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Let $\mathcal{P}$ be an operad and $A$ a $\mathcal{P}$-algebra. In Algebraic Operads(AO) by Loday-Vallette there are some cohomology theories defined for $A$:

  • Operadic cohomology = cohomology of $C_{Op}(A) = Der(\mathcal{P}_{\infty}, End(A))$ for some cofibrant replacement (or minimal model) $\mathcal{P}_\infty \overset{\sim}{\longrightarrow}\mathcal{P}$ (AO, §12.4.1)
  • André-Quillen cohomology = cohomology of $C_{AQ}(A) = Der(R,A)$ for some cofibrant replacemnt $R\overset{\sim}{\longrightarrow} A$ for $A$ as $\mathcal{P}$-algebra (AO,12.3.26).

Under suitable conditions these are shown to compute the same cohomology. In André-Quillen cohomology of algebras over an operad by J. Millès, it is shown for suitable operads $\mathcal{P}$ that $$HC_{Op}(A) = HC_{AQ}(A) = Ext_{Mod_A^{\mathcal{P}}}(\Omega_{\mathcal{P}}A,A)$$ where $Mod_A^{\mathcal{P}}$ is the abelian category of $A$-modules and $\Omega_{\mathcal{P}}A$ the Kähler module of differentials of $A$ (AO, § 12.3.19).

However, I wonder if there is work on comparisons with the more naïve derived $Ext$-functor? More precisely, whether there is work that shows that $$HC_{AQ}(A,A) = Ext_{Mod_A^{\mathcal{P}}}(A,A)$$ as every $\mathcal{P}$-algebra $A$ is also a $A$-module over itself. Or are there arguments to say that this is a rather naïve question?

My question is motivated by the classical case of Hochschild cohomology: for a unital associative algebra $A$, the operadic cohomology complex agrees with the Hochschild cocomplex on the nose and it computes $Ext_{A-A-bimod}(A,A)$ (as the bar complex is a free resolution of A). Does this mean that $\Omega_{uAs}(A)$ and $A$ are related in some sense? According to my computations following Algebraic Operads I obtain that $\Omega_{uAs}(A) \cong \frac{A \otimes A/k \otimes A }{\sim}$ where $a\otimes d(bb') \otimes c \sim a \otimes db \otimes bc + ab \otimes db' \otimes c$. I work here with the operad $uAs$ instead of $As$, encoding respectively unital and non-unital asssociative algebras, because what I know for non-unital associative algebras the Hochschild cocomplex will not necessarily compute the $Ext(A,A)$ (but please correct me if I'm wrong!).

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A sufficient criterion is contained in Theorem 17.3.4 of Modules over operads and functors by Fresse. In its simplest form when $\mathsf{P}$ is a non-dg operad, it implies one can compute (co)homology as an $\textsf{Ext}$ functor if the right $\mathsf{P}$-module $\mathsf{P}[1]$ (also known as the 'species derivative' of $\mathsf{P}$) and the right $\mathsf{P}$-module of operadic Kähler differentials $\Omega_\mathsf{P}^1$ are free. See also this paper where that criterion is applied in case $\mathsf{P} = \mathsf{PreLie}$.

Classically, the reason why things work in the case of unital $\mathsf{As}$-algebras is that one has the short exact sequence $0\longrightarrow\Omega_A^1 \longrightarrow A\otimes A \longrightarrow A \longrightarrow 0$ and the middle term is $A^e$-free, so one can apply the LES to conclude.

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  • $\begingroup$ Thanks for the answer, but the criterion by Fresse simply states the same result that Millès already showed, namely comparison with Ext(KählerDiff(A),A). In the linked paper, they note "contrary to the intuition coming from studying cohomology of Lie algebras and of associative algebras, cohomology of algebras over operads is not given by the Ext-functor over the universal multiplicative enveloping algebra". Do you think this phenomenon is particular to unital As-algebras (through the SES you give) or can this be phrased more operadically? $\endgroup$
    – Lilolance
    Aug 8 at 12:18

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