I am trying to understand Hochschild homology, in particular the Hochschild–Kostant–Rosenberg theorem. As far as I understand this result gives an isomorphism between the algebraic (Kähler) differential forms of a smooth commutative $k$-algebra and the Hochschild homology. This clearly implies that the homology of the complex is infinite-dimensional, which seems strange to me. How can Hochschild homology be a good invariant of an algebra if the dimensions of its homology groups are all infinite-dimensional?

Edit: Based on the answers below, perhaps the interesting invariant of HH is the length of the complex, not the dims of the cohomology groups?

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