On the Homological Dimension of Algebras of Differential Operators

Let A be a commutative algebra over a field k, and V_A be the k-subalgebra of End_k(A) generated by End_A(A) = A and all k-derivations of A. A study of the homological properties of V_A was initiated by Hochschild, Kostant, and Rosenberg in 5], and continued by Rinehart 8], 9], Roos 11], Björk 1], Rinehart and Rosenberg 10], and others. It was proved in 5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of V_A is between r and 2r. Moreover, if k has positive characteristic, then gl.dim V_A = 2r 8]. By a recent celebrated theorem of Roos 11], gl.dim V_A = r if k has characteristic zero and A = kx₁, …, x_r]; in this case V_A is the so-called “Weyl algebra on 2r variables”.