  We derive highly constraining no-go theorems for classical de Sitter
backgrounds of string theory, with parallel sources; this should impact the
embedding of cosmological models. We study ten-dimensional vacua of type II
supergravities with parallel and backreacted orientifold Op-planes and
Dp-branes, on four-dimensional de Sitter space-time times a compact manifold.
Vacua for p=3, 7 or 8 are completely excluded, and we obtain tight constraints
for p=4, 5, 6. This is achieved through the derivation of an enlightening
expression for the four-dimensional Ricci scalar. Further interesting
expressions and no-go theorems are obtained. The paper is self-contained so
technical aspects, including conventions, might be of more general interest.
