The cotangent complex of a morphism of schemes

Assume the Axiom of Choice inherited from the ring-map resolution comparison and affine quasi-coherent equivalence (The Axiom of Choice, Independence of the cotangent complex from the chosen simplicial resolution, Affine quasi-coherent sheaves are modules). Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a morphism of schemes (Morphisms of schemes, Schemes and morphisms over a base). For affine open subschemes $\mathrm{Spec}B=U\subseteq X$ (Affine schemes and their coordinate rings) and $\mathrm{Spec}A=V\subseteq S$ with $f(U)\subseteq V$ one has the ring-map cotangent complex $L_{B\mathrm{/}A}$ (The cotangent complex of a ring map), a complex of $B$-modules concentrated in cohomological degrees $\le 0$, hence bounded above. The cotangent complex $L_{X\mathrm{/}S}$ is the cotangent complex $L_{{\mathcal{O}}_X\mathrm{/}{f}^{-1}{\mathcal{O}}_S}$ of the morphism of Zariski ringed spaces: resolve the sheaf ${\mathcal{O}}_X$ by the standard simplicial polynomial $f^{-1}{\mathcal{O}}_S$-algebra resolution, take its relative differentials, and extend coefficients to ${\mathcal{O}}_X$. This is Stacks Definition 24.1 (tag 08T2), using its sheaf-ring definition 18.2 (tag 08SS). The affine comparison of Lemma 24.2 (tag 08T3) identifies its restriction to $U$ with the associated sheaf complex of $L_{B\mathrm{/}A}$, compatibly with smaller charts. Thus the chart complexes are restrictions of this one global complex; canonical isomorphisms in a derived category alone are not being used as a gluing construction. This also agrees with the discrete case of Derived schemes and the cotangent complex of a morphism. It is well defined up to canonical isomorphism in the derived category $D({\mathcal{O}}_X)$ (Derived category of an abelian category, Quasi-isomorphism), and $L_{X\mathrm{/}S}$ is a bounded-above complex of ${\mathcal{O}}_X$-modules. The cohomology sheaves $H^i(L_{X\mathrm{/}S})$ are quasi-coherent ${\mathcal{O}}_X$-modules, and $L_{X\mathrm{/}S}$ is a quasi-coherent derived ${\mathcal{O}}_X$-module (Quasi-coherent module on a scheme). If $X$ is quasi-compact (Quasi-compact and quasi-separated schemes) with affine diagonal (The diagonal morphism, Affine morphisms), or if $X$ is Noetherian (Locally Noetherian and Noetherian schemes), then $L_{X\mathrm{/}S}$ is represented by a complex of quasi-coherent sheaves concentrated in degrees $\le 0$, in particular bounded above. Functoriality: a commutative square of schemes $\begin{array}{ccc}{X}^{\mathrm{\prime }}& \stackrel{g^{\mathrm{\prime }}}{\to }& X\\ \downarrow & & \downarrow f\\ S^{\mathrm{\prime }}& \stackrel{g}{\to }& S\end{array}$ gives a canonical comparison map $\mathbf{L}{g}^{\mathrm{\prime }\ast }{L}_{X\mathrm{/}S}\to L_{X^{\mathrm{\prime }}\mathrm{/}{S}^{\mathrm{\prime }}}$, which is an isomorphism when the square is cartesian and tor-independent (for instance a flat base change). For composable scheme morphisms $X\stackrel{f}{\to }S\to T$ there is a distinguished transitivity triangle (Distinguished triangle, The shift of a chain complex) $\mathbf{L}{f}^{\ast }{L}_{S\mathrm{/}T}⟶L_{X\mathrm{/}T}⟶L_{X\mathrm{/}S}⟶(\mathbf{L}{f}^{\ast }{L}_{S\mathrm{/}T})[1]\mathrm{.}$ The absolute case $S=\mathrm{Spec}k$ for a field $k$ is written $L_{X\mathrm{/}k}$.

• Convention fixed. The definition is the discrete case of Definition 24.1 (tag 08T2) of Stacks, The Cotangent Complex: the cotangent complex of a morphism of ringed spaces, restricted to schemes. Lemma 24.2 (tag 08T3) supplies the canonical chart comparison $L_{B\mathrm{/}A}\to L_{X\mathrm{/}S}{\mathrm{\mid }}_U$ that is an isomorphism in $D({\mathcal{O}}_U)$, compatible with restrictions of the globally defined ringed-space complex. Quasi-coherent cohomology follows from this affine comparison; the additional global representative assertion uses the separate comparison theorems below. The absolute case is Lemma 24.3 (tag 08V6).

• Choice. The standing Axiom of Choice is inherited through the ring-map resolution comparison and the affine quasi-coherent equivalence: the standard resolution is itself constructed without choices, but the comparison of an arbitrary simplicial resolution with it uses the derived-tensor and resolution-independence suppliers declared in Independence of the cotangent complex from the chosen simplicial resolution. This inheritance is carried into every later item that computes with $L_{X\mathrm{/}S}$. The representative argument also uses the affine equivalence to identify kernels of maps of quasi-coherent sheaves with kernels of module maps.

• Construction route. The affine ring-map and derived-scheme suppliers now carry their actual comparison statements. The global sheaf-ring standard resolution is applied from the exact cited Stacks definitions; affine derived isomorphisms alone are not treated as effective descent data.

• Quasi-coherent representatives. Under either stated hypothesis on $X$, Stacks Proposition 36.7.5 (08DB) or Proposition 36.8.3 (09T4) gives an equivalence $D(\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}({\mathcal{O}}_X))\to D_{\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}}({\mathcal{O}}_X)$, where the target consists of complexes with quasi-coherent cohomology. Apply it to $L_{X\mathrm{/}S}$ to obtain a complex $K^{\bullet }$ of quasi-coherent sheaves. Since $H^i(K^{\bullet })=0$ for $i>0$, the canonical truncation ${\tau }^{\le 0}{K}^{\bullet }\to K^{\bullet }$ is a quasi-isomorphism (Canonical truncation of a complex, Canonical truncation is a complex and has the claimed cohomology). Its degree-zero term is $\ker (d^0)$, which is quasi-coherent: on each affine open, the affine equivalence identifies this kernel with the associated sheaf of the kernel of the corresponding module map (Affine quasi-coherent sheaves are modules). This gives the asserted bounded-above representative. The general global construction and quasi-coherent cohomology assertion impose no affine-diagonal or Noetherian hypothesis.

• Base change and transitivity. The standard sheaf-ring resolution is functorial in commutative squares of sheaf rings, and inverse image commutes with it (Stacks Section 92.18 and Lemma 92.18.3, 08SV), giving the displayed comparison for every commutative scheme square. For a cartesian square, take affine charts $U=\mathrm{Spec}B\to V=\mathrm{Spec}A$ and $V^{\mathrm{\prime }}=\mathrm{Spec}{A}^{\mathrm{\prime }}\to V$; their fibre product is $U^{\mathrm{\prime }}=\mathrm{Spec}(B{\otimes }_A{A}^{\mathrm{\prime }})$. Tor-independence means ${\mathrm{Tor}}_i^A(B,A^{\mathrm{\prime }})=0$ for all $i>0$ on such charts. Thus the derived-pushout criterion in Independence of the cotangent complex from the chosen simplicial resolution (Stacks 08QQ) makes the comparison an isomorphism on these charts, which cover $X^{\mathrm{\prime }}$; an affine-local quasi-isomorphism is a global quasi-isomorphism. Transitivity is Stacks Lemma 92.20.3 (08T4), applied to the underlying ringed spaces, as allowed by Definition 92.24.1 (08T2).