The arrow topos and relative Ext groups

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

A morphism of ringed topoi \(f\colon X\to Y\) is itself a ringed topos. Its arrow topos \(XY\) has as objects the triples of a sheaf \(E\) on \(X\), a sheaf \(F\) on \(Y\) and a map \(f^{-1}F\to E\); modules over it are pairs of modules with such a map. Derived Hom on \(XY\) combines derived Hom on \(X\) and on \(Y\). The relative Ext groups \(\operatorname{Ext}^i(Y/X;M,F)\) of complexes \(M\), \(F\) on \(Y\) are the Ext groups on \(XY\) from \(M\), placed over \(Y\), into the pair formed by \(Lf^*F\) and \(F\). They fit into a long exact sequence with \(\operatorname{Ext}^i(M,F)\) and \(\operatorname{Ext}^i(Lf^*M,Lf^*F)\), and for an open subspace they are the cohomology groups with supports in the closed complement. A commutative square of ringed topoi gives a morphism of arrow topoi whose cotangent complex restricts to the cotangent complexes of the two sides. The lesson proves:

Prerequisites. Fibred topoi and their total topoi (Examples 1.2 (a), Definition 2.1, Propositions 2.2 and 3.1, Theorem 4.2); Morphisms of fibred topoi and modules on total topoi (Definition 1.1, Propositions 2.1 and 4.2); Cohomology of total topoi (Examples 3.10 (a)); Filtered internal Hom and the formulas of Cartan (Proposition 4.1); Non-additive functors of complexes in a ringed topos (Lemma 4.2). From the AI-integrated Stacks project: localization of a topos at an object [Stacks, Tag 04GZ]; extension by zero [Tag 03DJ]; internal Hom and derived internal Hom restrict to localizations [Tag 0E8H, Tag 08JB]; right adjoints of exact functors preserve injectives [Tag 015Z]; resolutions by a cofinal class of injectives [Tag 05T6]; bounded below complexes of injectives are K-injective [Tag 070J]; K-flat resolutions [Tag 06YS]; derived pullback [Tag 06YY, Tag 0D6D], the canonical map of a derived functor [Tag 05SX] and the adjunction of derived pullback with \(Rf_*\) [Tag 07A6]; global derived Hom [Tag 0B6E]; the triangle of a short exact sequence [Tag 0152] and the long exact Hom sequences [Tag 0149]; the cotangent complex of a morphism of ringed topoi [Tag 08SU] and its transitivity triangle [Tag 08V1].

Conventions

Complexes are indexed cohomologically. For a ringed topos \((X,\mathcal O_X)\), \(D(X)\) is the derived category of \(\mathcal O_X\)-modules, \(D^+(X)\) and \(D^-(X)\) its full subcategories of complexes with cohomology bounded below, respectively above; \(\mathcal Hom\) and \(R\mathcal Hom\) are the internal Hom and its derived functor, \(\mathcal Ext^i=H^iR\mathcal Hom\), \(R\operatorname{Hom}=R\Gamma\circ R\mathcal Hom\) [Tag 0B6E] and \(\operatorname{Ext}^i=H^iR\operatorname{Hom}\), so that \(\operatorname{Ext}^i(K,L)=\operatorname{Hom}_{D(X)}(K,L[i])\). For a morphism of ringed topoi \(f\colon X\to Y\), \(f^{-1}\) is the inverse image of sheaves, \(f^*=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\) the inverse image of modules, \(Lf^*\) and \(Rf_*\) the derived functors. Every complex \(L\) with \(H^n(L)=0\) for \(n>b\) receives a quasi-isomorphism from a complex of flat modules vanishing in degrees \(>b\) (Lemma 4.2 (1) of the lesson on non-additive functors), and \(Lf^*\) is computed by such complexes [Tag 06YY]. A bounded below complex of injective modules is K-injective [Tag 070J], so it computes \(R\mathcal Hom\) into it from any complex.

1. The arrow topos

Throughout, \(f\colon X\to Y\) is a morphism of topoi, and from 1.3 on a morphism of ringed topoi.

Definition 1.1. The arrow topos \(XY=\operatorname{Top}(f)\) is the total topos of the fibred topos over \(\{0\to1\}\) defined by \(f\) (Examples 1.2 (a) and Definition 2.1 of the lesson on fibred topoi). An object is a triple of \(E\in X\), \(F\in Y\) and a map \(\varphi\colon f^{-1}F\to E\), written \((E\xleftarrow{\varphi}F)\) or \((E\leftarrow F)\). A morphism \((E\leftarrow F)\to(E'\leftarrow F')\) is a pair \(a\colon E\to E'\), \(b\colon F\to F'\) with \(a\circ\varphi=\varphi'\circ f^{-1}(b)\). By Theorem 4.2 of the lesson on fibred topoi, \(XY\) is a topos.

Proposition 1.2.

