  This paper is a first step in the study of nonstandard graded algebras having
Poincar\'e duality and their Lefschetz properties. We prove the equivalence
between the toric setup and the G-graded one, generalize Macaulay-Matlis
duality, introduce Lefschetz properties and prove a Hessian criteria in the
G-graded setup. We prove a special case of the Codimension One Conjecture of
Cattani-Cox-Dickenstein.
