The exterior algebra $\Lambda^*_kM$ [can be defined][1] for a $k$-module $M$, where $k$ is a commutative ring. A number of sources mention, without condition or proof, a (canonical) isomorphism $$(\Lambda^*_kM)^\vee\cong\Lambda^*_k(M^\vee),$$ where $M^\vee:=\operatorname{Hom}_k(M,k)$ is the dual of $M$. Any proofs I can find, however, are for $M$ a finitely-generated vector space and $k$ a field, with no discussion of other cases. Which, if any, of these conditions are needed for the isomorphism? Several proofs construct a homomorphism without these conditions, but argue in terms of a finite basis to show isomorphy. 

Examples of the result stated without conditions or proof: in their 1961 paper [Differential forms on regular affine algebras](https://doi.org/10.1090/S0002-9947-1962-0142598-8), Hochschild–Konstant–Rosenberg state that "dual of exterior algebra $\simeq$ exterior algebra over dual". Similarly stated in Fulton–Harris' [Representation Theory][3]. They work over a field, but make no mention of finite-generation. 

Examples of proofs/proof sketches over a finitely-generated vector space over a a field: Stack Exchange questions [1][4], [2][5], [3][6], this [Math Overflow question][7], [Conrad's review][8].


  [1]: https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf "Conrad - Exterior powers"
  [2]: https://www.ams.org/journals/tran/1962-102-03/S0002-9947-1962-0142598-8/S0002-9947-1962-0142598-8.pdf
  [3]: https://mat.uab.cat/~pitsch/ReadingSeminar/Fulton-Harris.pdf
  [4]: https://math.stackexchange.com/questions/1591751/dual-space-of-exterior-power-and-exterior-power-of-dual-space "Dual space of exterior power and exterior power of dual space"
  [5]: https://math.stackexchange.com/questions/18595/exterior-power-of-dual-space/18628#18628 "Answer by Qiaochu Yuan to 'Exterior power of dual space'"
  [6]: https://math.stackexchange.com/questions/44179/signs-in-the-natural-map-lambdak-v-otimes-lambdak-v-to-bbbk/44183#44183 "Answer by Qiaochu Yuan to 'Signs in the natural map \$\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk\$'"
  [7]: https://mathoverflow.net/questions/68004/natural-pairings-between-exterior-powers-of-a-vector-space-and-its-dual "\"Natural\" pairings between exterior powers of a vector space and its dual"
  [8]: https://virtualmath1.stanford.edu/~conrad/diffgeomPage/handouts/tensor.pdf "Conrad - Tensor algebras, tensor pairings, and duality"