  1. The restrictions \(i_X^{-1}(E\leftarrow F)=E\) and \(i_Y^{-1}(E\leftarrow F)=F\) are the inverse images of morphisms of topoi \(i_X\colon X\to XY\) and \(i_Y\colon Y\to XY\), with \[ \begin{gathered}i_{X*}L=(L\leftarrow f_*L),\\ i_{X!}L=(L\leftarrow\varnothing),\\ i_{Y*}M=(e_X\leftarrow M),\\ i_{Y!}M=(f^{-1}M\leftarrow M),\end{gathered} \] where \(f^{-1}f_*L\to L\) is the counit, \(e_X\) is the final object of \(X\), and the map of \(i_{Y!}M\) is the identity.
  2. \(i_X\) is the localization of \(XY\) at the object \((e_X\leftarrow\varnothing)\).
  3. \(i_{Y!}\) is the inverse image of a morphism of topoi \(\pi\colon XY\to Y\), with \(\pi_*=i_Y^{-1}\). There are canonical isomorphisms \(\pi\circ i_Y\cong\mathrm{id}_Y\) and \(\pi\circ i_X\cong f\).
  4. The objects \((U\leftarrow\varnothing)\), \(U\in X\), and \((f^{-1}V\leftarrow V)\), \(V\in Y\), generate \(XY\).

Proof. (1) Proposition 3.1 of the lesson on fibred topoi, for \(I=\{0\to1\}\): there are no arrows \(1\to0\), so the coproduct defining \(i_{X!}L\) at level \(1\) is empty and the product defining \(i_{Y*}M\) at level \(0\) is empty, while the only arrow \(0\to1\) gives \(f^{-1}M\) and \(f_*L\). As \(i_X^{-1}\) and \(i_Y^{-1}\) have left adjoints, they preserve limits; with their right adjoints they are inverse images of morphisms of topoi.

(2) If \((E\leftarrow F)\) maps to \((e_X\leftarrow\varnothing)\), then \(F\) maps to the initial object \(\varnothing\) of \(Y\), so \(F\) is initial: over every object \(V\), \(\varnothing(V)\) is empty or, when \(V\) is covered by the empty family, a point, and in the second case \(F(V)\) is a point as well, \(F\) being a sheaf. So the objects over \((e_X\leftarrow\varnothing)\) are the \((E\leftarrow\varnothing)\), and \(XY/(e_X\leftarrow\varnothing)\) is identified with \(X\) [Tag 04GZ]. The inverse image of the localization morphism is \(W\mapsto W\times(e_X\leftarrow\varnothing)=(i_X^{-1}W\leftarrow\varnothing)\), products being computed levelwise (Proposition 2.2 of the lesson on fibred topoi), and under the identification this is \(i_X^{-1}\).

(3) Limits in \(XY\) are computed levelwise, and \(f^{-1}\) preserves finite limits, so \(i_{Y!}\) preserves finite limits; it preserves colimits as a left adjoint. So \((i_{Y!},i_Y^{-1})\) is a morphism of topoi. Then \(i_Y^{-1}\pi^{-1}M=M\) and \(i_X^{-1}\pi^{-1}M=f^{-1}M\).

(4) By adjunction, \(\operatorname{Hom}((U\leftarrow\varnothing),W)=\operatorname{Hom}(U,i_X^{-1}W)\) and \(\operatorname{Hom}((f^{-1}V\leftarrow V),W)=\) \(\operatorname{Hom}(V,i_Y^{-1}W)\), naturally in \(W\). So two morphisms \(W\to W'\) that agree after composition with every morphism from these objects have the same components over \(X\) and over \(Y\), and a morphism \((a,b)\) of \(XY\) is determined by \(a\) and \(b\). \(\square\)

1.3. Rings and modules. Let \(f\colon(X,\mathcal O_X)\to(Y,\mathcal O_Y)\) be a morphism of ringed topoi. Then \(\mathcal O=(\mathcal O_X\leftarrow\mathcal O_Y)\), with the structure map \(f^{-1}\mathcal O_Y\to\mathcal O_X\), is a ring of \(XY\). By Proposition 4.2 (a) of the lesson on morphisms of fibred topoi, an \(\mathcal O\)-module is a pair \((E\leftarrow F)\) of an \(\mathcal O_X\)-module \(E\) and an \(\mathcal O_Y\)-module \(F\) with an \(\mathcal O_X\)-linear map \(\varphi\colon f^*F\to E\). Since \(i_X^{-1}\mathcal O=\mathcal O_X\) and \(i_Y^{-1}\mathcal O=\mathcal O_Y\), \(i_X\) and \(i_Y\) are morphisms of ringed topoi with exact inverse images \(i_X^*=i_X^{-1}\) and \(i_Y^*=i_Y^{-1}\) on modules. The map \(\pi^{-1}\mathcal O_Y=(f^{-1}\mathcal O_Y\leftarrow\mathcal O_Y)\to\mathcal O\) makes \(\pi\) a morphism of ringed topoi, and tensor products of \(\mathcal O\)-modules are computed levelwise, \[ \begin{aligned}&(E\leftarrow F)\otimes_{\mathcal O}(E'\leftarrow F')\\&\quad=(E\otimes E'\leftarrow F\otimes F'),\end{aligned} \] since inverse images commute with tensor products. Hence \(\pi^*M=(f^*M\xleftarrow{\mathrm{id}}M)\), and \(\pi_*=i_Y^*\) on modules, an exact functor. On modules, \(i_{X!}L=(L\leftarrow0)\) is the extension by zero [Tag 03DJ], \(i_{X*}L=(L\leftarrow f_*L)\) and \(i_{Y*}M=(0\leftarrow M)\).

Lemma 1.4. Let \(L\) be an \(\mathcal O_X\)-module, \(M\) an \(\mathcal O_Y\)-module and \((E\xleftarrow{\varphi}F)\) an \(\mathcal O\)-module.

