Direct-sum cancellation for modules over real quadratic orders

Abstract

Let R be an order in a real quadratic number field. We say that R has mixed cancellation, respectively, torsion-free cancellation if L ⊕ M ≅ L ⊕ N ⇒ M ≅ N holds for all finitely generated R -modules M , N and L , respectively, for all finitely generated torsion-free R -modules M , N and L . We derive criteria for real quadratic orders to have mixed cancellation. For instance, we prove that torsion-free cancellation holds and mixed cancellation fails for all orders R p ≔ Z [ p 1 + p 2 ] , where p is a prime satisfying 13 ≤ p ≤ 10 11 and p ≡ 1 mod 4 . Our considerations show that if the Ankeny–Artin–Chowla conjecture turned out to be true, then R p would have torsion-free cancellation but not mixed cancellation for every prime p ≥ 13 with p ≡ 1 mod 4 .