On the decomposition of lattices over orders
Let \(R\) be a Dedekind domain with field of fractions \(K\), and let \(\Lambda\) be an order in a finite dimensional \(K\)-algebra \(A\) that need not be separable. For two \(\Lambda\) lattices \(L\) and \(L'\), write \(KL\gg KL'\) if each \(A\)-indecomposable direct summand of \(KL'\) occurs strictly more often in \(KL\) than in \(KL'\). Write \(M\sim L\) if the \(P\)-adic completions \(M_P\cong L_P\) for all maximal ideals \(P\). Two theorems are proved. The first is a form of the Roiter-Jacobinski Divisibility Theorem. It says that if \(M\sim L'\oplus L''\), then \(M\cong M'\oplus M''\), where \(M'\sim L'\) and \(M''\sim L''\). Further, if \(KM\gg KL'\), then \(M\cong L'\oplus M''\). The second theorem is a form of Jacobinski-Swan cancellation. Namely, let \(G(L)\) denote the set of lattices \(M\) with \(M\sim L\), and write \(nL\) for the direct sum of \(n\) copies of \(L\). Then the mapping \(G(L)\to G(nL)\) given by \(X\mapsto X\oplus(n-1)L\) is injective. -- The proofs avoid the strong approximation methods in the classical versions by applying methods of K-theory at the adelic level.
scientific article; zbMATH DE number 1172200