  An affine oriented matroid is a combinatorial abstraction of an affine
hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a
squarefree monomial ideal in a polynomial ring, called an oriented matroid
ideal, and got beautiful results. Developing their theory, we will show the
following.
  (1) If an oriented matroid ideal is Cohen-Macaulay, then the bounded complex
(a regular CW complex associated with it) of the corresponding affine oriented
matroid is a contractible homology manifold with boundary. This is closely
related to Dong's theorem, which used to be "Zaslavsky's conjecture".
  (2) We characterize the affine oriented matroid whose corresponding ideal is
Cohen-Macaulay.
  (3) In the Cohen-Macaulay case, we give a description of the canonical module
of the residue class ring by an oriented matroid ideal.
