Ext groups of the cotangent complex

Assume the Axiom of Choice; it implies the Dependent Choice hypothesis of the derived-Hom supplier (The Axiom of Choice, AC implies DC implies countable choice). Let $X$ be a scheme, let $L$ be a bounded-above complex of ${\mathcal{O}}_X$-modules with quasi-coherent cohomology (for instance $L=L_{X\mathrm{/}S}$ of The cotangent complex of a morphism of schemes), and let $M$ be a quasi-coherent ${\mathcal{O}}_X$-module, regarded as a complex concentrated in degree $0$ (Quasi-coherent module on a scheme). Define ${\mathrm{Ext}}_{{\mathcal{O}}_X}^i(L,M):=H^i(\mathbf{R}{\mathrm{Hom}}_{{\mathcal{O}}_X}(L,M)),$ the cohomology of the derived Hom computed in the abelian category of ${\mathcal{O}}_X$-modules via Derived hom in the bounded setting. Then ${\mathrm{Ext}}_{{\mathcal{O}}_X}^i(L,M)\cong {\mathrm{Hom}}_{D({\mathcal{O}}_X)}(L,M[i])$ for every $i$ (Cohomology of derived hom is ext, Ext is hom in the derived category, The shift of a chain complex), so the groups are natural in $L$ and $M$ and vanish for $i<0$ when $L$ is concentrated in cohomological degrees $\le 0$. If $L$ is concentrated in degree $0$ and is a module $F$, then ${\mathrm{Ext}}_{{\mathcal{O}}_X}^i(F,M)$ is the global sheaf-Ext group of Sheaf Ext of coherent modules. For a two-term complex $N^{-1}\stackrel{d}{\to }{N}^0$ of ${\mathcal{O}}_X$-modules, Ext is still computed by $\mathbf{R}\mathrm{Hom}(N,M)$. If both terms are projective objects of the chosen abelian module category, then the ordinary Hom complex computes derived Hom and hence ${\mathrm{Ext}}^0(N,M)=\ker (\mathrm{Hom}(N^0,M)\to \mathrm{Hom}(N^{-1},M)),{\mathrm{Ext}}^1(N,M)=\mathrm{coker}(\mathrm{Hom}(N^0,M)\to \mathrm{Hom}(N^{-1},M)),$ the map being composition with $d$. Without that hypothesis one must retain derived Hom. For finite locally free terms on a scheme, one may instead take derived global sections of the internal Hom complex; ordinary global Hom gives the displayed formula only when its terms are acyclic for global sections. Ring analogue: for a ring map $A\to B$, a bounded-above complex $L$ of $B$-modules and a $B$-module $M$, ${\mathrm{Ext}}_B^i(L,M)$ is defined in the same way in the abelian category of $B$-modules.

• Reference conventions. The notation ${\mathrm{Ext}}^i(L_{X\mathrm{/}S},-)$ is the one used by Illusie, Complexe cotangent et deformations I, Chapitre II, and by Stacks, The Cotangent Complex, Sections 92.16 and 92.21, where the same groups carry the obstruction class, the torsor structure and the automorphism groups of deformations. The identification with ${\mathrm{Hom}}_D(L,M[i])$ is the published derived-Hom comparison Cohomology of derived hom is ext, whose Dependent Choice hypothesis is supplied by the declared Axiom of Choice.
• Two-term computation. The ordinary Hom formula requires the projectivity or acyclicity hypotheses just stated. A module in degree zero has arbitrary positive Ext in general; boundedness of the ordinary Hom complex alone does not make it a representative of derived Hom.
• Open supplier note. The derived-Hom and resolution suppliers used here are published except for the in-run ring-map cotangent complex The cotangent complex of a ring map and the comparison Independence of the cotangent complex from the chosen simplicial resolution; the consumer steps that rely on them are recorded in the pair report.