Localization morphisms and the cotangent complex

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

A map of rings $A\to B$ of a topos is a localization morphism if $B$ is obtained from $A$ by inverting a subsheaf of sections. Such maps are flat epimorphisms of rings, and a flat epimorphism has a vanishing cotangent complex. This lesson develops localization morphisms in a topos and draws the consequences for cotangent complexes of ringed topoi and of schemes. The cotangent complex of a ringed topos is unchanged by localization. On an affine scheme it is the quasi-coherent module associated with the cotangent complex of the rings, so its homology sheaves are quasi-coherent, and coherent in the noetherian case of finite type. It satisfies Tor-independent base change, for rings of a topos and for ringed topoi whose structure rings are localizations of tensor products, as they are for fibre products of schemes.

Prerequisites. Cotangent complexes of simplicial rings (Proposition 1.3, Proposition 2.5, Corollary 3.5); Derived tensor products of simplicial modules (Lemma 2.1, Proposition 3.4); The cotangent complex (Section 10).

Conventions

$T$ is a topos, $T=\operatorname{Sh}(\mathcal C)$ for a site $\mathcal C$, and $a$ is sheafification; it is exact [Stacks, Tag 03EI] and commutes with modules of differentials [Tag 08TP], and the tensor product of sheaves of modules is the sheaf associated with the presheaf tensor product. Rings are commutative with $1$. For a ring $B$, $B^*\subseteq B$ is the subsheaf of units: $B^*(U)=B(U)^*$. It is the image of the finite limit $\{(b,c)\colon bc=1\}\subseteq B\times B$ under the first projection, which is a monomorphism. For a map of rings $A\to B$, $L_{B/A}$ is the cotangent complex [Stacks, Tag 08SS], a degreewise free simplicial $B$-module, regarded as an object of $D(B)$. For a morphism of ringed topoi $f\colon X\to Y$, $L_{X/Y}=L_{\mathcal O_X/f^{-1}\mathcal O_Y}$ [Stacks, Tag 08SU], and $f^*M=f^{-1}M\otimes_{f^{-1}\mathcal O_Y}\mathcal O_X$. Inverse images of rings along morphisms of topoi commute with cotangent complexes [Stacks, Tag 08SV], and maps $A\to B\to C$ of rings give the distinguished triangle $L_{B/A}\otimes_BC\to L_{C/A}\to L_{C/B}\to$ in $D(C)$ [Stacks, Tag 08SZ]; the tensor product is underived because $L_{B/A}$ is flat.

1. Rings of fractions and localization morphisms

Definition 1.1. Let $A$ be a ring of $T$ and $X\subseteq A$ a subsheaf of sets. A ring of fractions of $A$ with respect to $X$ is a map of rings $\iota\colon A\to X^{-1}A$ carrying $X$ into $(X^{-1}A)^*$, such that every map of rings $A\to R$ carrying $X$ into $R^*$ factors uniquely through $\iota$.

Lemma 1.2. For every subsheaf of sets $X\subseteq A$, a ring of fractions exists. It is the sheaf associated with the presheaf $P\colon U\mapsto S(U)^{-1}A(U)$, where $S(U)$ is the multiplicative monoid generated by $X(U)$. The map $\iota$ is flat and an epimorphism of rings, so $X^{-1}A\otimes_AX^{-1}A\to X^{-1}A$ is an isomorphism.

Proof. Universal property. $P$ is a presheaf of rings under $A$ in which the sections of $X$ are invertible, hence so is $aP$. Let $g\colon A\to R$ carry $X$ into $R^*$. For every $U$, $g(U)$ carries $X(U)$ into $R(U)^*$, so it factors uniquely through $S(U)^{-1}A(U)$, compatibly with restrictions; the resulting map of presheaves $P\to R$ factors uniquely through $aP$, as $R$ is a sheaf. A factorization through $aP$ is determined by the composite $P\to aP\to R$, which is determined by $g$. So $aP$ is a ring of fractions.

Flatness. Let $N'\to N$ be an injective map of $A$-modules. For every $U$, the map from $N'(U)\otimes_{A(U)}P(U)$ to $N(U)\otimes_{A(U)}P(U)$ is injective, because $S(U)^{-1}A(U)$ is a flat $A(U)$-module. Sheafifying, which is exact and carries the presheaf tensor products to $N'\otimes_AaP$ and $N\otimes_AaP$, gives an injective map. So $aP$ is flat over $A$.

