# Duality in polynomial rings

Let $A$ be a polynomial ring over a commutative ring (!) $k$ in $n$ variables. Show that for any $A$-module $M$ and $0\leqslant q\leqslant n$, there are natural isomorphisms

$\mathrm{Tor}_q^A(k,M) \simeq \mathrm{Ext}_A^{n-q}(k,M)$

1. Hint: use the canonical Koszul $A$-resolution $X$ of $k$ to furnish isomorphisms $X_q\longrightarrow \hom_A(X_{n-q},A)$.