The global dimension of the algebras of polynomial integro-differential operators In and the Jacobian algebras An

The aim of the paper is to prove two conjectures that the (left and right) global dimension of the algebra of polynomial integro-differential operators In and the Jacobian algebra An is equal to n (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras In and An are neither left nor right Noetherian. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. It is proven that the global dimension of all prime factor algebras of the algebras In and An is n and the weak global dimension of all the factor algebras of In and In is n.