# Filtered transitivity triangles and the filtered Atiyah triangle

*Written by Claude Opus 5.5 (Anthropic) on 23 October 2026. Self-checked by the writing AI. Public domain (CC0).*

For ring maps \(A\to B\to C\) of a topos, the transitivity triangle of cotangent complexes is the triangle of a short exact sequence of complexes of flat \(C\)-modules. Regarded as a complex with a two-step filtration, this sequence is an object of the filtered derived category, and it carries more information than its triangle: it can be truncated, pulled back and descended as a single object, whereas the truncations of a distinguished triangle need not form a distinguished triangle. This lesson applies this to the triangles of the lessons on equivariant cotangent complexes and on torsors. Truncations of filtered complexes are made from the quotients \(L/F^pL\). They do not commute with graded pieces, but the defect is computed, and it disappears in the projective systems of truncations. A dévissage over graded pieces shows that an exact functor of abelian categories which is fully faithful on a class of objects of the derived category stable under shifts is fully faithful on the filtered objects whose graded pieces lie in that class, with the expected essential image. For the nerve functor of a classifying category, this gives the equivariant transitivity triangle as a projective system of bounded filtered objects on the classifying category; its triangles and connecting maps give the maps and the equivariant Kodaira–Spencer class of the lesson on equivariant cotangent complexes. For descent along a torsor, it gives the filtered Atiyah triangle, a projective system of bounded filtered objects on the base whose triangles are the Atiyah triangles of the lesson on torsors.

The lesson proves:

- truncations of filtered complexes define functors on the filtered derived category; the graded pieces of a truncation are computed, and the first filtration steps of the truncations form a projective system isomorphic to the truncations of the first filtration step;
- for an exact functor \(\Phi\) of abelian categories, fully faithful on a class of objects of the derived category stable under shifts: \(\Phi\) is fully faithful on the filtered objects whose graded pieces lie in that class, its essential image consists of the filtered objects whose graded pieces lie in the essential image, and the same holds for projective systems;
- the filtered transitivity complex of ring maps of a topos and of morphisms of diagrams of schemes: a filtered complex with flat graded pieces whose triangle is the transitivity triangle;
- the equivariant filtered transitivity triangle \(t_{[}L^G_{X/Y/Z}\), a projective system of bounded filtered objects on the classifying category, and the equivariant Kodaira–Spencer class as its connecting maps;
- filtered descent along a torsor, on big fppf and on Zariski sites, for bounded filtered objects and their projective systems; the filtered Atiyah triangle \(\ell_{X/Y/S}\), whose triangles are the Atiyah triangles, and its relations with direct images, with sections of the torsor and with the equivariant filtered transitivity triangle.

**Prerequisites.** [*Filtered derived categories*](filtered-derived-categories.md) (Section 1 (truncations and cones), Lemmas 1.2, 2.1, 3.1 and 3.4, Propositions 2.2 and 3.5, Definitions 3.2 and 5.1); [*Filtered tensor products on a ringed topos*](filtered-tensor-products-on-a-ringed-topos.md) (Definitions 1.3 and 2.1, Lemma 2.2, Propositions 2.3 and 4.1); [*Filtered internal Hom and the formulas of Cartan*](filtered-internal-hom-and-the-formulas-of-cartan.md) (Proposition 4.1); [*Quasi-coherent and perfect complexes on big fppf sites*](quasi-coherent-and-perfect-complexes-on-big-fppf-sites.md) (Proposition 2.1, Construction 4.1, Lemma 4.2); [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md) (Conventions, Proposition 1.3); [*Equivariant cotangent complexes*](equivariant-cotangent-complexes.md) (its notation, (0.1) and (0.2), Propositions 1.5, 2.3 and 3.3, Lemma 3.1, Theorem 2.4); [*Torsors, co-Lie complexes and the Atiyah class*](torsors-co-lie-complexes-and-the-atiyah-class.md) (its notation, Proposition 1.3, Construction 2.1, Proposition 2.2, Theorems 2.4, 2.5 and 3.3, Corollary 2.6, Lemmas 3.1 and 3.2). From the programme lesson [*The cotangent complex*](course:AG-HP/the-cotangent-complex#2-independence-of-the-resolution): Lemma 2.1, Theorem 2.2 with its functoriality map (2.6), [Theorem 4.1](course:AG-HP/the-cotangent-complex#4-the-fundamental-triangle) and its proof, with the sequence (4.6) and the product formula (4.4), and the sheaf constructions of its [Section 10](course:AG-HP/the-cotangent-complex#10-morphisms-of-schemes). From the AI-integrated Stacks project: localization [[Stacks, Tag 04VG](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/categories.html#categories-lemma-properties-left-localization)] and its morphisms [[Tag 05Q4](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/categories.html#categories-remark-right-localization-morphisms-colimit)]; Verdier quotients [[Tag 05RJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-universal-property-quotient)]; the triangle of a short exact sequence of complexes [[Tag 0152](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-derived-canonical-delta-functor)]; the long exact sequence of a triangle [[Tag 0149](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-representable-homological)]; morphisms of triangles [[Tag 014A](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-third-isomorphism-triangle)]; truncations [[Tag 0118](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/homology.html#homology-section-truncations)] and maps into truncated objects [[Tag 06XS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-negative-exts)]; flat modules [[Tag 03EY](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-flat-ses)] and K-flat complexes [[Tag 06YQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-bounded-flat-K-flat)]; derived adjunction [[Tag 07A6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-adjoint)].

