Let A be a commutative ring and let B be a commutative A-algebra.

Question

Let A be a commutative ring and let B be a commutative A-algebra. Let d be a positive integer, and let M be an A-module satisfying $\operatorname{Tor}_A^n(B,M)=0 ~~$for $~0\lt n\le d~.$ Prove that for any B-module N there exists an isomorphism $\operatorname{Ext}_B^m(B\otimes_A M,N)\cong \operatorname{Ext}_A^m(M,N)$ for every $m\leq d$