  In this paper, we develop the theory of Jacobian rings of open complete
intersections, which mean a pair $(X,Z)$ where $X$ is a smooth complete
intersection in the projective space and and $Z$ is a simple normal crossing
divisor in $X$ whose irreducible components are smooth hypersurface sections on
$X$. Our Jacobian rings give an algebraic description of the cohomology of the
open complement $X-Z$ and it is a natural generalization of the Poincar\'e
residue representation of the cohomology of a hypersurface originally invented
by Griffiths. The main results generalize the Macaulay's duality theorem and
the Donagi's symmetrizer lemma for usual Jacobian rings for hypersurfaces. A
feature that distinguishes our generalized Jacobian rings from usual ones is
that there are instances where duality fails to be perfect while the defect can
be controlled explicitly by using the defining equations of $Z$ in $X$. Two
applications of the main results are given: One is the infinitesimal Torelli
problem for open complete intersections. Another is an explicit bound for
Nori's connectivity in case of complete intersections. The results have been
applied also to study of algebraic cycles in several other works.
