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)

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Duality in polynomial rings

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