## Conventions

\(\mathcal A\) and \(\mathcal B\) are abelian categories. The notation for filtered complexes is that of [*Filtered derived categories*](filtered-derived-categories.md): \(CF\), \(KF\), \(DF\), \(F^p\), \(F_p=(-)/F^p\), \(\operatorname{gr}^p\), the twists \(L(n)\), and \(DF^{[a,b]}\), \(D^bF\) (Definition 3.2 and Proposition 3.5 there). For \(K\in D(\mathcal A)\) and \(p\in\mathbf Z\), \(\iota_pK\) is \(K\) with \(F^iK=K\) for \(i\le p\) and \(F^iK=0\) for \(i>p\). So \(\iota_0\) is the functor \(\iota\) of Proposition 3.5 (2) there, \((\iota_pK)(p)=\iota_0K\), \(\operatorname{gr}^p\iota_pK=K\), and \(\iota_p\colon D(\mathcal A)\to DF(\mathcal A)\) is exact. Truncations of complexes are \(t_{[n}=\tau_{\geq n}\) [Tag 0118].

*The triangle of a filtered object.* Let \(X\in DF^{[0,1]}(\mathcal A)\), represented by a filtered complex with filtration in \([0,1]\) (Proposition 3.5 (1) there). The short exact sequence \(0\to F^1X\to X\to\operatorname{gr}^0X\to0\) gives the distinguished triangle of [Tag 0152]
\[
\begin{aligned}&F^1X\longrightarrow X\longrightarrow\operatorname{gr}^0X\\&\quad\xrightarrow{\ \partial_X\ }F^1X[1]\end{aligned}
\tag{0.1}
\]
in \(D(\mathcal A)\). Here \(\partial_X=\partial^0_X\) of Definition 5.1 there, as \(F^0X/F^2X=X\); the triangle is natural in \(X\in DF^{[0,1]}(\mathcal A)\). We call (0.1) the *triangle of \(X\)*.

*Projective systems.* A projective system in a category \(\mathcal C\) is a family \((X_n)_{n\in\mathbf Z}\) of objects with maps \(X_n\to X_{n+1}\), its transition maps; for instance \(t_{[}K=(t_{[n}K)_n\). We regard projective systems as pro-objects, indexed by \(n\to-\infty\):
\[
\begin{aligned}&\operatorname{Hom}\bigl((X_n),(Y_m)\bigr)\\&\quad=\varprojlim_m\ \operatorname*{colim}_{n\le m}\ \operatorname{Hom}(X_n,Y_m),\end{aligned}
\]
the colimit along the transition maps of \(X\) and the limit along those of \(Y\), with the evident composition. For a fixed \(k\geq0\), a family of maps \(X_n\to Y_{n+k}\) compatible with the transition maps defines a morphism. If maps \(u_n\colon X_n\to Y_n\) and \(v_n\colon Y_n\to X_{n+1}\) satisfy \(v_nu_n=(X_n\to X_{n+1})\) and \(u_{n+1}v_n=(Y_n\to Y_{n+1})\), then \((u_n)\) and \((v_n)\) are inverse isomorphisms of pro-objects, since a transition map represents the identity. By the formula for morphisms, a fully faithful functor is fully faithful on projective systems.