Epimorphism. Two maps of rings $X^{-1}A\rightrightarrows R$ that agree on $A$ agree, by the uniqueness in Definition 1.1. Applied to the two maps $X^{-1}A\to X^{-1}A\otimes_AX^{-1}A$, $b\mapsto b\otimes1$ and $b\mapsto1\otimes b$, this gives $b\otimes1=1\otimes b$. Then $b\otimes c=(b\otimes1)(c\otimes1)=bc\otimes1$, so $b\mapsto b\otimes1$ is inverse to the multiplication map. $\square$

Definition 1.3. A map of rings $u\colon A\to B$ of $T$ is a localization morphism if there is a subsheaf of sets $X\subseteq A$ with $u(X)\subseteq B^*$ such that the induced map $X^{-1}A\to B$ is an isomorphism.

If $u$ is a localization morphism, then $X_u=u^{-1}(B^*)$ has the same property, and it is the largest such subsheaf. Indeed $X\subseteq X_u$, and the elements of $X_u$ are invertible in $B\cong X^{-1}A$, so $\iota\colon A\to X^{-1}A$ factors through $X_u^{-1}A$; the universal properties show that $X^{-1}A\to X_u^{-1}A$ and the factorization are inverse to each other. By Lemma 1.2, a localization morphism is flat and an epimorphism of rings.

Lemma 1.4.

Let $u\colon A\to B$ be a map of presheaves of rings on $\mathcal C$ such that every $u(U)\colon A(U)\to B(U)$ is a localization of rings. Then $au\colon aA\to aB$ is a localization morphism.
Let $C\to A\xrightarrow{u}B$ be maps of rings of $T$ with $u$ a localization morphism. Then $u\otimes_CC'\colon A\otimes_CC'\to B\otimes_CC'$ is a localization morphism for every $C$-algebra $C'$.
Let $A\xrightarrow{u}B\xrightarrow{v}C$ be maps of rings of $T$. If $u$ and $v$ are localization morphisms, so is $vu$. If $vu$ is a localization morphism and $u$ is an epimorphism of rings, then $v$ is a localization morphism.
Let $f\colon T'\to T$ be a morphism of topoi and $u\colon A\to B$ a map of rings of $T$. If $u$ is a localization morphism, so is $f^{-1}u$. If $f^{-1}u$ is a localization morphism and $f^{-1}$ is faithful, then $u$ is a localization morphism.

Proof. (1) Let $X\subseteq A$ be the subpresheaf $U\mapsto u(U)^{-1}(B(U)^*)$, so that $u(U)$ induces an isomorphism $X(U)^{-1}A(U)\cong B(U)$ for every $U$. Let $R$ be a sheaf of rings. A map of rings $aB\to R$ is a map of presheaves $B\to R$, that is, a map of presheaves $A\to R$ carrying $X$ into $R^*$. Since $R^*$ is a sheaf, this is a map $aA\to R$ carrying $aX$ into $R^*$. So $au\colon aA\to aB$ is a ring of fractions of $aA$ with respect to $aX$.

(2) Let $X\subseteq A$ induce $X^{-1}A\cong B$, and let $X'$ be the image of $X$ under $a\mapsto a\otimes1$. A map of rings $A\otimes_CC'\to R$ carrying $X'$ into $R^*$ is a pair of maps $A\to R$, $C'\to R$ agreeing on $C$, the first carrying $X$ into $R^*$; it factors uniquely through $B\otimes_CC'$. So $X'^{-1}(A\otimes_CC')\cong B\otimes_CC'$.

(3) Suppose $u$ and $v$ are localization morphisms. Put $Y=v^{-1}(C^*)$ and $X=u^{-1}(Y)$. Let $g\colon A\to R$ carry $X$ into $R^*$. It carries $u^{-1}(B^*)\subseteq X$ into $R^*$, so it factors uniquely as $\bar gu$ with $\bar g\colon B\to R$. Every section $b$ of $Y$ is locally of the form $u(x)^{-1}u(a)$ with $x\in u^{-1}(B^*)$. Then $vu(a)=v(b)\,vu(x)$ is a unit, so $a\in X$ and $\bar g(b)=g(x)^{-1}g(a)$ is a unit. Hence $\bar g$ factors uniquely through $Y^{-1}B=C$. So $C$ is a ring of fractions of $A$ with respect to $X$.