  1. \(\mathcal Hom((L\leftarrow0),(E\leftarrow F))=\) \(i_{X*}\mathcal Hom(L,E)\), that is \((\mathcal Hom(L,E)\leftarrow f_*\mathcal Hom(L,E))\).
  2. \(\mathcal Hom(\pi^*M,(E\leftarrow F))=\) \((\mathcal Hom(f^*M,E)\leftarrow\) \(\mathcal Hom(M,F))\), with the map \(f^*\mathcal Hom(M,F)\to\) \(\mathcal Hom(f^*M,f^*F)\to\) \(\mathcal Hom(f^*M,E)\) given by \(\varphi\).

Proof. For \(\mathcal O\)-modules \(A\), \(B\), \(i_X^*\mathcal Hom(A,B)=\mathcal Hom(i_X^*A,i_X^*B)\), by Proposition 1.2 (2) and [Tag 0E8H]; this gives the levels over \(X\). For \(V\in Y\), the objects of \(XY\) over \(\pi^{-1}V=(f^{-1}V\leftarrow V)\) are the triples over \(X/f^{-1}V\) and \(Y/V\), so the localization \(XY/\pi^{-1}V\) is the arrow topos of the restriction \(f_V\colon X/f^{-1}V\to Y/V\). The sections of \(i_Y^*\mathcal Hom(A,B)=\pi_*\mathcal Hom(A,B)\) over \(V\) are the morphisms \(A\to B\) over this localization. For \(A=(L\leftarrow0)\) these are the morphisms \(L|_{f^{-1}V}\to E|_{f^{-1}V}\), by the adjunction of \(i_{X!}\) and \(i_X^*\) for \(f_V\), that is the sections of \(f_*\mathcal Hom(L,E)\) over \(V\). For \(A=\pi^*M\) they are the morphisms \(M|_V\to F|_V\), by the adjunction of \(\pi^*\) and \(\pi_*\) for \(f_V\), that is the sections of \(\mathcal Hom(M,F)\) over \(V\). The map from the level over \(Y\) to the level over \(X\) sends such a morphism to its component over \(X\): in (1) this is the counit \(f^{-1}f_*\mathcal Hom(L,E)\to\mathcal Hom(L,E)\); in (2) a morphism is determined by its component \(\beta\colon M|_V\to F|_V\), its component over \(X\) being \(\varphi\circ f^*\beta\), as stated. \(\square\)

2. Derived Hom on the arrow topos

Call an \(\mathcal O\)-module special injective if it is \(i_{X*}I\oplus i_{Y*}J=(I\leftarrow f_*I\oplus J)\) with \(I\) an injective \(\mathcal O_X\)-module and \(J\) an injective \(\mathcal O_Y\)-module.

Lemma 2.1. Special injective modules are injective, every \(\mathcal O\)-module is a submodule of one, and every complex in \(D^+(XY)\) is quasi-isomorphic to a bounded below complex of special injective modules. The restriction of a special injective module to \(X\) is injective.

Proof. \(i_{X*}\) and \(i_{Y*}\) are right adjoint to the exact functors \(i_X^*\), \(i_Y^*\), so they preserve injectives [Tag 015Z]. The units give a map \((L\leftarrow M)\to i_{X*}L\oplus i_{Y*}M\), which over \(X\) is \(L\to L\oplus0\) and over \(Y\) is \(M\to f_*L\oplus M\), the second component the identity; it is injective on both levels, hence injective. Embed \(L\subseteq I\) and \(M\subseteq J\) in injectives; \(i_{X*}\) and \(i_{Y*}\) preserve monomorphisms. The resolutions exist by [Tag 05T6]. Finally \(i_X^*(i_{X*}I\oplus i_{Y*}J)=I\). \(\square\)

Proposition 2.2. For \(C=(E\leftarrow F)\in D^+(XY)\), restriction to \(Y\) is an isomorphism \(R\Gamma(XY,C)\to R\Gamma(Y,F)\).

Proof. For every \(\mathcal O\)-module, \(\Gamma(XY,(E\leftarrow F))=\Gamma(Y,F)\), a global section being determined by its component over \(Y\) (Examples 3.10 (a) of the lesson on cohomology of total topoi). Represent \(C\) by a bounded below complex \(K\) of special injective modules (Lemma 2.1). Then \(R\Gamma(XY,C)=\Gamma(XY,K)=\Gamma(Y,i_Y^*K)\). The terms of \(i_Y^*K\) are \(f_*I\oplus J\); here \(Rf_*I=f_*I\), so \(H^q(Y,f_*I)=H^q(X,I)=0\) for \(q>0\), and \(J\) is injective. So \(i_Y^*K\) is a bounded below complex of \(\Gamma(Y,-)\)-acyclic modules representing \(F\), and \(\Gamma(Y,i_Y^*K)\) represents \(R\Gamma(Y,F)\). \(\square\)

Proposition 2.3. Let \(L\in D^-(X)\) and \(C=(E\leftarrow F)\in D^+(XY)\). There are canonical isomorphisms \[ \begin{aligned}&i_X^*R\mathcal Hom((L\leftarrow0),C)\cong R\mathcal Hom(L,E),\\&i_Y^*R\mathcal Hom((L\leftarrow0),C)\\&\quad\cong Rf_*R\mathcal Hom(L,E),\\&R\operatorname{Hom}((L\leftarrow0),C)\cong R\operatorname{Hom}(L,E).\end{aligned} \]

