Derived deformations of Artin stacks

Abstract

Abstract. We generalise the techniques of [Pri5] to describe derived deformations in simplicial categories. This allows us to consider deformation problems with higher automorphisms, such as chain complexes (which have homotopies) and stacks (which have 2-automorphisms). We also give a general approach for studying deformations of diagrams.

Constructing derived moduli stacks

“…The only differences lie in a straightforward check that Del g (E r ) ♯ is locally of finite presentation, and in the calculation of cohomology groups. For the comonad ⊥ := Symm + • F ∂ U ∂ on cNAlg(A), we get a canonical simplicial resolution ⊥ • S, given by ⊥ n S := ⊥ n+1 S. For A ∈ Alg R , the proof of [Pri3] Lemma 5.7 then shows that A ⊕ (⊥ • S) m is a cofibrant resolution of A ⊕ S m for all m, whenever S m is projective as an A-module. If we set L ⊥ • (S) := Ω((A ⊕ ⊥ • S)/A), this means that the simplicial complex L ⊥ • (S) m is a model for the cotangent complex of A ⊕ S m .…”

“…In fact, we can go further than this. By [Pri3] §5.1, the monad Symm distributes over ⊤ ∂ , so the composite monad Symm • ⊤ ∂ is another monad. Moreover, cAlg(A) ≃ c + Mod(A) (Symm•⊤ ∂ ) .…”

“…If we set L ⊥ • (S) := Ω((A ⊕ ⊥ • S)/A), this means that the simplicial complex L ⊥ • (S) m is a model for the cotangent complex of A ⊕ S m . If S is levelwise projective as an A-module, then by [Pri3] Lemma 5.8, L ⊥ n (S) is a projective object of cMod(S). Thus, for S ∈ E ♯ r (A), Lemma 5.47 gives that D i…”

The derived deformation theory of a point

“…This appears as e.g. [DN15, 4.2.6] or [Pri15,4.6]. The basic idea of the proof is the same as that of the set-valued case 5.3.3; we give a sketch.…”

“…The basic idea of the proof is the same as that of the set-valued case 5.3.3; we give a sketch. The proof of [Pri15,3.13] shows that the simplicial groupoid of deformations of X is the same as the simplicial subgroupoid of Del(E) on constant objects, which we will denote C. By [Pri15, §4.1], C and Del(E) are both derived deformation functors. The inclusion induces a map on tangent spaces which is a weak equivalence and so the two functors are weakly equivalent by an obstruction-theoretic argument.…”