Conversely, suppose $vu$ is a localization morphism, with $Z=(vu)^{-1}(C^*)$, and $u$ is an epimorphism. Put $Y=v^{-1}(C^*)$. Let $g\colon B\to R$ carry $Y$ into $R^*$. Since $u(Z)\subseteq Y$, $gu$ carries $Z$ into $R^*$, so $gu=\bar g\,vu$ for a unique $\bar g\colon C\to R$. As $u$ is an epimorphism, $\bar gv=g$. If also $\bar g'v=g$, then $\bar g'vu=gu$ and $\bar g'=\bar g$. So $C$ is a ring of fractions of $B$ with respect to $Y$.

(4) The functor $f^{-1}$ is exact, so it carries $B^*$ to $(f^{-1}B)^*$ and inverse images to inverse images. Let $X\subseteq A$ induce $X^{-1}A\cong B$. By adjunction, a map of rings $f^{-1}A\to R'$ carrying $f^{-1}X$ into $R'^*$ is a map $A\to f_*R'$ carrying $X$ into $f_*(R'^*)=(f_*R')^*$. It factors uniquely through $B$, and by adjunction again through $f^{-1}B$. So $(f^{-1}X)^{-1}f^{-1}A\cong f^{-1}B$. Conversely, put $X=u^{-1}(B^*)$, so that $u$ factors as $A\to X^{-1}A\xrightarrow{\bar u}B$. Then $f^{-1}X=(f^{-1}u)^{-1}((f^{-1}B)^*)$, and $f^{-1}\bar u$ is the isomorphism of the localization $f^{-1}u$, by the first part and the uniqueness of rings of fractions. An exact faithful functor reflects isomorphisms, so $\bar u$ is an isomorphism. $\square$

2. Flat epimorphisms and the cotangent complex

Proposition 2.1. Let $u\colon A\to B$ be a map of rings of $T$ which is flat and an epimorphism of rings. Then $L_{B/A}\cong0$ in $D(B)$.

Proof. Let $P=P_A(B)$ be the standard resolution, with augmentation $\varepsilon\colon P\to B$, and $Q=P\otimes_AB$, a simplicial $B$-algebra with $Q_n=B[P_{n-1}]$ degreewise free ($P_{-1}=B$). The map $Q=P\otimes_AB\xrightarrow{\varepsilon\otimes B}B\otimes_AB\xrightarrow{\ \mu\ }B$ is a quasi-isomorphism. Indeed $\varepsilon\otimes B$ is the tensor product over the ring $A$ of the quasi-isomorphism $\varepsilon$ with the flat $A$-module $B$, a quasi-isomorphism by Lemma 2.1 of the lesson on derived tensor products, and the multiplication $\mu$ is an isomorphism by the epimorphism property. By Corollary 3.5 of the lesson on cotangent complexes of simplicial rings, $\Omega^1_{Q/B}\otimes_QB\cong L_{B/B}$. And $L_{B/B}\cong0$: by Proposition 1.3 there, applied to $B=\operatorname{Sym}_B(0)$, it is quasi-isomorphic to $\Omega^1_{B/B}=0$.

Differentials commute with base change, degree by degree: $\Omega^1_{Q/B}=\Omega^1_{P/A}\otimes_AB=\Omega^1_{P/A}\otimes_PQ$. The composite $P\to Q\to B$ is $p\mapsto\mu(\varepsilon(p)\otimes1)=\varepsilon(p)$, so $\Omega^1_{Q/B}\otimes_QB=\Omega^1_{P/A}\otimes_PB=L_{B/A}.$ Hence $L_{B/A}\cong0$. $\square$

Corollary 2.2. Let $\begin{array}{ccc}B&\longrightarrow&B'\\ \uparrow&&\uparrow\\ A&\longrightarrow&A'\end{array}$ be a commutative square of rings of $T$ whose horizontal maps are flat epimorphisms of rings. Then the canonical map $L_{B/A}\otimes_BB'\to L_{B'/A'}$ is a quasi-isomorphism.

Proof. By Proposition 2.1, $L_{B'/B}\cong0$ and $L_{A'/A}\cong0$. The triangle of $A\to B\to B'$ shows that $L_{B/A}\otimes_BB'\to L_{B'/A}$ is an isomorphism in $D(B')$, and the triangle of $A\to A'\to B'$ shows that $L_{B'/A}\to L_{B'/A'}$ is one. The composite is the canonical map. $\square$

By Lemma 1.2, Proposition 2.1 and Corollary 2.2 apply to localization morphisms: a localization morphism $A\to B$ has $L_{B/A}\cong0$.

