A comparison of Hochschild homology in algebraic and smooth settings

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Abstract: Consider a complex affine variety

i l d e V

and a real analytic Zariski-dense submanifold V of

i l d e V

. We compare modules over the ring

O (i l d e V)

of regular functions on

i l d e V

with modules over the ring

C^{i} n f t y (V)

of smooth complex valued functions on V. Under a mild condition on the tangent spaces, we prove that

C^{i} n f t y (V)

is flat as a module over

O (i l d e V)

. From this we deduce a comparison theorem for the Hochschild homology of finite type algebras over

O (i l d e V)

and the Hochschild homology of similar algebras over

C^{i} n f t y (V)

. We also establish versions of these results for functions on

i l d e V

(resp. V) that are invariant under the action of a finite group G. As an auxiliary result, we show that

C^{i} n f t y (V)

has finite rank as module over

C^{i} n f t y (V)^{G}

.

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