*Exact functors.* An exact functor \(\Phi\colon\mathcal A\to\mathcal B\) is applied to complexes termwise. It carries filtered complexes to filtered complexes, commutes with \(F^p\), \(F_p\), \(\operatorname{gr}^p\), cones and filtered homotopies, and preserves filtered quasi-isomorphisms (Lemma 2.1 there). So it induces exact functors \(KF(\mathcal A)\to KF(\mathcal B)\) and \(DF(\mathcal A)\to DF(\mathcal B)\) [Tag 05RJ], again denoted \(\Phi\). They commute with \(F^p\), \(F_p\), \(\operatorname{gr}^p\), the underlying objects, twists, the functors \(\iota_p\) and the triangles (0.1).

## 1. Truncations of filtered objects

Recall from Section 1 of the lesson on filtered derived categories that for \(L\in CF(\mathcal A)\) the truncation \(\tau_{\geq n}L\) is filtered by \(F^p(\tau_{\geq n}L)=\operatorname{Ker}(\tau_{\geq n}L\to\tau_{\geq n}(F_pL))\), so that \(F_p(\tau_{\geq n}L)=\tau_{\geq n}(F_pL)\), and that \(L\to\tau_{\geq n}L\) is a filtered morphism. In degrees \(\geq n+1\) the filtration of \(\tau_{\geq n}L\) is that of \(L\); in degree \(n\) it is the image filtration of \(\operatorname{Coker}d^{n-1}\).

**Lemma 1.1.** Let \(n\in\mathbf Z\).

1. \(\tau_{\geq n}\) carries filtered homotopies to filtered homotopies and filtered quasi-isomorphisms to filtered quasi-isomorphisms. It induces a functor \(t_{[n}\colon DF(\mathcal A)\to DF(\mathcal A)\) with a natural transformation \(L\to t_{[n}L\), and \(t_{[n+1}t_{[n}=t_{[n+1}\), compatibly with these transformations.
2. \(F_pt_{[n}L=t_{[n}F_pL\), naturally in \(L\), and the underlying object of \(t_{[n}L\) is \(t_{[n}\) of the underlying object of \(L\).
3. A morphism \(\varphi\) of \(DF(\mathcal A)\) is an isomorphism if and only if every \(F_p\varphi\) is an isomorphism of \(D(\mathcal A)\).
4. If \(H^q(\operatorname{gr}^pL)=0\) for all \(p\) and all \(q<n\), then \(L\to t_{[n}L\) is an isomorphism.
5. An exact functor \(\Phi\colon\mathcal A\to\mathcal B\) commutes with \(t_{[n}\) and with these transformations.

**Proof.** (1) Let \(h\) be a filtered homotopy between \(f,g\colon L\to M\), so \(f-g=dh+hd\). Define \(h'\) on the truncations by \(h'^k=h^k\) for \(k\geq n+2\), by \(h'^{n+1}=\pi\circ h^{n+1}\), where \(\pi\colon M^n\to\operatorname{Coker}d_M^{n-1}\), and by \(h'^k=0\) for \(k\leq n\). In degrees \(\geq n+1\) the identity \(\tau_{\geq n}f-\tau_{\geq n}g=dh'+h'd\) follows from that for \(h\), as \(d^n_M\) factors through \(\pi\). In degree \(n\), on the class \(\bar x\) of \(x\in L^n\), \((f-g)(\bar x)\) is the class of \(dh(x)+hd(x)\), which is \(h'^{n+1}(d\bar x)\) because \(dh(x)\) lies in the image of \(d^{n-1}_M\). As \(\pi\) carries \(F^pM^n\) into the image filtration, \(h'\) is filtered. If \(f\) is a filtered quasi-isomorphism, every \(F_pf\) is a quasi-isomorphism (Lemma 2.1 there), hence so is every \(F_p\tau_{\geq n}f=\tau_{\geq n}F_pf\), and \(\tau_{\geq n}f\) is a filtered quasi-isomorphism. So \(\tau_{\geq n}\) induces a functor on \(KF(\mathcal A)\) which inverts the filtered quasi-isomorphisms, hence a functor on \(DF(\mathcal A)=FQis^{-1}KF(\mathcal A)\) [Tag 04VG]. The maps \(L\to\tau_{\geq n}L\) are natural. On complexes \(\tau_{\geq n+1}\tau_{\geq n}=\tau_{\geq n+1}\), and the filtration of a truncation is determined by its quotients \(F_p\), which agree.

(2) This holds by construction, and \(F_pL=L\) for large \(p\).