Proof. Represent \(L\) by a bounded above complex \(P\) of flat \(\mathcal O_X\)-modules and \(C\) by a bounded below complex \(K\) of special injective modules. Since \(i_{X!}\) is exact, \((P\leftarrow0)\) represents \((L\leftarrow0)\), and, by Lemma 1.4 (1), \[ \begin{aligned}&R\mathcal Hom((L\leftarrow0),C)\\&\quad=\mathcal Hom^\bullet((P\leftarrow0),K)\\&\quad=i_{X*}\mathcal Hom^\bullet(P,i_X^*K).\end{aligned} \] Here \(i_X^*K\) is a bounded below complex of injectives representing \(E\), so \(\mathcal Hom^\bullet(P,i_X^*K)\) represents \(R\mathcal Hom(L,E)\). Its terms are finite products of the modules \(\mathcal Hom(P^p,i_X^*K^q)\), which are injective, \(P^p\) being flat; and it is bounded below. So \(f_*\) of it represents \(Rf_*R\mathcal Hom(L,E)\). This gives the first two isomorphisms. The third follows by applying \(R\Gamma(XY,-)\), which is \(R\Gamma(Y,i_Y^*-)\) by Proposition 2.2, and \(R\Gamma(Y,Rf_*-)=R\Gamma(X,-)\). \(\square\)

Proposition 2.4. For \(L\in D(Y)\) and \(M\in D(X)\) there are canonical isomorphisms \[ \begin{aligned}&R\mathcal Hom(L,Rf_*M)\\&\quad\cong Rf_*R\mathcal Hom(Lf^*L,M),\\&R\operatorname{Hom}(L,Rf_*M)\\&\quad\cong R\operatorname{Hom}(Lf^*L,M).\end{aligned} \] If \(L\) is a flat \(\mathcal O_Y\)-module and \(I\) an injective \(\mathcal O_X\)-module, then \(\mathcal Ext^i(L,f_*I)=0\) and \(\operatorname{Ext}^i(L,f_*I)=0\) for \(i>0\).

Proof. The isomorphisms are Proposition 4.1 (3) of the lesson on filtered internal Hom, applied to objects with the trivial filtration and read on \(\operatorname{gr}^0\). For the vanishing, \(Rf_*I=f_*I\) and \(Lf^*L=f^*L\) [Tag 06YY]. The module \(\mathcal Hom(f^*L,I)\) is injective, as \(f^*L\) is flat and \(I\) injective, so \(R\mathcal Hom(f^*L,I)=\mathcal Hom(f^*L,I)\), and both \(Rf_*\mathcal Hom(f^*L,I)\) and \(R\Gamma(X,\mathcal Hom(f^*L,I))\) are concentrated in degree \(0\). By the two isomorphisms, so are \(R\mathcal Hom(L,f_*I)\) and \(R\operatorname{Hom}(L,f_*I)\). \(\square\)

Lemma 2.5. Let \(P\) be a bounded above complex and \(N\) a bounded below complex of \(\mathcal O_Y\)-modules with \(\mathcal Ext^k(P^p,N^q)=0\) for all \(p\), \(q\) and all \(k\geq1\). Then \(\mathcal Hom^\bullet(P,N)\) represents \(R\mathcal Hom(P,N)\).

Proof. Let \(N\to I\) be a quasi-isomorphism to a bounded below complex of injectives, and \(Z\) its cone, a bounded below acyclic complex whose terms satisfy the same vanishing. Write \(B^q\subseteq Z^q\) for the cycles, so that \(0\to B^q\to Z^q\to B^{q+1}\to0\) is exact and \(B^q=0\) for \(q\) small. By induction on \(q\), \(\mathcal Ext^k(P^p,B^q)=0\) for \(k\geq1\): the long exact sequence gives \(\mathcal Ext^k(P^p,B^{q+1})\cong\mathcal Ext^{k+1}(P^p,B^q)=0\). So \(\mathcal Hom(P^p,-)\) is exact on these sequences and \(\mathcal Hom(P^p,Z)\) is acyclic for every \(p\). The complex \(\mathcal Hom^\bullet(P,Z)\) is the total complex of a double complex with acyclic rows and finitely many terms in each total degree, so it is acyclic. Hence \(\mathcal Hom^\bullet(P,N)\to\mathcal Hom^\bullet(P,I)\) is a quasi-isomorphism, and the second complex represents \(R\mathcal Hom(P,N)\). \(\square\)

Definition 2.6. For a complex \(C=(L\xleftarrow{\varphi}M)\) of \(\mathcal O\)-modules, the map \(u_C\colon Lf^*M\to L\) defined by \(C\) is the composite of the canonical map \(Lf^*M\to f^*M\) of the derived functor [Tag 05SX] with \(\varphi\colon f^*M\to L\). For a quasi-isomorphism \(s\colon P\to M\) from a K-flat complex with flat terms [Tag 06YS], or from a bounded above complex of flat modules when \(M\) has bounded above cohomology, \(Lf^*M=f^*P\) and \(u_C=\varphi\circ f^*(s)\) [Tag 06YY]. The map \(u_C\) is natural in \(C\), so it is defined for objects of \(D(XY)\). An object \(C=(L\leftarrow M)\) of \(D^-(XY)\) is quasi-cartesian if \(u_C\) is an isomorphism of \(D(X)\).