Proposition 2.3. Let $f\colon X\to Y$ be a morphism of ringed topoi, $V$ an object of $Y$ and $U$ an object of $X$ over $f^{-1}V$. Let $X'=X_{/U}$ and $Y'=Y_{/V}$, with the induced rings, so that the localization morphisms $j\colon X'\to X$ and $j'\colon Y'\to Y$ and the morphism $f'\colon X'\to Y'$ induced by $f$ satisfy $fj=j'f'$. Then the canonical map $j^*L_{X/Y}\to L_{X'/Y'}$ is an isomorphism.

Proof. The map $f'^{-1}\mathcal O_{Y'}\to\mathcal O_{X'}$ is $j^{-1}$ of the map $f^{-1}\mathcal O_Y\to\mathcal O_X$, since $j^{-1}$ is restriction to $U$ and $j'^{-1}$ restriction to $V$. By [Stacks, Tag 08SV], $j^{-1}L_{X/Y}=L_{X'/Y'}$, and $j^*=j^{-1}$ on $\mathcal O_X$-modules because $\mathcal O_{X'}=j^{-1}\mathcal O_X$. $\square$

Corollary 2.4. Let $g\colon X\to Z$ and $p\colon Z\to Y$ be morphisms of ringed topoi such that $p^{-1}\mathcal O_Y\to\mathcal O_Z$ is a localization morphism. Then the canonical map $L_{X/Y}\to L_{X/Z}$ is a quasi-isomorphism.

Proof. By Lemma 1.4 (4), $g^{-1}p^{-1}\mathcal O_Y\to g^{-1}\mathcal O_Z$ is a localization morphism, so its cotangent complex vanishes. The triangle of $g^{-1}p^{-1}\mathcal O_Y\to g^{-1}\mathcal O_Z\to\mathcal O_X$ gives the claim. $\square$

3. Schemes

For a ring $A$ and a scheme $X$, $A_X$ is the constant sheaf with value $A$.

Proposition 3.1. Let $X=\operatorname{Spec}A$. The canonical map $A_X\to\mathcal O_X$ is a localization morphism.

Proof. For a point $x\in X$ with prime $\mathfrak p$, the stalk map $A\to\mathcal O_{X,x}=A_{\mathfrak p}$ is a localization of rings. The stalk functors form the inverse image of a morphism of topoi from the disjoint union of the points of $X$ to $X$; this inverse image is exact and faithful, because a map of sheaves on a topological space is determined by its stalks. Lemma 1.4 (4) applies. $\square$

Let $u\colon A\to B$ be a map of rings, $f\colon X=\operatorname{Spec}B\to Y=\operatorname{Spec}A$ the corresponding morphism, and $M\mapsto\widetilde M$ the quasi-coherent module associated with a $B$-module. The natural map $M_X\otimes_{B_X}\mathcal O_X\to\widetilde M$ is an isomorphism: at a point with prime $\mathfrak q$ of $B$, both stalks are $M_{\mathfrak q}$. Applied degree by degree, and with $(L_{B/A})_X=L_{B_X/A_X}$ by [Stacks, Tag 08SV] for the morphism from $X$ to the punctual topos, this gives $(L_{B/A})^\sim=L_{B_X/A_X}\otimes_{B_X}\mathcal O_X.$ The commutative square of rings of $X$ $\begin{array}{ccc}B_X&\longrightarrow&\mathcal O_X\\ \uparrow&&\uparrow\\ A_X&\longrightarrow&f^{-1}\mathcal O_Y\end{array}$ then gives a canonical map $\lambda\colon(L_{B/A})^\sim\to L_{X/Y}$.

Proposition 3.2. The map $\lambda\colon(L_{B/A})^\sim\to L_{X/Y}$ is a quasi-isomorphism.

Proof. The top map of the square is a localization morphism by Proposition 3.1. The bottom map is $f^{-1}$ of $A_Y\to\mathcal O_Y$, a localization morphism by Proposition 3.1 and Lemma 1.4 (4); note $f^{-1}A_Y=A_X$. Corollary 2.2 applies. $\square$

Corollary 3.3. Let $f\colon X\to Y$ be a morphism of schemes. The homology sheaves of $L_{X/Y}$ are quasi-coherent. If $Y$ is locally noetherian and $f$ is locally of finite type, they are coherent.

