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The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V$ , $d$ $)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an L$(\infty)$ quasi-isomorphism between the two.

We can think of the derivation d as corresponding to a vector-field v. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism:$$ (T^{poly},[v,) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of L$(\infty)$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an L$(\infty)$ quasi-isomorphism $(T^{poly}[[t]],[tv,]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same (though I believe it remains a quasi-iso). Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V , d)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an $L_\infty$ quasi-isomorphism between the two.

We can think of the derivation $d$ as corresponding to a vector-field $v$. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism: $$ (T^{poly},[v,-]) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of $L_\infty$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an $L_\infty$ quasi-isomorphism $(T^{poly}[[t]],[tv,-]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same (though I believe it remains a quasi-iso). Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V$ , $d$ $)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an L$(\infty)$ quasi-isomorphism between the two.

We can think of the derivation d as corresponding to a vector-field v. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism:$$ (T^{poly},[v,) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of L$(\infty)$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an L$(\infty)$ quasi-isomorphism $(T^{poly}[[t]],[tv,]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same or even remains a quasi-iso. Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V$ , $d$ $)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an L$(\infty)$ quasi-isomorphism between the two.

We can think of the derivation d as corresponding to a vector-field v. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism:$$ (T^{poly},[v,) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of L$(\infty)$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an L$(\infty)$ quasi-isomorphism $(T^{poly}[[t]],[tv,]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same (though I believe it remains a quasi-iso). Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V$ , $d$ $)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an L$(\infty)$ quasi-isomorphism between the two.

We can think of the derivation d as corresponding to a vector-field v. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism:$$ (T^{poly},[v,) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of L$(\infty)$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an L$(\infty)$ quasi-isomorphism $(T^{poly}[[t]],[tv,]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think this works in general for some simple counting reasons that make the above formula converge. Any confirmation or help would be great. Otherwise, I'll check my reasoning and update again.

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V$ , $d$ $)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an L$(\infty)$ quasi-isomorphism between the two.

We can think of the derivation d as corresponding to a vector-field v. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism:$$ (T^{poly},[v,) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of L$(\infty)$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an L$(\infty)$ quasi-isomorphism $(T^{poly}[[t]],[tv,]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same or even remains a quasi-iso. Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.