Proposition 2.7. Let \(C=(L\leftarrow M)\in D^-(XY)\) be quasi-cartesian and \(C'=(E\leftarrow F)\in D^+(XY)\). The restriction maps \[ \begin{aligned}i_X^*R\mathcal Hom(C,C')&\to R\mathcal Hom(L,E),\\ i_Y^*R\mathcal Hom(C,C')&\to R\mathcal Hom(M,F)\end{aligned} \] are isomorphisms.

Proof. Represent \(C\) by a complex \((L\xleftarrow{\varphi}M)\), and let \(s\colon P\to M\) be a quasi-isomorphism from a bounded above complex of flat modules, so that \(u_C=\varphi\circ f^*(s)\). The pair \((\varphi\circ f^*(s),s)\) is a morphism of complexes \(\pi^*P=(f^*P\leftarrow P)\to C\); it is a quasi-isomorphism over \(Y\), and over \(X\) because \(C\) is quasi-cartesian. Quasi-isomorphisms of \(\mathcal O\)-modules are detected on the two levels, \(i_X^*\) and \(i_Y^*\) being exact and jointly faithful. So we may take \(C=\pi^*P\), and \(C'\) a bounded below complex \(K\) of special injective modules. By Lemma 1.4 (2), \[ \begin{aligned}&\mathcal Hom^\bullet(\pi^*P,K)\\&\quad=\bigl(\mathcal Hom^\bullet(f^*P,i_X^*K)\\&\qquad\leftarrow\mathcal Hom^\bullet(P,i_Y^*K)\bigr),\end{aligned} \] and this complex represents \(R\mathcal Hom(C,C')\), \(K\) being K-injective. Over \(X\), \(i_X^*K\) is a bounded below complex of injectives, so \(\mathcal Hom^\bullet(f^*P,i_X^*K)\) represents \(R\mathcal Hom(L,E)\). Over \(Y\), the terms \(f_*I\oplus J\) of \(i_Y^*K\) satisfy \(\mathcal Ext^k(P^p,f_*I\oplus J)=0\) for \(k\geq1\) (Proposition 2.4), so \(\mathcal Hom^\bullet(P,i_Y^*K)\) represents \(R\mathcal Hom(M,F)\) by Lemma 2.5. Both identifications are the restriction maps. \(\square\)

Proposition 2.8. Let \(M\in D^-(Y)\) and \(C=(E\leftarrow F)\in D^+(XY)\), and let \(\epsilon\colon F\to Rf_*E\) be adjoint to the map \(Lf^*F\to E\) defined by \(C\) [Tag 07A6].

  1. \(i_X^*R\mathcal Hom((0\leftarrow M),C)=0\).
  2. There is a distinguished triangle \[ \begin{aligned}&i_Y^*R\mathcal Hom((0\leftarrow M),C)\\&\quad\to R\mathcal Hom(M,F)\\&\quad\to R\mathcal Hom(M,Rf_*E)\to\end{aligned} \] whose first map is restriction to \(Y\) and whose second map is \(R\mathcal Hom(M,\epsilon)\); and \(R\mathcal Hom(M,Rf_*E)\cong\) \(Rf_*R\mathcal Hom(Lf^*M,E)\) by Proposition 2.4.
  3. There is a distinguished triangle \[ \begin{aligned}&R\operatorname{Hom}((0\leftarrow M),C)\\&\quad\to R\operatorname{Hom}(M,F)\\&\quad\to R\operatorname{Hom}(Lf^*M,E)\to\end{aligned} \] whose second map sends \(b\colon M\to F[i]\) to the composite of \(Lf^*b\) with the map \(Lf^*F\to E\) defined by \(C\).

Proof. Let \(P\to M\) be a quasi-isomorphism from a bounded above complex of flat modules. The sequence of complexes of \(\mathcal O\)-modules \[ \begin{aligned}&0\to(f^*P\leftarrow0)\to\pi^*P\\&\quad\to(0\leftarrow P)\to0\end{aligned} \] is exact on both levels, hence exact, and gives a distinguished triangle [Tag 0152]; \(R\mathcal Hom(-,C)\) carries it to a distinguished triangle \[ \begin{aligned}&R\mathcal Hom((0\leftarrow P),C)\to R\mathcal Hom(\pi^*P,C)\\&\quad\to R\mathcal Hom((f^*P\leftarrow0),C)\to.\end{aligned} \] Take \(C\) to be a bounded below complex \(K\) of special injective modules, so that the terms are \(\mathcal Hom^\bullet\) complexes. Over \(X\), by Propositions 2.7 and 2.3 the second map becomes the identity of \(R\mathcal Hom(f^*P,E)\), the inclusion being the identity over \(X\); so the first term vanishes over \(X\). Over \(Y\), Proposition 2.7 and Lemma 1.4 (1) give the triangle with second map \(\mathcal Hom^\bullet(P,i_Y^*K)\to\) \(f_*\mathcal Hom^\bullet(f^*P,i_X^*K)=\) \(\mathcal Hom^\bullet(P,f_*i_X^*K)\), which sends \(b\) to the adjoint of the composite of \(f^*b\) with the structure map of \(K\), that is to the composite of \(b\) with \(i_Y^*K\to f_*i_X^*K\). This map represents \(\epsilon\), since \(f_*i_X^*K=Rf_*E\). This proves (1) and (2). Applying \(R\Gamma(XY,-)=R\Gamma(Y,i_Y^*-)\) (Proposition 2.2) and the global form of Proposition 2.4 gives (3); under that isomorphism, composing with \(\epsilon\) becomes composing \(Lf^*\) with the adjoint of \(\epsilon\). \(\square\)