Proof. By Proposition 2.3, applied to affine opens $U\subseteq X$ and $V\subseteq Y$ with $f(U)\subseteq V$, the question is local, so we may assume $X=\operatorname{Spec}B$ and $Y=\operatorname{Spec}A$. By Proposition 3.2, $L_{X/Y}$ is quasi-isomorphic to $(L_{B/A})^\sim$, and $M\mapsto\widetilde M$ is exact, so $H_i(L_{X/Y})=H_i(L_{B/A})^\sim$ is quasi-coherent. If $A$ is noetherian and $B$ of finite type over $A$, there is a quasi-isomorphism $P\to B$ of simplicial $A$-algebras with every $P_n$ a polynomial algebra over $A$ in finitely many variables [Stacks, Tag 08PX]. By Corollary 3.5 of the lesson on cotangent complexes of simplicial rings, $L_{B/A}\cong\Omega^1_{P/A}\otimes_PB$, a simplicial $B$-module whose terms are finite free. As $B$ is noetherian, its homology modules are finite, so $H_i(L_{X/Y})$ is coherent. $\square$

4. Tor-independent base change

Proposition 4.1. Let $A\to B$ and $A\to A'$ be maps of rings of $T$ with $\operatorname{Tor}_i^A(B,A')=0$ for $i>0$, and put $B'=B\otimes_AA'$. Then the canonical map $L_{B/A}\otimes_BB'\to L_{B'/A'}$ is a quasi-isomorphism.

Proof. Let $\varepsilon\colon P=P_A(B)\to B$ be the standard resolution and $Q=P\otimes_AA'$, a simplicial $A'$-algebra with $Q_n=A'[P_{n-1}]$ degreewise free. The map $\varepsilon\otimes A'\colon Q\to B'$ is a quasi-isomorphism: $P$ is a degreewise free, hence flat, simplicial $A$-module quasi-isomorphic to $B$, so $P\otimes_AA'$ represents $B\otimes^\ell_AA'$, whose homotopy sheaves are the sheaves $\operatorname{Tor}_i^A(B,A')$ (Section 3 of the lesson on derived tensor products, with Proposition 3.4 there). These vanish for $i>0$, and $\pi_0(P\otimes_AA')=B\otimes_AA'=B'$. By Corollary 3.5 of the lesson on cotangent complexes of simplicial rings, $L_{B'/A'}\cong\Omega^1_{Q/A'}\otimes_QB'$. As in the proof of Proposition 2.1, $\Omega^1_{Q/A'}=\Omega^1_{P/A}\otimes_PQ$, so $\Omega^1_{Q/A'}\otimes_QB'$ is $\Omega^1_{P/A}\otimes_PB'=L_{B/A}\otimes_BB'$, and the isomorphism is the canonical map. $\square$

Let $\begin{array}{ccc}X'&\xrightarrow{\ u\ }&X\\ {\scriptstyle f'}\downarrow&&\downarrow{\scriptstyle f}\\ S'&\xrightarrow{\ v\ }&S\end{array}$ be a commutative square of ringed topoi, and $h=fu=vf'$. On $X'$ there are the rings $h^{-1}\mathcal O_S\to u^{-1}\mathcal O_X$, $h^{-1}\mathcal O_S\to f'^{-1}\mathcal O_{S'}$, and the canonical map $\kappa\colon u^{-1}\mathcal O_X\otimes_{h^{-1}\mathcal O_S}f'^{-1}\mathcal O_{S'}\to\mathcal O_{X'}.$

Theorem 4.2. Suppose that $\kappa$ is a localization morphism and that $\operatorname{Tor}_i^{h^{-1}\mathcal O_S}(u^{-1}\mathcal O_X,f'^{-1}\mathcal O_{S'})=0$ for $i>0$. Then the canonical maps $\begin{aligned}&u^*L_{X/S}\to L_{X'/S'},\\&f'^*L_{S'/S}\to L_{X'/X},\\&u^*L_{X/S}\oplus f'^*L_{S'/S}\to L_{X'/S}\end{aligned}$ are isomorphisms in $D(\mathcal O_{X'})$.

