure Appl. Math. 55, 153-223 (1980; Zbl 0478.12010)]. The classical finiteness theorems for \(Cl(\Lambda)\) are concerned with the case where K is a global field and R some ring of integers in K. General conjectures (Bass) would suggest that Cl(\(\Lambda)\) is finitely generated when R is a regular integral affine k-algebra over a field k finitely generated over its prime field.
In the paper under review, the author shows that \(Cl(\Lambda)\) is finite if \(R=k[t]\), \(K=k(t)\) (t a variable), the index n of A is square-free and k is a number field and more generally (modulo a few extra assumptions) when k is finitely generated, as a field, over its prime field. The proof involves a good deal of algebraic K-theory and a use of one of the theorems of A. S. Merkur’ev and A. A. Suslin [Izv. Akad. Nauk SSSR, Ser. Mat. 46, 1011-1046 (1982; Zbl 0525.18008)]. In the introduction, the author says he has been inspired by papers of S. Bloch [Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 41-59 (1981; Zbl 0524.14006)] and of J.-J. Sansuc and the reviewer [Duke Math. J. 48, 421-447 (1981; Zbl 0479.14006)]. The point is that for \(n=2\) there is a very useful isomorphism, due to the author [K-theory of orders and their Brauer-Severi schemes, thesis, Göteborg (1985)] between Cl(\(\Lambda)\) and the Chow group \(CH_ 0(X)\) of 0-dimensional cycles modulo rational equivalence on ”the” conic bundle surface \(X=X(\Lambda)\) over Spec R\(=A^ 1_ k\) associated to \(\Lambda\) [for the correspondance \(X\to X(\Lambda)\), see J. Brzezinski, Brauer-Severi schemes of orders, preprint, Göteborg, (1985)]. Such an isomorphism actually holds over an arbitrary Dedekind scheme in place of Spec R, orders being replaced by sheaves of orders. Bloch’s method for studying \(CH_ 0(X)\) (for X over \(P^ 1_ k)\) is to view \(CH_ 0(X)\) as \(CH^ 2(X)\) (codimension 2 cycles), and then use the Galois cohomology of the global sections of the Quillen resolution of the sheaf \(\underline{K}_ 2\) over \(\bar X=X\times_ k\bar k\) (\(\bar k=\) algebraic closure of k). It seems difficult to extend this sheaf-theoretic approach to study \(CH^ n(X)\) for Severi-Brauer schemes (with degeneracies) of relative dimension n-1 over \(P^ 1_ k.\)
The author obviates this difficulty by staying over \(A^ 1_ k\) and using the ”first approach” to \(K_ 1\) (the infinite linear group, elementary matrices). It nevertheless requires many tools and a great amount of skill to achieve all the ”analogues” of the various steps in the two quoted papers. By Galois descent, the author first defines a characteristic homomorphism \(\Phi: Cl(\Lambda)\to H^ 1(G,K_ 1(\Lambda')),\) where \(G=Gal(k'/k)\), k’ is a finite Galois extension of k such that \(k'(t)\) splits A, and \(\Lambda '=\Lambda \otimes_ kk'\). The image of \(\Phi\) is shown to be finite by using the K-theory of tilted orders [M. E. Keating, J. Algebra 43, 193-197 (1976; Zbl 0341.16017); Lect. Notes Math. 1046, 178-192 (1984; Zbl 0526.16015)] and the classical finiteness theorems (Dirichlet, Mordell-Weil-Néron). Finiteness of \(Cl(\Lambda)\) will follow if one shows \(\Phi\) to be injective. The author identifies \(Cl(\Lambda)\) with the group \(K^*_{dn}/k^*Nred A\), where \(K^*_{dn}\) denotes the subgroup of \(K^*\) consisting of elements which are everywhere locally on \(A^ 1_ k\) the product of a unit by a reduced norm from A, and he further identifies \(\Phi\) as a map to a certain group depending on the ramification points of A. Now one of the Merkur’ev-Suslin theorems tells us that an element \(\alpha\) of \(K^*\) is a reduced norm from A if the cup-product \(\alpha \cup A\in H^ 3(K,\mu_ n^{\otimes 2})\) vanishes. But the above identification implies that an element \(\alpha \in K^*_{dn}\) whose image under \(\Phi\) is zero gives rise to an unramified element of \(H^ 3(K,\mu_ n^{\otimes 2})\). That \(\Phi\) is injective now follows from a suitable use of the localization sequence for étale cohomology on the line together with a few more tricks.
This is an important paper, and one can only regret that the author was not offered more room to display his arguments, which are very concise in places. In the light of recent results of K. Kato [J. Reine Angew. Math. 366, 142-183 (1986; Zbl 0576.12012)] which have already been used by M. Gros [to appear in J. Reine Angew. Math.] to show that \(CH_ 0(X)\) is a finitely generated group when X is a conic bundle over a curve of arbitrary genus over a number field, one may expect the author’s result to extend to a finite generation statement for Cl(\(\Lambda)\) when k[t] is replaced by the coordinate ring of any smooth affine curve.
The author informs me that the proof of Lemma (4.1) only works for prime index but that this does not affect the ensuing arguments (replace \(H^ 1(G,E(\Lambda '))\) everywhere by its image in \(H^ 1(G,SL(\Lambda')))\).

MSC:
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16E20 Grothendieck groups, \(K\)-theory, etc.
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
16P10 Finite rings and finite-dimensional associative algebras
14C25 Algebraic cycles
11R70 \(K\)-theory of global fields
14C05 Parametrization (Chow and Hilbert schemes)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
11R34 Galois cohomology