(3) The sequences \(0\to\operatorname{gr}^pX\to F_{p+1}X\to F_pX\to0\) give distinguished triangles in \(D(\mathcal A)\), natural in \(X\in DF(\mathcal A)\). If all \(F_p\varphi\) are isomorphisms, so are all \(\operatorname{gr}^p\varphi\) [Tag 014A], and \(\varphi\) is an isomorphism by Proposition 2.2 (4) there. The converse is clear.

(4) This is Lemma 3.1 there.

(5) \(\Phi\) preserves kernels and cokernels, so \(\Phi(\tau_{\geq n}F_pL)=\tau_{\geq n}F_p\Phi(L)\), and the filtrations are determined by the quotients \(F_p\). \(\square\)

**Lemma 1.2 (graded pieces of truncations).** Let \(L\in CF(\mathcal A)\) and \(n,p\in\mathbf Z\).

1. \(\operatorname{gr}^p(\tau_{\geq n}L)\) vanishes in degrees \(<n\). The map \(\operatorname{gr}^pL\to\operatorname{gr}^p(\tau_{\geq n}L)\) induces isomorphisms on \(H^q\) for \(q\geq n+1\), and \(H^n(\operatorname{gr}^p\tau_{\geq n}L)\) is the image of \(H^n(\operatorname{gr}^pL)\) in \(H^n(F_{p+1}L)\).
2. Let \(L\) have filtration in \([0,1]\). Then \(\operatorname{gr}^0\tau_{\geq n}L=\tau_{\geq n}\operatorname{gr}^0L\) and \(\operatorname{gr}^1\tau_{\geq n}L=F^1\tau_{\geq n}L\). The map \(F^1L\to F^1\tau_{\geq n}L\) factors through a morphism of complexes
\[
\psi_n\colon\tau_{\geq n}F^1L\longrightarrow F^1\tau_{\geq n}L,
\]
and the map \(\theta_n\colon F^1\tau_{\geq n}L\to\tau_{\geq n+1}F^1L\) which is zero in degree \(n\), the projection \(F^1L^{n+1}\to\) \(\operatorname{Coker}(F^1L^n\to F^1L^{n+1})\) in degree \(n+1\) and the identity in higher degrees is a morphism of complexes. The composites \(\theta_n\psi_n\) and \(\psi_{n+1}\theta_n\) are the transition maps \(\tau_{\geq n}F^1L\to\tau_{\geq n+1}F^1L\) and \(F^1\tau_{\geq n}L\to F^1\tau_{\geq n+1}L\). So \((\psi_n)_n\) is an isomorphism of projective systems of complexes from \(t_{[}F^1L\) to \((F^1\tau_{\geq n}L)_n\), with inverse \((\theta_n)_n\).
3. In (2), \(H^q(\psi_n)\) is an isomorphism for \(q\neq n\), and \(H^n(\psi_n)\) is surjective, with kernel the image of the connecting map \(H^{n-1}(\operatorname{gr}^0L)\to H^n(F^1L)\) of \(0\to F^1L\to L\to\operatorname{gr}^0L\to0\). In particular \(\psi_n\) is a quasi-isomorphism if \(H^{n-1}(\operatorname{gr}^0L)=0\).

**Proof.** (1) Since \(\tau_{\geq n}\) preserves epimorphisms, \(\tau_{\geq n}F_{p+1}L\to\tau_{\geq n}F_pL\) is surjective, with kernel \(\operatorname{gr}^p\tau_{\geq n}L\). The filtered map \(L\to\tau_{\geq n}L\) gives a morphism to the exact sequence \(0\to\) \(\operatorname{gr}^p\tau_{\geq n}L\to\) \(\tau_{\geq n}F_{p+1}L\to\) \(\tau_{\geq n}F_pL\to\) \(0\) from \(0\to\operatorname{gr}^pL\to F_{p+1}L\to F_pL\to0\). The maps \(H^q(F_rL)\to H^q(\tau_{\geq n}F_rL)\) are isomorphisms for \(q\geq n\), so the five lemma on the segments
\[
\begin{aligned}&H^{q-1}(F_{p+1})\to H^{q-1}(F_p)\\&\quad\to H^q(\operatorname{gr}^p)\to H^q(F_{p+1})\\&\quad\to H^q(F_p)\end{aligned}
\]
of the long exact sequences gives the isomorphisms for \(q\geq n+1\). As \(H^{n-1}(\tau_{\geq n}F_pL)=0\), \(H^n(\operatorname{gr}^p\tau_{\geq n}L)\) is the kernel of \(H^n(F_{p+1}L)\to H^n(F_pL)\), which is the image of \(H^n(\operatorname{gr}^pL)\) by the long exact sequence of \(L\). The truncation vanishes in degrees \(<n\), and so do its subquotients.

