Derived schemes and the cotangent complex of a morphism

Assume the Axiom of Choice inherited from the cotangent-comparison suppliers used in the cotangent portion (The Axiom of Choice).

A derived scheme is a pair $(X,{\mathcal{O}}_X)$ consisting of a topological space $X$ and a sheaf of simplicial commutative rings ${\mathcal{O}}_X$ (Simplicial objects, simplicial commutative rings and homotopy groups) such that $(X,{\pi }_0{\mathcal{O}}_X)$ is a scheme (Schemes) and each ${\pi }_i{\mathcal{O}}_X$, $i>0$, is a quasi-coherent module on that scheme (Quasi-coherent module on a scheme). Thus the truncation $t_0(X,{\mathcal{O}}_X)=(X,{\pi }_0{\mathcal{O}}_X)$ is an ordinary scheme and the higher homotopy sheaves are quasi-coherent modules on it.

Morphisms of derived schemes are taken in the homotopical category of derived locally ringed spaces: a morphism is a morphism of the underlying simplicially ringed spaces which is local on the truncations, and the mapping spaces are the derived enriched mapping spaces constructed from the model structures on simplicial commutative rings and their modules (Model structures for variable simplicial modules and algebras, Replacement-invariant derived enriched mapping spaces) together with the strict simplicial algebra adjunctions and the Dold-Kan equivalence (The strict simplicial algebra adjunctions underlying the cotangent construction, Dold-Kan equivalence for simplicial modules with explicit inverse). The global category is the full subcategory of derived locally ringed spaces of Toën, Definition 2.5 (cited survey, PDF page 33). Affine computations may be performed in the strict projective diagram models of Projective models for simplicial and variable-module diagrams and the span comparison of The projective span model computes the homotopy-pushout mapping property. The local diagram models alone are not a construction of the global category; the cited derived locally ringed-space construction supplies that category and its homotopical gluing.

A scheme embeds as the constant (discrete) derived scheme via $i\text{\hspace{0.17em}⁣}:Sch\to dSch$: a scheme $Y$ is sent to the pair with the constant simplicial structure sheaf; this is fully faithful. The truncation $t_0(X)=(X,{\pi }_0{\mathcal{O}}_X)$ is right adjoint to this inclusion, ${\mathrm{Map}}_{dSch}(iY,X)\simeq {\mathrm{Hom}}_{Sch}(Y,t_{0}X),$ with the right-hand side discrete: for a constant derived scheme a morphism to $X$ is determined by its truncation, and every morphism $Y\to t_{0}X$ lifts. The counit of the adjunction is the canonical morphism $j_X\text{\hspace{0.17em}⁣}:i(t_{0}X)\to X$.

For a morphism of derived schemes $f\text{\hspace{0.17em}⁣}:X\to Y$, the cotangent complex $L_{X\mathrm{/}Y}$ is a quasi-coherent derived ${\mathcal{O}}_X$-module, obtained by gluing the affine derived cotangent complexes: on charts $\mathrm{Spec}B\to \mathrm{Spec}A$ of derived rings it is the $B$-module cotangent complex representing relative derived derivations in the fixed-base sense of The fixed-base simplicial cotangent module represents derived derivations; the global existence and descent of these modules uses HAG II, Theorem 1.3.7.2 (QCoh is a stack), in the simplicial-ring context of Section 2.2.1, and Corollary 2.2.3.3 for the relative cotangent complex. This agrees with the affine-local construction of Toën (survey, PDF pages 38-40). The local comparisons use the projective module-diagram models, the contractible cosimplicial evaluation criterion, coefficient and category change, and the bounded-above flat tensor compatibility (Module diagrams have projective representables and computable derived colimits, Contractible cosimplicial evaluation computes diagram derived colimits, Derived colimit commutes with coefficient change and admissible category change, Bounded above flat tensor complexes preserve quasi isomorphisms), with the affine higher-cohomology vanishing of Affine acyclicity of quasi-coherent sheaves under its stated AC used for the bounded section computation. For discrete ordinary $A$ and $B$ this is the ordinary ring-map complex (The cotangent complex of a ring map), whose resolution and base-change comparisons are Independence of the cotangent complex from the chosen simplicial resolution. The derived pullback $j_X^{\ast }{L}_{X\mathrm{/}Y}$ is a complex of ${\pi }_0{\mathcal{O}}_X$-modules on $t_{0}X$ and must be distinguished from the full derived ${\mathcal{O}}_X$-module $L_{X\mathrm{/}Y}$; its homology and shifts use Homology object of a chain complex and The shift of a chain complex, and quasi-isomorphisms are those of Quasi-isomorphism.

These constructions require genuine homotopical module and cotangent comparisons: they are not obtained by applying the ordinary-ring definition of The cotangent complex of a ring map to non-discrete simplicial rings, and the definition therefore imports only the interfaces listed above.

Source applications. The source passages above define the global category, establish descent of quasi-coherent derived modules, and supply the relative cotangent complex. The exact HAG II statements and printed proofs at PDF pages 33-34, 96-97, 111-112, 142-146 and 160-161, together with the cited Toën survey pages 20, 33 and 38-40, were checked. The strict diagram suppliers are used for affine computations and do not replace the global descent theorem.