## Derived algebraic geometry and virtual fundamental cycles: cotangent complexes

I have been thinking of a way to apply the derived algebraic geometry of Toen-Vezzosi to construct virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. This seems to be the natural setting for which to define symplectic topological invariants when the relevant moduli spaces aren’t necessarily cut out transversally. Indeed, Joyce adopts this philosophy in his “D-orbifolds” project. There, Joyce defines a d-orbifold to be a kind of Deligne-Mumford $$C^{\infty}$$– stack equipped with a sequence of quasi-coherent sheaves on this stack that should be thought of a cotangent complex. Should one expect that this “cotangent complex” is a literal cotangent complex for some derived geometric $$C^{\infty}$$ – stack? More precisely – should there correspond, to a D-orbifold, a derived stack $$\mathcal{X}^{der}$$ which is a derived extension of some geometric stack $$\mathcal{X}$$, and where this derived structure is the analog of a perfect obstruction theory?

## Cotangent complex and its distinguished triangle- a generalisation?

Associated to any ring maps $$A\to B\to C$$ there is the distinguished triangle $$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\longrightarrow} \$$ in $$D(C)$$. The cotangent complex $$\mathbf{L}_{C/A}$$ is (the value at $$C$$ of) the Quillen left derived functor of $$C\otimes_-\Omega^1_{-/A} \ : \ \text{simplicial }A\text{ rings over }C\ \longrightarrow \ \text{simplicial }C\text{ modules}$$ Note that $$D(C)$$ is the homotopy category of simplicial $$C$$ modules, by Dold-Kan.

Is there a deeper reason for the triangle, or is it something very special about $$\mathbf{L}$$? i.e. can we say anything about when a derived Quillen functor (say between stable model categories) admits a long exact sequence like this?

The question must be fairly subtle because already the result fails if we replace rings with general schemes in the above, except under certain conditions on the maps $$X\to Y\to Z$$.

## Descent for the cotangent complex along faithfully flat SCRs

By Theorem 3.1 of Bhatt-Morrow-Scholze II (https://arxiv.org/pdf/1802.03261.pdf), we know that for $$R$$ a commutative ring, $$\wedge^{i}L_{(-)/R}$$ satisfies descent for faithfully flat maps $$A \rightarrow B$$ of (ordinary) $$R$$-algebras. Can this descent statement be promoted to, say, faithfully flat maps of simplicial commutative $$R$$-algebras (where $$R$$ is either discrete or a simplicial commutative ring)? I am primarily interested in the case where $$i=1$$, as the proof of this claim along with an induction argument using a filtration, can probably give the $$i>1$$ case.