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 from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra 𝕀 n : = K 〈 x 1 , ... , x n , ∂ ∂ x 1 , ... , ∂ ∂ x n , ∫ 1 , ... , ∫ n 〉 of polynomial integro-differential operators and the Jacobian algebra 𝔸 n is equal to n (over a field of characteristic zero). The algebras 𝕀 n and 𝔸 n are neither left nor right Noetherian and 𝕀 n ⊂ 𝔸 n . Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert's Syzygy Theorem is proven for the algebras 𝕀 n , 𝔸 n and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras 𝕀 n and 𝔸 n is n and the weak global dimension of all the factor algebras of 𝕀 n and 𝔸 n is n. [ABSTRACT FROM AUTHOR]