3. Relative Ext groups

Definition 3.1. For \(F\in D(Y)\) put \((Lf^*F\leftarrow F)=L\pi^*F\in D(XY)\). Since \(\pi\circ i_X\cong f\), \(\pi\circ i_Y\cong\mathrm{id}\) and \(i_X^*\), \(i_Y^*\) are exact, its restrictions are \(Lf^*F\) and \(F\) [Tag 0D6D]; for a K-flat resolution \(Q\to F\) with flat terms it is \(\pi^*Q=(f^*Q\leftarrow Q)\) [Tag 06YY]. Let \(M\in D^-(Y)\) and \(F\in D^+(Y)\) with \(Lf^*F\in D^+(X)\), for instance \(f\) flat, or \(F\) of finite tor amplitude. The relative derived Hom and relative Ext groups are \[ \begin{aligned}&R\mathcal Hom(Y/X;M,F)\\&\quad=i_Y^*R\mathcal Hom((0\leftarrow M),L\pi^*F),\\&R\operatorname{Hom}(Y/X;M,F)\\&\quad=R\operatorname{Hom}((0\leftarrow M),L\pi^*F),\end{aligned} \] with cohomology \(\mathcal Ext^i(Y/X;M,F)\) and \(\operatorname{Ext}^i(Y/X;M,F)\). For abelian sheaves, that is \(\mathcal O_X=\mathbf Z\), \(\mathcal O_Y=\mathbf Z\), the condition holds for every \(F\in D^+(Y)\), \(f^{-1}\) being exact, and the relative cohomology is \(R\Gamma(Y/X;F)=R\operatorname{Hom}(Y/X;\mathbf Z,F)\), with \(H^i(Y/X;F)=H^iR\Gamma(Y/X;F)\).

Proposition 3.2. Let \(M\), \(F\) be as in Definition 3.1, and \(\eta\colon F\to Rf_*Lf^*F\) the unit. There are distinguished triangles \[ \begin{aligned}&R\mathcal Hom(Y/X;M,F)\to R\mathcal Hom(M,F)\\&\quad\to R\mathcal Hom(M,Rf_*Lf^*F)\to,\\ &R\operatorname{Hom}(Y/X;M,F)\to R\operatorname{Hom}(M,F)\\&\quad\to R\operatorname{Hom}(Lf^*M,Lf^*F)\to,\end{aligned} \] the second maps being \(R\mathcal Hom(M,\eta)\) and the functor \(Lf^*\). In particular there are long exact sequences \[ \begin{aligned}&\cdots\to\operatorname{Ext}^{i-1}(Lf^*M,Lf^*F)\\&\quad\to\operatorname{Ext}^i(Y/X;M,F)\to\operatorname{Ext}^i(M,F)\\&\quad\to\operatorname{Ext}^i(Lf^*M,Lf^*F)\to\cdots,\end{aligned} \] and the same with the sheaves \(\mathcal Ext\) and \(\mathcal Ext^i(M,Rf_*Lf^*F)\cong\) \(\mathcal H^iRf_*R\mathcal Hom(Lf^*M,Lf^*F)\) in the last place.

Proof. Proposition 2.8 for \(C=L\pi^*F\), whose map \(Lf^*F\to Lf^*F\) is the identity, so that \(\epsilon=\eta\). By the adjunction [Tag 07A6], a map \(b\colon M\to F[i]\) followed by \(\eta\) corresponds to \(Lf^*b\), since the composite \(Lf^*F\to Lf^*Rf_*Lf^*F\to Lf^*F\) of the unit and the counit is the identity. \(\square\)

Proposition 3.3. Let \(D'\) be the category whose objects are triples \((E,F,u)\) with \(E\in D(X)\), \(F\in D(Y)\) and \(u\colon Lf^*F\to E\) in \(D(X)\), and whose morphisms \((E,F,u)\to(E',F',u')\) are pairs \(a\colon E\to E'\), \(b\colon F\to F'\) with \(a\circ u=u'\circ Lf^*b\).

  1. If \(Lf^*F\neq0\) for some \(F\in D(Y)\), then \((0,\mathrm{id}_F)\colon(Lf^*F,F,\mathrm{id})\to(0,F,0)\) is an epimorphism of \(D'\) without a section. So \(D'\) admits no structure of triangulated category.
  2. The functor \(\Phi\colon D(XY)\to D'\), \(C\mapsto(i_X^*C,i_Y^*C,u_C)\) with \(u_C\) the map of Definition 2.6, is conservative.
  3. Let \(M\in D^-(Y)\) and \(F\) be as in Definition 3.1. The map \[ \begin{aligned}&\operatorname{Ext}^0(Y/X;M,F)\\&\quad=\operatorname{Hom}_{D(XY)}((0\leftarrow M),L\pi^*F)\\&\quad\to\operatorname{Hom}_{D'}(\Phi(0\leftarrow M),\Phi L\pi^*F)\end{aligned} \] induced by \(\Phi\) is surjective onto \(\operatorname{Ker}(\operatorname{Ext}^0(M,F)\to\) \(\operatorname{Ext}^0(Lf^*M,Lf^*F))\), and its kernel is the image of \(\operatorname{Ext}^{-1}(Lf^*M,Lf^*F)\).