Proof. Write $A=h^{-1}\mathcal O_S$, $B=u^{-1}\mathcal O_X$, $A'=f'^{-1}\mathcal O_{S'}$, $B_1=B\otimes_AA'$ and $B'=\mathcal O_{X'}$. By [Stacks, Tag 08SV], $u^*L_{X/S}=L_{B/A}\otimes_BB'$ and $f'^*L_{S'/S}=L_{A'/A}\otimes_{A'}B'$. By Proposition 4.1 and its symmetric form, $L_{B/A}\otimes_BB_1\cong L_{B_1/A'}$ and $L_{A'/A}\otimes_{A'}B_1\cong L_{B_1/B}$. The map $\kappa\colon B_1\to B'$ is a localization morphism, so $L_{B'/B_1}\cong0$, and the triangles of $A'\to B_1\to B'$ and $B\to B_1\to B'$ give $\begin{aligned}L_{B/A}\otimes_BB'&\cong L_{B'/A'},\\ L_{A'/A}\otimes_{A'}B'&\cong L_{B'/B}.\end{aligned}$ These are the first two maps. For the third, the triangle of $A\to A'\to B'$ reads $L_{A'/A}\otimes_{A'}B'\to L_{B'/A}\to L_{B'/A'}\to$. The composite of the canonical map $L_{B/A}\otimes_BB'\to L_{B'/A}$, coming from $A\to B\to B'$, with $L_{B'/A}\to L_{B'/A'}$ is the first isomorphism. So the triangle splits, and the sum of the two canonical maps into $L_{B'/A}$ is an isomorphism [Stacks, Tag 05QT]. $\square$

Corollary 4.3. Let $S$ be a ringed topos, $X\to S$ and $Y\to S$ morphisms of ringed topoi, and $Z$ a ringed topos with morphisms $p\colon Z\to X$, $q\colon Z\to Y$ forming a commutative square over $S$, with $h\colon Z\to S$. If $p^{-1}\mathcal O_X\otimes_{h^{-1}\mathcal O_S}q^{-1}\mathcal O_Y\to\mathcal O_Z$ is a localization morphism and $\operatorname{Tor}_i^{h^{-1}\mathcal O_S}(p^{-1}\mathcal O_X,q^{-1}\mathcal O_Y)=0$ for $i>0$, then the canonical maps $p^*L_{X/S}\to L_{Z/Y}$ and $p^*L_{X/S}\oplus q^*L_{Y/S}\to L_{Z/S}$ are isomorphisms.

Proof. Theorem 4.2 with $X'=Z$, $S'=Y$, $u=p$ and $f'=q$. $\square$

Proposition 4.4. Let $X\to S$ and $Y\to S$ be morphisms of schemes and $Z=X\times_SY$, with projections $p$, $q$ and $h\colon Z\to S$. Then $p^{-1}\mathcal O_X\otimes_{h^{-1}\mathcal O_S}q^{-1}\mathcal O_Y\to\mathcal O_Z$ is a localization morphism.

Proof. The question is local on $Z$, by Lemma 1.4 (4) applied to the localizations at the members of a covering, whose inverse images form a faithful family. So let $W=\operatorname{Spec}(B\otimes_RC)\subseteq Z$ for affine opens $\operatorname{Spec}R\subseteq S$, $\operatorname{Spec}B\subseteq X$ and $\operatorname{Spec}C\subseteq Y$ over it. On $W$ consider $\begin{aligned}B_W\otimes_{R_W}C_W&\xrightarrow{\ w\ }p^{-1}\mathcal O_X\otimes_{h^{-1}\mathcal O_S}q^{-1}\mathcal O_Y\\&\xrightarrow{\ \kappa\ }\mathcal O_W.\end{aligned}$ The map $w$ is the tensor product of $p^{-1}$ of $B_X\to\mathcal O_X$ with $q^{-1}$ of $C_Y\to\mathcal O_Y$, over $R_W\to h^{-1}\mathcal O_S$, on $W$; the maps $B_X\to\mathcal O_X$, $C_Y\to\mathcal O_Y$ and $R_S\to\mathcal O_S$ are localization morphisms on the affine opens by Proposition 3.1. By Lemma 1.4 (4), (2) and (3), $B_W\otimes_{R_W}C_W\to p^{-1}\mathcal O_X\otimes_{R_W}q^{-1}\mathcal O_Y$ is a localization morphism. Its target is $p^{-1}\mathcal O_X\otimes_{h^{-1}\mathcal O_S}q^{-1}\mathcal O_Y$: the map $R_W\to h^{-1}\mathcal O_S$ is an epimorphism of rings, so $h^{-1}\mathcal O_S\otimes_{R_W}h^{-1}\mathcal O_S=h^{-1}\mathcal O_S$, and tensor products of $h^{-1}\mathcal O_S$-modules over $R_W$ and over $h^{-1}\mathcal O_S$ agree. So $w$ is a localization morphism, in particular an epimorphism. The composite $\kappa w$ is $(B\otimes_RC)_W\to\mathcal O_W$, a localization morphism by Proposition 3.1. By the second part of Lemma 1.4 (3), $\kappa$ is a localization morphism. $\square$

