Representability theorem in derived analytic geometry

Derived Algebraic Geometry XIV: Representability Theorems

2 Properties of Moduli Functors 22 2.1 Nilcomplete, Cohesive, and Integrable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Relativized Properties of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Finiteness Conditions on Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Moduli of Spectral Deligne-Mumford St...

Grothendieck's Existence Theorem in Analytic Geometry and Related Results

We state and prove several kinds of analytification theorems of formal objects (such as coherent sheaves and formal complex spaces) which are in the spirit of Grothendieck's algebraization theorem in [EGA, III]. The formulation of the results was derived from deformation theory and especially M. Artin's work on representability of functors. The methods of proof depend heavily on a deeper study ...

Failure of Brown Representability in Derived Categories

Let T be a triangulated category with coproducts, T ⊂ T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors {T}op −→ Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this ...

Failure of Brown Representability in Derived Categories

Let T be a triangulated category with coproducts, T ⊂ T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors {T} −→ Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this ge...

Derived Algebraic Geometry V: Structured Spaces Introduction: Bezout's Theorem