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Follow Matt E's construction in http://math.stackexchange.com/questions/51146/hodge-number-jump-in-family-example to produce an example of a family over $W(k)$ where Hochschild-Kostant-Rosenberg and Hodge=>de Rham degeneration hold at the special fiber, and where the ranks of Hodge cohomology jump. This ensures that the ranks of some Hochschild homology must also jump, violating flatness.

Follow Matt E's construction in https://math.stackexchange.com/questions/51146/hodge-number-jump-in-family-example to produce an example of a family over $W(k)$ where Hochschild-Kostant-Rosenberg and Hodge=>de Rham degeneration hold at the special fiber, and where the ranks of Hodge cohomology jump. This ensures that the ranks of some Hochschild homology must also jump, violating flatness.

Lemma 2: If $A$ is smooth and proper over $R$, then there is a perfect pairing (in the sense of perfect $R$-modules) $$ HH_\bullet^R(A) \otimes HH_\bullet^R(A^{op}) \longrightarrow R $$.

Sketch: This should follow from the general "field theory" formalism. Smooth and proper categories $R$-algebras are fully dualizable objects in a certain $(\infty,2)$-category, give rise to field theories that assign the Hochschild homology to the circle with one orientation, etc. Something similar should appear in Caldararu and Willerton's papers on the Mukai pairing. $\square$

Lemma 2: If $A$ is smooth and proper over $R$, then there is a perfect pairing of perfect $R$-module $$ HH_\bullet^R(A) \otimes HH_\bullet^R(A^{op}) \longrightarrow R $$.

Sketch: Smooth and proper $R$-algebras are fully dualizable objects in the Morita $\infty$-category of $R$-algebras (and perfect bi-modules), and $HH^R_\bullet(-)$ gives a symmetric monoidal functor from this Morita $\infty$-category to the $\infty$-category of complexes of $R$-modules localized at quasi-isomorphisms. When $A$ is smooth and proper, its dual in the Morita category is $A^{op}$ -- and thus the result follows! (Presumably similar discussion happens in Caldararu and Willerton's papers on the Mukai pairing.) $\square$

But you asked for "projective," so we must ask furthermore whether the homologies of this perfect complex are flat. I believe that this is not generally true, at least if $A$ is allowed to be a dg-algebra (and I don't see an obvious reason why the non-dg case would be different).

But you asked for "projective." There are two things to say:

1. If $A$ is an ordinary algebra, then I will sketch an argument at the bottom for why this is true.
2. If $A$ is allowed to be a dg-algebra, then I believe that this is not true and I'll sketch how to get a counterexample.

The case where $A$ is discrete (I'll use the word "discrete" to indicate that something that was dg is actually concentrated in degree $0$.)

The argument is via three Lemmas:

Lemma 1: If $A$ is a discrete $R$-algebra (here $R$ is a discrete commutative ring), then $HH_\bullet^R(A)$ is a connective $R$-module. (Recall $M$ is connective if $H^i M = 0$ for $i>0$.)

Pf: Derived tensor products, over connective rings, preserve conective objects. $\square$

Lemma 2: If $A$ is smooth and proper over $R$, then there is a perfect pairing (in the sense of perfect $R$-modules) $$ HH_\bullet^R(A) \otimes HH_\bullet^R(A^{op}) \longrightarrow R $$.

Sketch: This should follow from the general "field theory" formalism. Smooth and proper categories $R$-algebras are fully dualizable objects in a certain $(\infty,2)$-category, give rise to field theories that assign the Hochschild homology to the circle with one orientation, etc. Something similar should appear in Caldararu and Willerton's papers on the Mukai pairing. $\square$

Lemma 3: Suppose that $P$ is a perfect $R$-module, and that both $P$ and its dual $RHom_R(P, R)$ are connective. Then, $P$ is a discrete projective $R$-module.

Pf: If $R$ is a field, this is clear. Thus, it follows that $P \stackrel{L}\otimes_R k$ is discrete for every map $R \to k$ from $R$ to a field. Now this is a standard criterion for a perfect complex to be a discrete projective module.$\square$