(2) Here \(F_1L=\operatorname{gr}^0L\) and \(F_2L=L\), so \(\operatorname{gr}^0\tau_{\geq n}L=F_1\tau_{\geq n}L=\tau_{\geq n}\operatorname{gr}^0L\), and \(F^2\tau_{\geq n}L=0\). The complex \(F^1\tau_{\geq n}L\) vanishes in degrees \(<n\), so \(F^1L\to F^1\tau_{\geq n}L\) factors uniquely through \(\tau_{\geq n}F^1L\). In degree \(n\), \(F^1\tau_{\geq n}L\) is the image of \(F^1L^n\) in \(\operatorname{Coker}d^{n-1}_L\), and in degrees \(\geq n+1\) it is \(F^1L\). The differential of the class of \(a\in F^1L^n\) is \(da\in F^1L^{n+1}\), whose image in \(\operatorname{Coker}(F^1L^n\to F^1L^{n+1})\) vanishes; so \(\theta_n\) is a morphism of complexes. In degree \(n+1\), \(\theta_n\psi_n\) is the projection \(F^1L^{n+1}\to\) \(\operatorname{Coker}(F^1L^n\to F^1L^{n+1})\), and \(\psi_{n+1}\theta_n\) sends \(x\in F^1L^{n+1}\) to its class in \(\operatorname{Coker}d^n_L\); in degree \(n\) both transition maps vanish, and above they are identities. The last assertion is the criterion of the Conventions.

(3) For \(q<n\) both sides vanish. For \(q\geq n+1\), \(H^q(\psi_n)\) composed with the isomorphism \(H^q(F^1L)\cong H^q(\tau_{\geq n}F^1L)\) is \(H^q(F^1L)\to H^q(F^1\tau_{\geq n}L)\), an isomorphism by (1) for \(p=1\). For \(q=n\), (1) for \(p=1\) identifies \(H^n(F^1\tau_{\geq n}L)\) with the image of \(H^n(F^1L)\to H^n(L)\), compatibly with \(H^n(\psi_n)\); the kernel of \(H^n(F^1L)\to H^n(L)\) is the image of the connecting map. \(\square\)

So \(\operatorname{gr}\) does not commute with \(t_{[n}\), even for a two-step filtration (Exercise 6.1); the defect lies in the degree \(n\) of \(F^1\) and is invisible in the projective systems.

**Lemma 1.3 (truncated triangles).** Let \(L\in CF(\mathcal A)\) have filtration in \([0,1]\), let \(u\colon F^1L\to L\), \(v\colon L\to\operatorname{gr}^0L\) and \(\partial_L\) be the maps of its triangle (0.1), and let \(n\in\mathbf Z\). For a morphism \(w\) of \(D(\mathcal A)\) write \(t_{[n}(w)\) for its image under the functor \(t_{[n}\) of \(D(\mathcal A)\), so that \(t_{[n}(\partial_L)\colon\) \(t_{[n}\operatorname{gr}^0L\to\) \(t_{[n}(F^1L[1])=\) \((t_{[n+1}F^1L)[1]\). The filtered map \(L\to\tau_{\geq n}L\) induces a morphism from the triangle of \(L\) to that of \(\tau_{\geq n}L\), so that
\[
\begin{aligned}&\quad \partial_{\tau_{\geq n}L}\\ &\quad \circ(\operatorname{gr}^0L\to\tau_{\geq n}\operatorname{gr}^0L)\\ &\quad \quad\\ &\quad =(F^1L\to F^1\tau_{\geq n}L)[1]\\ &\quad \circ\partial_L,\end{aligned} \tag{1.1}
\]
and the maps of the triangle of \(\tau_{\geq n}L\) satisfy
\[
\begin{gathered}(F^1\tau_{\geq n}L\to\tau_{\geq n}L)\circ\psi_n\\ =t_{[n}(u),\\ (\tau_{\geq n}L\to\operatorname{gr}^0\tau_{\geq n}L)\\ =t_{[n}(v),\\ \theta_n[1]\circ\partial_{\tau_{\geq n}L}=t_{[n}(\partial_L).\end{gathered}
\]

