One of the first results that one needs to prove in the theory of chiral algebras is a de Rham version of Cohen's theorem on the homology of $C_n$ spaces. This is achieved in Beilinson-Drinfeld's book Chiral Algebras in Theorem 3.1.5. Roughly speaking, one is interested in studying Lie algebras in the category of $D_X$-modules for a smooth curve $X$. The theorem asserts that $\omega_X$, the canonical bundle, is naturally such an algebra in an unique way.

(Slightly) more precisely, one constructs a (colored) operad structure in the category of right $D_X$-modules and the theorem asserts that the full subcategory whose only object is $\omega_X$ is canonically identified with the operad $\mathcal{L}ie$.

In recent years we have seen many developments that parallel Beilinson and Drinfeld's constructions in the homotopy world. Lurie defines factorizable sheaves and topological chiral homology in Chapter 5 of his Higher Algebra book. Francis and Gaitsgory defined something like an $\infty$-operad structure in the $\infty$-category of $D_X$-modules for $X$ a higher dimensional scheme (they defined two compatible monoidal structures on $D$-modules over the Ran space of $X$).

Q1: is Cohen-Beilinson-Drinfeld theorem valid in this situation?

By the Koszul duality proved in Francis-Gaitsgory's paper chiral algebras (or Lie algebras) in this setting correspond to commutative coalgebras (or factorization algebras). That theorem above would translate to the fact that the structure sheaf on $Ran(X)$ is the unit (co)algebra in an unique way. This is essentially by definition of the monoidal structure they define, so by their equivalence of categories I suppose the Q1 above would be answered.

Q2: Is the above paragraph correct?

What strikes me as odd though is that Theorem 3.1.5 is an integral part of the (non-$\infty$) equivalence between chiral algebras and factorization algebras, namely, one strongly uses that the Cousin resolution of $\omega_X$ is a factorization algebra because it is the Chevalley complex of the unique chiral algebra $\omega_X$. I could not find the analog of this in Francis-Gaitsgory's proof nor in Lurie's works. Perhaps in Costello's?

Finally and more ambitiously, if the answer to Q1 is yes

Q3: Can Francis-Gaitsgory's construction be generalized to derived schemes? (as in Lurie's DAG)

The factorization side seems to be well defined, the Lie algebra world I have no clue what it would/can be.


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