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On the existence of maximal orders. (English) Zbl 1258.11096
It is well known that maximal orders over a Dedekind domain exist whenever the ambient algebra is separable. In the complete case, “separable” can be replaced by “semi-simple”. For orders over a regular domain, existence criteria for maximal orders were given by {\it M. Auslander} and {\it O. Goldman} [Trans. Am. Math. Soc. 97, 1--24 (1960; Zbl 0117.02506)]. Using the methods of EGA IV, a very general and most satisfactory criterion is given in the paper. For an arbitrary Noetherian integral domain $R$ with quotient field $K$, it is shown that maximal orders exist in every semisimple $K$-algebra if and only if $R$ is a Japanese ring. The result is applied to abelian varieties, whose endomorphism ring is an order in a semisimple $\Bbb Q$-algebra. The author proves that up to isogeny, this $\Bbb Z$-order can be assumed to be maximal.

##### MSC:
 11S45 Algebras and orders, and their zeta functions 16H15 Commutative orders 11G10 Abelian varieties of dimension $> 1$
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