So for schemes, Corollary 4.3 holds whenever $\operatorname{Tor}_i^{h^{-1}\mathcal O_S}(p^{-1}\mathcal O_X,q^{-1}\mathcal O_Y)=0$ for $i>0$.

Remark 4.5. Stalks are exact and commute with tensor products, so the stalk of $\operatorname{Tor}_i^{h^{-1}\mathcal O_S}(p^{-1}\mathcal O_X,q^{-1}\mathcal O_Y)$ at $z\in Z$ is $\operatorname{Tor}_i^{\mathcal O_{S,s}}(\mathcal O_{X,x},\mathcal O_{Y,y})$, with $x=p(z)$, $y=q(z)$, $s=h(z)$. Every pair of points $x\in X$, $y\in Y$ over the same point of $S$ is the image of a point of $Z$ [Stacks, Tag 01JT]. So the hypothesis means that $\operatorname{Tor}_i^{\mathcal O_{S,s}}(\mathcal O_{X,x},\mathcal O_{Y,y})=0$ for $i>0$ for all such pairs: $X$ and $Y$ are Tor-independent over $S$.

5. Exercises

Exercise 5.1. Let $A$ be a ring and $f\in A$. Show that the map of constant sheaves $A_X\to(A_f)_X$ on any topological space $X$ is a localization morphism, and compute $L_{(A_f)_X/A_X}$.

Exercise 5.2. Let $k$ be a field, $S=\operatorname{Spec}k[t]$ and $X=S'=\operatorname{Spec}k$, mapped to $S$ by $t\mapsto0$. Show that the first conclusion of Theorem 4.2 fails for the fibre product $X'=X\times_SS'$, and identify the hypothesis that fails.

6. Complete solutions

Solution 5.1. The constant presheaves $U\mapsto A$ and $U\mapsto A_f$ have the sheafifications $A_X$ and $(A_f)_X$, and the map between them is, on every $U$, the localization $A\to A_f$ with respect to $\{f\}$. Lemma 1.4 (1) applies. Alternatively, apply Lemma 1.4 (4) to the morphism from $X$ to the punctual topos. A localization morphism has vanishing cotangent complex (Proposition 2.1), so $L_{(A_f)_X/A_X}\cong0$.

Solution 5.2. Here $X'=\operatorname{Spec}(k\otimes_{k[t]}k)=\operatorname{Spec}k$, and $\kappa$ is the identity of $k$, a localization morphism. But $\operatorname{Tor}_1^{k[t]}(k,k)=k\neq0$, from the resolution $0\to k[t]\xrightarrow{t}k[t]\to k\to0$; the Tor-independence fails. And the conclusion fails: $L_{X'/S'}=L_{k/k}=0$, while $u^*L_{X/S}=L_{k/k[t]}\otimes k$. Since $t$ is a nonzerodivisor, $L_{k/k[t]}$ is $(t)/(t^2)$ placed in homological degree $1$ [Stacks, Tag 08SL], which is nonzero.

Sources

Localization morphisms in a topos, the vanishing of the cotangent complex of a flat epimorphism, the comparison on affine schemes, the quasi-coherence and coherence of the homology of the cotangent complex of a morphism of schemes, and the base change statements under localization and Tor-independence hypotheses are due to Illusie, in his work on the cotangent complex; he states the base change for the relative schemes of Hakim, whose structure rings are localizations of tensor products. The proof of the localization property of fibre products follows the argument Illusie attributes to Hakim. The resolution by polynomial algebras of finite type and the cotangent complex of a morphism of ringed topoi are taken from the AI-integrated Stacks project.