Adjunction of the associated sheaf functor of non-commutative rings Original Research Article

We first extend the notion of structure sheaf for left noetherian rings in the sense of Van Oystaéyen to non-noetherian case; and then, by choosing a suitable sheaf category whose restriction to commutative rings yields the classical one, we prove that, for a vast class of non-necessarily commutative rings (including all commutative rings, all left stable left noetherian rings and all biregular rings) the structure sheaf functor admits a right adjoint, and is therefore exact.