Proof. (1) A morphism \((a,b)\) out of \((0,F,0)\) with \((a,b)\circ(0,\mathrm{id}_F)=0\) has \(b=0\), and \(a\) has source \(0\); so \((0,\mathrm{id}_F)\) is an epimorphism. A section would be a morphism \((a',\mathrm{id}_F)\colon(0,F,0)\to(Lf^*F,F,\mathrm{id})\), which requires \(a'\circ0=\mathrm{id}\circ Lf^*(\mathrm{id}_F)\), that is \(\mathrm{id}_{Lf^*F}=0\). In a triangulated category every epimorphism \(e\colon A\to B\) has a section: in a distinguished triangle \(A\xrightarrow{e}B\xrightarrow{g}D\to\), \(g\circ e=0\) forces \(g=0\), and the exact sequence \(\operatorname{Hom}(B,A)\to\operatorname{Hom}(B,B)\to\) \(\operatorname{Hom}(B,D)\) [Tag 0149] gives \(s\) with \(e\circ s=\mathrm{id}_B\).

(2) A morphism of \(D(XY)\) is an isomorphism if its two restrictions are, since cohomology sheaves are computed on the two levels.

(3) A morphism of \(D'\) from \((0,M,0)\) to \((Lf^*F,F,\mathrm{id})\) is a pair \((0,b)\) with \(\mathrm{id}\circ Lf^*b=0\); so these morphisms form the stated kernel. The map induced by \(\Phi\) sends a morphism to its restriction to \(Y\), the restriction to \(X\) being zero, its source vanishing there. By Proposition 3.2 the image of \(\operatorname{Ext}^0(Y/X;M,F)\to\operatorname{Ext}^0(M,F)\) is that kernel, and the kernel of this map is the image of \(\operatorname{Ext}^{-1}(Lf^*M,Lf^*F)\). \(\square\)

So the triples of Proposition 3.3 do not form a triangulated category, and \(\Phi\) loses information: Exercise 5.2 gives a case where \(\Phi\) is not faithful.

4. Squares of ringed topoi

Proposition 4.1. Let \(f\colon X\to Y\), \(g\colon Z\to T\), \(p\colon X\to Z\) and \(q\colon Y\to T\) be morphisms of ringed topoi with an isomorphism \(c\colon q\circ f\cong g\circ p\). There is a morphism of ringed topoi \((p,q)\colon XY\to ZT\) with \[ (p,q)^{-1}(E\xleftarrow{\psi}F)=(p^{-1}E\leftarrow q^{-1}F), \] the map being \(p^{-1}(\psi)\) composed with the isomorphism \(f^{-1}q^{-1}F\cong p^{-1}g^{-1}F\) given by \(c\), and the same formula for \((p,q)^*\) on modules. It satisfies \((p,q)\circ i_X\cong i_Z\circ p\) and \((p,q)\circ i_Y\cong i_T\circ q\).

Proof. The levels \(p\), \(q\) with \(c\) form a morphism of fibred topoi over the identity of \(\{0\to1\}\) (Definition 1.1 of the lesson on morphisms of fibred topoi), which induces a morphism of the total topoi with the stated inverse image (Proposition 2.1 there). The maps \(p^{-1}\mathcal O_Z\to\mathcal O_X\) and \(q^{-1}\mathcal O_T\to\mathcal O_Y\) are compatible with the structure maps, so they define \((p,q)^{-1}\mathcal O_{ZT}\to\mathcal O_{XY}\). The two isomorphisms hold on inverse images, by restriction to the levels. \(\square\)

Proposition 4.2. In the situation of Proposition 4.1, the transitivity triangles give isomorphisms \[ \begin{gathered}i_X^*L_{XY/ZT}\cong L_{X/Z},\\ i_Y^*L_{XY/ZT}\cong L_{Y/T}.\end{gathered} \] Similarly, for morphisms \(f\colon X\to Y\), \(p\colon X\to Z\), \(q\colon Y\to Z\) of ringed topoi with \(q\circ f\cong p\), the morphism \((p,q)\colon XY\to Z\) with \((p,q)^{-1}G=(p^{-1}G\leftarrow q^{-1}G)\) satisfies \(i_X^*L_{XY/Z}\cong L_{X/Z}\) and \(i_Y^*L_{XY/Z}\cong L_{Y/Z}\).