**Proof.** The filtered map \(L\to\tau_{\geq n}L\) is a morphism of short exact sequences, so it gives a morphism of triangles [Tag 0152], which is (1.1). The next two equalities hold for the morphisms of complexes, by the uniqueness of factorizations through \(\tau_{\geq n}\). Composing (1.1) with \(\theta_n[1]\), and using that \(\theta_n\circ(F^1L\to F^1\tau_{\geq n}L)=\) \(\theta_n\psi_n\circ(F^1L\to\tau_{\geq n}F^1L)\) is the truncation map \(F^1L\to\tau_{\geq n+1}F^1L\), we find that \(\theta_n[1]\circ\partial_{\tau_{\geq n}L}\) and \(t_{[n}(\partial_L)\) have the same composite with \(\operatorname{gr}^0L\to t_{[n}\operatorname{gr}^0L\). Their target lies in \(D^{\geq n}\), so maps into it from \(t_{[n}\operatorname{gr}^0L\) are determined by this composite [Tag 06XS], as in (0.1) of the lesson on equivariant cotangent complexes. \(\square\)

**Lemma 1.4 (pullbacks and direct images).** Let \(h\colon(T',A')\to(T,A)\) be a morphism of ringed topoi, and \(Lh^*\colon DF(A)\to DF(A')\), \(Rh_*\colon DF(A')\to DF(A)\) the filtered derived functors (Proposition 4.1 of [*Filtered tensor products on a ringed topos*](filtered-tensor-products-on-a-ringed-topos.md) and Proposition 4.1 of [*Filtered internal Hom and the formulas of Cartan*](filtered-internal-hom-and-the-formulas-of-cartan.md)).

1. \(Lh^*\) and \(Rh_*\) commute with \(F^p\), \(F_p\) and \(\operatorname{gr}^p\), and under these identifications \(F_p\) of the unit \(L\to Rh_*Lh^*L\) is the unit of \(F_pL\).
2. For a morphism \(k\colon(T'',A'')\to(T',A')\), \(L(hk)^*\cong Lk^*Lh^*\), compatibly with (1). If \(h^*\) is exact on modules, then \(Lh^*L\) is \(h^*L\), filtered by the subcomplexes \(h^*F^pL\), for every \(L\in CF(A)\).
3. For \(L\in D^-F(A)\) and \(n\in\mathbf Z\), the map induced by \(L\to t_{[n}L\) is an isomorphism \(t_{[n}Lh^*L\cong t_{[n}Lh^*t_{[n}L\).

**Proof.** (1) Let \(P\to L\) be a filtered K-flat resolution (Definition 2.1 and Proposition 2.3 of the lesson on filtered tensor products). By Lemma 2.2 (1) there, \(F^pP\) and \(P/F^pP\) are K-flat with flat terms, and \(h^*F^pP\subset h^*P\) with quotient \(h^*(P/F^pP)\) (Proposition 4.1 (1) there); so \(F^p\), \(F_p\) and \(\operatorname{gr}^p\) of \(h^*P\) compute \(Lh^*\) of \(F^pL\), \(F_pL\) and \(\operatorname{gr}^pL\). For a filtered K-injective resolution \(M\to I\), \(F^pI\) and \(I/F^pI\) are K-injective with injective terms and the filtration of every term of \(I\) splits (Lemma 2.2 (4) there), so \(h_*F^pI=F^ph_*I\) and \(h_*(I/F^pI)=h_*I/h_*F^pI\) compute \(Rh_*\) of \(F^pM\) and \(F_pM\), and likewise for \(\operatorname{gr}^pM\). With a filtered K-injective resolution \(h^*P\to J\), the unit is represented by \(P\to h_*h^*P\to h_*J\), and \(F_p\) of these models are a K-flat resolution of \(F_pL\) and a K-injective resolution of \(h^*F_pP\); so \(F_p\) of the composite represents the unit of \(F_pL\) [Tag 07A6].

(2) \(h^*\) carries filtered K-flat complexes to filtered K-flat complexes (Proposition 4.1 (1) there), so \(L(hk)^*P=(hk)^*P=k^*h^*P\) computes \(Lk^*Lh^*L\). If \(h^*\) is exact, it commutes with \(\operatorname{gr}^p\) and preserves quasi-isomorphisms, so \(h^*P\to h^*L\) is a filtered quasi-isomorphism.

