Cech hypercohomology of an affine cover computes Ext of the cotangent complex

The global/internal derived-Hom comparison in [F1] can be computed explicitly. Let $M\to I^{\bullet }$ be a bounded-below module-injective resolution. Since restriction preserves injectives (its left adjoint, extension by zero, is exact), the bounded-below complex $K_0=\mathcal{H}o{m}^{\bullet }(L,I^{\bullet })$ computes internal derived Hom on every open. Each term is a finite product of sheaves $\mathcal{H}om(L^a,I^b)$, because $L$ is bounded above. These sheaves are flasque: a morphism $L{\mathrm{\mid }}_U\to I{\mathrm{\mid }}_U$ extends over $V\supset U$ by injectivity of $I{\mathrm{\mid }}_V$ applied to $j_{!}(L{\mathrm{\mid }}_U)\hookrightarrow L{\mathrm{\mid }}_V$. Thus $K_0$ has global-section-acyclic terms and the bounded-below hypercohomology comparison [F2] identifies $\mathbf{R}\mathrm{\Gamma }(X,K_0)$ with $\mathrm{\Gamma }(X,K_0)={\mathrm{Hom}}^{\bullet }(L,I^{\bullet })$, the complex computing global derived Hom. This establishes [F1].

Resolve $M$ injectively and compute the internal derived Hom, then replace the resulting bounded-below complex $K$ by a bounded-below injective complex $J$, using Stacks tag 013K and the enough-injectives assertion of Injective modules are flasque and Ext from the structure sheaf is cohomology through its module-injective supplier. Boundedness below follows from $L$ being bounded above and $M$ being in degree zero. The augmented sheaf Cech complex of each $J^q$ is exact: near a point choose one cover member containing it, shrink inside that member, and insert its index in the alternating Cech differential to obtain a contracting homotopy of the augmentation. Restriction of an injective sheaf of modules to an open is injective, since extension by zero is its exact left adjoint. Thus every intersection has no higher cohomology for $J^q$. For an intersection inclusion $j$, the module $j_{\ast }(J^q{\mathrm{\mid }}_U)$ is injective because $j_{\ast }$ is right adjoint to the exact restriction functor. Thus the augmented sheaf Cech complex is an injective resolution of the injective $J^q$ and splits into short exact sequences, so applying global sections preserves its exactness. The double-complex filtration therefore gives a quasi-isomorphism from $\mathrm{\Gamma }(X,J)$ to the total complex of $\mathrm{\Gamma }(U_{i_0\dots i_p},J)$. The cover direction is finite, so totalization and its filtration converge in every degree. Each column computes $\mathbf{R}\mathrm{\Gamma }(U_{i_0\dots i_p},K)$ by injectivity, and $\mathrm{\Gamma }(X,J)$ computes $\mathbf{R}\mathrm{\Gamma }(X,K)$. Taking cohomology and [F1] proves the derived Cech assertion.

For a bounded-below representative $K^{\mathrm{\prime }}$ with acyclic terms on every intersection, the first hypercohomology spectral sequence [F2] collapses to the ordinary section complex on each intersection. Replacing the derived columns in step 1.2 by these section complexes therefore preserves the total cohomology; the finite cover filtration again ensures convergence. Affine quasi-coherent terms are one sufficient case of the required acyclicity by [F4]. Finally, when the local deformation data are represented by the indicated truncation of derived Hom, their degree-one descent cocycles and degree-two obstruction cocycles have precisely the total cohomology just computed. This last application requires the stated representation of local deformation data; cohomology comparison by itself does not construct that representation.