Proof. The cotangent complex of a morphism of ringed topoi \(h\colon X'\to Y'\) is \(L_{X'/Y'}=L_{\mathcal O_{X'}/h^{-1}\mathcal O_{Y'}}\) [Tag 08SU]. Since \(i_X^{-1}\mathcal O_{XY}=\mathcal O_X\), \(L_{X/XY}=L_{\mathcal O_X/\mathcal O_X}=0\); likewise \(L_{Z/ZT}=0\). The transitivity triangles [Tag 08V1] of \(X\xrightarrow{i_X}XY\xrightarrow{(p,q)}ZT\) and of \(X\xrightarrow{p}Z\xrightarrow{i_Z}ZT\), whose composites agree by Proposition 4.1, give isomorphisms \(Li_X^*L_{XY/ZT}\to L_{X/ZT}\to L_{X/Z}\), and \(Li_X^*=i_X^*\), \(i_X^*\) being exact. The same argument applies over \(Y\) and to \((p,q)\colon XY\to Z\), whose inverse image is a morphism of topoi as in Proposition 4.1 for the square with \(g=\mathrm{id}_Z\), composed with the projection \(ZZ\to Z\). \(\square\)

5. Exercises

Exercise 5.1. Let \(Y\) be a topological space, \(j\colon U\to Y\) the inclusion of an open subset and \(Z=Y\smallsetminus U\), with the sheaves of abelian groups. Show that \(H^i(Y/U;F)\cong H^i_Z(Y,F)\) canonically for \(F\in D^+(Y)\).

Exercise 5.2. In Exercise 5.1 let \(Y=\mathbf R\), \(U=\mathbf R\smallsetminus\{0\}\) and \(F=\mathbf Z[1]\). Show that \(\operatorname{Ext}^0(Y/U;\mathbf Z,F)\cong\mathbf Z\), while the corresponding group of morphisms of the category \(D'\) of Proposition 3.3 vanishes.

Exercise 5.3. For the identity \(f=\mathrm{id}_X\) of a ringed topos, show that \(R\operatorname{Hom}(X/X;M,F)=0\) for \(M\in D^-(X)\), \(F\in D^+(X)\).

6. Complete solutions

Solution 5.1. Here \(j^{-1}=j^*\) is exact, so \(\pi^*G=(j^{-1}G\leftarrow G)\) is exact and \(L\pi^*=\pi^*\). For a sheaf \(G\) on \(Y\), a morphism \((0\leftarrow\mathbf Z)\to\pi^*G\) is a section \(s\in\Gamma(Y,G)\) whose image in \(\Gamma(U,G)\) vanishes, by the compatibility with the maps of the two objects; so \(\operatorname{Hom}((0\leftarrow\mathbf Z),\pi^*G)=\Gamma_Z(Y,G)\), naturally in \(G\). The functor \(\pi^*\) has the exact left adjoint \(\lambda(A_U\xleftarrow{\psi}A_Y)=(A_Y\oplus j_!A_U)/j_!j^{-1}A_Y\), the quotient by the image of \((\iota,-j_!\psi)\), with \(\iota\colon j_!j^{-1}A_Y\to A_Y\) the inclusion: a morphism \(A\to\pi^*G\) is a pair of maps \(A_Y\to G\) and \(j_!A_U\to G\) (adjoint to \(A_U\to j^{-1}G\)) that agree on \(j_!j^{-1}A_Y\). \(\lambda\) is exact, being the cokernel of a natural monomorphism between exact functors. So \(\pi^*\) preserves injectives [Tag 015Z]. For \(F\) represented by a bounded below complex of injectives \(I\), \(\pi^*I\) is a bounded below complex of injectives representing \(L\pi^*F\), and \(R\operatorname{Hom}(Y/U;\mathbf Z,F)\) is represented by \(\operatorname{Hom}^\bullet((0\leftarrow\mathbf Z),\pi^*I)=\Gamma_Z(Y,I)\), which represents \(R\Gamma_Z(Y,F)\).

Solution 5.2. By Exercise 5.1, \(\operatorname{Ext}^0(Y/U;\mathbf Z,\mathbf Z[1])=H^1_{\{0\}}(\mathbf R,\mathbf Z)\). In the exact sequence \[ \begin{aligned}&H^0(\mathbf R,\mathbf Z)\to H^0(U,\mathbf Z)\\&\quad\to H^1_{\{0\}}(\mathbf R,\mathbf Z)\to H^1(\mathbf R,\mathbf Z)=0,\end{aligned} \] the first map is the diagonal \(\mathbf Z\to\mathbf Z^2\), \(U\) having two connected components, so \(H^1_{\{0\}}(\mathbf R,\mathbf Z)\cong\mathbf Z^2/\mathbf Z\cong\mathbf Z\). The morphisms of \(D'\) form the kernel of \(\operatorname{Ext}^0(\mathbf Z,\mathbf Z[1])=H^1(\mathbf R,\mathbf Z)=0\to\) \(\operatorname{Ext}^0(\mathbf Z_U,\mathbf Z_U[1])\), which is \(0\) (Proposition 3.3 (3)). So \(\Phi\) is not injective on this Hom group: its kernel is the image \(\mathbf Z\) of \(\operatorname{Ext}^{-1}(\mathbf Z_U,\mathbf Z_U[1])=H^0(U,\mathbf Z)\).

Solution 5.3. For \(f=\mathrm{id}\), \(Lf^*\) is the identity functor, so in the triangle of Proposition 3.2 the map \(R\operatorname{Hom}(M,F)\to R\operatorname{Hom}(Lf^*M,Lf^*F)\) is the identity; hence the first term vanishes.

Sources

The arrow topos, relative Ext groups and their exact sequences, and the computation of the cotangent complex of a morphism of arrow topoi are due to Illusie, in his work on the cotangent complex, where they serve to deform morphisms together with their target; there the relative Ext groups are credited to Grothendieck, who defined them through the triangle of Proposition 3.2.