(3) By (1) and Lemma 1.1 (2), \(F_p\) of the map is the map \(t_{[n}Lh^*F_pL\to t_{[n}Lh^*t_{[n}F_pL\), an isomorphism by (0.2) of the lesson on equivariant cotangent complexes; Lemma 1.1 (3) concludes. \(\square\)

## 2. Dévissage

**Lemma 2.1 (pure filtered objects).** Let \(K,M\in D(\mathcal A)\) and \(p,q\in\mathbf Z\).

1. If \(p\le q\), the functor "underlying object" is a bijection \(\operatorname{Hom}_{DF}(\iota_pK,\iota_qM)\to\) \(\operatorname{Hom}_D(K,M)\).
2. If \(p>q\), then \(\operatorname{Hom}_{DF}(\iota_pK,\iota_qM)=0\).

**Proof.** Twisting by \(p\), an automorphism of \(DF(\mathcal A)\) which preserves the underlying objects, carries \(\iota_pK\) and \(\iota_qM\) to \(\iota_0K\) and \(\iota_{q-p}M\); so let \(p=0\). A morphism \(\iota_0K\to Y\) of \(DF(\mathcal A)\) is a roof \(fs^{-1}\) with \(s\colon L'\to\iota_0K\) a filtered quasi-isomorphism and \(f\colon L'\to Y\) in \(KF(\mathcal A)\) [Tag 05Q4]. The graded pieces \(\operatorname{gr}^iL'\), \(i\neq0\), are acyclic, so by Lemma 3.4 of the lesson on filtered derived categories the subcomplex \(L''=\iota_0(F^0L')\) maps to \(L'\) by a filtered morphism inducing quasi-isomorphisms on all graded pieces. As in the proof of Proposition 3.5 (1) there, the roofs with \(L'=\iota_0K'\) are therefore cofinal, and then \(s=\iota_0(s_0)\) for a quasi-isomorphism \(s_0\colon K'\to K\). If \(q\geq0\), every morphism of complexes \(K'\to M\) is a filtered morphism \(\iota_0K'\to\iota_qM\), since \(F^i\iota_qM=M\) for \(i\le0\), and every homotopy is filtered; so the morphisms are the colimit of \(\operatorname{Hom}_K(K',M)\) over the quasi-isomorphisms \(K'\to K\), which is \(\operatorname{Hom}_D(K,M)\), and the identification is the underlying roof. If \(q<0\), a filtered morphism \(\iota_0K'\to\iota_qM\) carries \(F^0=K'\) into \(F^0\iota_qM=0\), so it vanishes. \(\square\)

**Lemma 2.2.** Let \(X\in CF(\mathcal A)\) have filtration in \([a,b]\) and \(a\le p\le b\). Filter \(F^{p+1}X\) by \(F^i(F^{p+1}X)=F^{\max(i,p+1)}X\), and \(F^pX\) likewise. There is a distinguished triangle in \(DF(\mathcal A)\)
\[
\begin{aligned}&F^{p+1}X\longrightarrow F^pX\\&\quad\longrightarrow\iota_p(\operatorname{gr}^pX)\longrightarrow F^{p+1}X[1],\end{aligned}
\]
whose first map is the inclusion and whose second is the projection; for \(p=a\) its middle term is \(X\).

**Proof.** Let \(u\) be the inclusion and \(C(u)\) its cone, with \(C(u)^k=F^pX^k\oplus F^{p+1}X^{k+1}\). The map \(C(u)\to\iota_p(\operatorname{gr}^pX)\), \((x,y)\mapsto x\bmod F^{p+1}X\), is a filtered morphism of complexes, as \(y\in F^{p+1}X\). On \(\operatorname{gr}^i\) it is \(C(\mathrm{id})\to0\) for \(i\geq p+1\), the identity of \(\operatorname{gr}^pX\) for \(i=p\), and \(0\to0\) for \(i<p\); so it is a filtered quasi-isomorphism, and the cone triangle of \(u\) (Proposition 2.2 (1) there) gives the triangle. \(\square\)

**Theorem 2.3 (dévissage).** Let \(\Phi\colon\mathcal A\to\mathcal B\) be an exact functor, \(D'\subset D(\mathcal A)\) a strictly full subcategory stable under shifts such that \(\operatorname{Hom}_D(K,M)\to\operatorname{Hom}_D(\Phi K,\Phi M)\) is bijective for all \(K,M\in D'\), and \(E\subset D(\mathcal B)\) the class of objects isomorphic to some \(\Phi K\), \(K\in D'\). For an interval \([a,b]\), let \(D'F^{[a,b]}(\mathcal A)\subset DF^{[a,b]}(\mathcal A)\) be the full subcategory of the \(X\) with \(\operatorname{gr}^pX\in D'\) for all \(p\), and \(EF^{[a,b]}(\mathcal B)\) the analogous subcategory of \(DF^{[a,b]}(\mathcal B)\). Then
\[
\Phi\colon D'F^{[a,b]}(\mathcal A)\longrightarrow DF^{[a,b]}(\mathcal B)
\]
is fully faithful, with essential image \(EF^{[a,b]}(\mathcal B)\).

