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!).
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.