  The classical theorems relating integral binary quadratic forms and ideal
classes of quadratic orders have been of tremendous importance in mathematics,
and many authors have given extensions of these theorems to rings other than
the integers. However, such extensions have always included hypotheses on the
rings, and the theorems involve only binary quadratic forms satisfying further
hypotheses. We give a complete statement of the relationship between binary
quadratic forms and modules for quadratic algebras over any base ring, or in
fact base scheme. The result includes all binary quadratic forms, and commutes
with base change. We give global geometric as well as local explicit
descriptions of the relationship between forms and modules.
