# Cohomology of algebra over an operad as naive Ext-functor instead of with Kähler module of differentials

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.