Big pure projective modules over commutative noetherian rings: comparison with the completion

A module over a ring $R$ is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module $M$, we consider $\mathrm{Add}\, (M)$, which consists of direct summands of direct sums of copies of $M$. We are primarily interested in the case where $R$ is a one-dimensional, local domain, and in torsion-free (or Cohen-Macaulay) modules. We show that, even in this case, $\mathrm{Add}\, (M)$ can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of $M$ and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behavior of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid $V^*(M)$, of isomorphism classes of countably generated modules in $\mathrm{Add}\, (M)$ with the addition induced by the direct sum. We show that $V^*(M)$ is a submonoid of $V^*(M\otimes _R \widehat R)$, this allows us to make computations with examples and to prove some realization results.