Abstract Goerss-Hopkins theory

Dirac geometry I: Commutative algebra

“…The map φ : R → A is even, by Theorem 4.25, so it is the base-change of the map φ ev : R ev → A ev along the inclusion j : R ev → R. Since φ ev is étale, and hence, flat, this is also the derived base-change. The E k -cotangent complex satisfies base-change along symmetric monoidal left adjoint functors between symmetric monoidal stable ∞-categories such as j * ≃ R ⊗ L R ev −; see [26,Lemma 7.6]. 5 In the case at hand, we conclude that the base-change map…”

“…Instead, we employ the obstruction theory of Goerss-Hopkins [13] in the refined form of the second author and VanKoughnett [26]. This, in turn, is based on the second author's synthetic deformation of the stable ∞-category Mod A (Sp), which has the derived ∞-category D(π * (A)) as its special fiber.…”

“…By [26,Section 5] (which assumes the background prestable ∞-category Syn R to be symmetric monoidal, but E k -monoidal is sufficient for our purposes), we have a tower of ∞-categories of potential n-stages…”

“…G G M 0 with the following properties: 6 We use motivic indexing, but the motivic weight j is allowed to be a half-integer, as opposed to an integer. In terms of the indexing used in [26], we have Σ i,j X = Σ i−2j X[2j]. 7 The map τ here is a tensor square-root of the map τ in motivic homotopy theory.…”

“…We will employ Goerss-Hopkins theory following [26]. This takes as an input an appropriate complete prestable ∞-category, which appears in [24, Section 6.5], but we recall the construction here.…”