Genera and decompositions of lattices over orders
H. Jacobinski · Acta Mathematica · 1968
Let k be an algebraic number field of finite degree, A/k a semi-simple finite-dimensional algebra over k and o a Dedekind ring with quotient field k.We consider o-orders R in A, that is, subrings with kR=A and 1 ER, such that R is finitely generated as an o-module.An important example is the group ring oG of a finite group G, which is an order in kG.An R-lattice M is a finitely generated (unital) R-module, which is torsion-flee as an omodule.The category of R-lattices we denote by CR.For every prime ideal p in o, let o v be the p-adic completion and put R~=%| M~=op| etc. Then R~ is an o~-order in A n and M v is an R~-lattice.Two Rv-lattices M and N belong to the same genus--notation M~ N if My ~ Nv as Rv-modules for every p.By ~R we denote the category of genera of R-lattices.It is wellknown, that M~_N" does not in general imply M~N; but the number of isomorphism classes in a genus is finite.In the present paper, we first give a classification of these isomorphism classes by means of ideal classes in the integral closure over o of the center of A (Theorem 2.2).This generalizes results of an earlier paper (Jacobinski [9]).The proof makes use of the classical theory of maximal orders and here we need the assumption, that k is an algebraic number field and o Dedekind.For a very small exceptional class of R-lattices, the classification is not complete.This is due to the fact, that maximal orders in a totally definite skew-field of index 2 have rather irregular properties.We then use our result on the isomorphism classes in a genus to study various properties of R-lattices.As an immediate consequence we obtain an upper bound---depending only on R--for the number of isomorphism classes in a genus (Prop.2.7), An R-lattice X is called a local direct factor of M, if for every p, Xr is isomorphic to a direct factor of My.We show (Theorem 3.3), that then M has a decomposition of the form M=X'| with X'NX, In a very special case--R=oG the group ring of a finite 1 --682903 Acta mathematica.121.