Exercise 3, § 9

Let $n$ be an integer, $k$ a ring, $A = k[X_1, \ldots, X_n]$. Now prove that there exists for every $A$-module $M$ an isomorphism of graded $A$-modules

such that for every homomorphism of A-modules $u : M \to N$, one has $\operatorname{Ext}(1_k, u) \circ \delta_M = \delta_N \circ \operatorname{Tor}(1_k, u)$.