DERIVED ∞-STACKS // AFFINE E∞ SCHEMES // COTANGENT COMPLEX
A derived stack is an ∞-sheaf on the site of simplicial commutative rings (or E_∞-ring spectra). Under Lurie's spectral algebraic geometry, these replace classical algebraic stacks with homotopy-coherent gluing data.
X: CAlg^cn → S (∞-sheaf on connective E∞-rings)
Given an E_∞-ring spectrum R, the spectral affine scheme Spec R has an underlying topological space |Spec π_0 R| with a sheaf of E_∞-rings. The functor Spec: (CAlg^cn)^op → SpSch is fully faithful.
For a map f: X → Y of spectral stacks, the cotangent complex L_{X/Y} is a quasi-coherent sheaf encoding the derived deformation theory. It satisfies a fiber sequence: f*L_{Y/Z} → L_{X/Z} → L_{X/Y}.