**Proof.** *Full faithfulness.* For \(X,Y\in D'F^{[a,b]}(\mathcal A)\) we show that \(\operatorname{Hom}(X[i],Y)\to\operatorname{Hom}(\Phi X[i],\Phi Y)\) is bijective for all \(i\), by induction on the lengths of the filtration intervals. Represent \(X\) with filtration in \([a,b]\). If \(a<b\), Lemma 2.2 for \(p=a\) gives a distinguished triangle \(F^{a+1}X\to X\to\iota_a\operatorname{gr}^aX\to F^{a+1}X[1]\) whose outer terms lie in \(D'F^{[a+1,b]}(\mathcal A)\) and \(D'F^{[a,a]}(\mathcal A)\), and \(\Phi\) carries it to the corresponding triangle of \(\Phi X\). The long exact sequences of \(\operatorname{Hom}(-,Y)\) and \(\operatorname{Hom}(-,\Phi Y)\) [Tag 0149] and the five lemma reduce the claim to the outer terms. So it suffices to treat \(X=\iota_pK\) with \(K\in D'\), and in the same way, with the sequences of \(\operatorname{Hom}(\iota_pK[i],-)\), \(Y=\iota_qM\) with \(M\in D'\). For \(p\le q\), Lemma 2.1 identifies both sides with \(\operatorname{Hom}_D(K[i],M)\) and \(\operatorname{Hom}_D(\Phi K[i],\Phi M)\), compatibly with \(\Phi\), which commutes with the underlying objects; the map is bijective as \(K[i],M\in D'\). For \(p>q\) both sides vanish.

*Essential image.* \(\Phi\) commutes with \(\operatorname{gr}^p\), so it maps into \(EF^{[a,b]}(\mathcal B)\). Conversely let \(Z\in EF^{[a,b]}(\mathcal B)\), represented with filtration in \([a,b]\); we argue by induction on \(b-a\). If \(a=b\), then \(Z=\iota_a\operatorname{gr}^aZ\cong\iota_a\Phi K=\Phi(\iota_aK)\). If \(a<b\), consider the triangle \(F^{a+1}Z\to Z\to\iota_a\operatorname{gr}^aZ\xrightarrow{\delta}F^{a+1}Z[1]\) of Lemma 2.2. By induction \(F^{a+1}Z\cong\Phi X'\) with \(X'\in D'F^{[a+1,b]}(\mathcal A)\), and \(\iota_a\operatorname{gr}^aZ\cong\Phi(\iota_aK)\) with \(K\in D'\). By full faithfulness, \(\delta\) corresponds to \(\Phi(\delta_0)\) for a unique \(\delta_0\colon\iota_aK\to X'[1]\). Choose a distinguished triangle \(X'\to X\to\iota_aK\xrightarrow{\delta_0}X'[1]\). The isomorphisms \(\Phi X'\cong F^{a+1}Z\) and \(\Phi(\iota_aK)\cong\iota_a\operatorname{gr}^aZ\) are compatible with \(\Phi(\delta_0)\) and \(\delta\); after rotating the triangles, they extend to a morphism from the image of this triangle under \(\Phi\) to the triangle of \(Z\), which is an isomorphism [Tag 014A]. In particular \(\Phi X\cong Z\). The graded pieces of the triangle show \(\operatorname{gr}^aX\cong K\) and \(\operatorname{gr}^pX\cong\operatorname{gr}^pX'\) for \(p\neq a\); so \(X\in DF^{[a,b]}(\mathcal A)\) (Proposition 3.5 (1) there) and \(X\in D'F^{[a,b]}(\mathcal A)\). \(\square\)

The stability of \(D'\) under shifts cannot be omitted (Exercise 6.3).

**Corollary 2.4.** In the situation of Theorem 2.3, \(\Phi\) is fully faithful on projective systems in \(D'F^{[a,b]}(\mathcal A)\), and for every projective system \((Z_n)\) in \(EF^{[a,b]}(\mathcal B)\) there is a projective system \((X_n)\) in \(D'F^{[a,b]}(\mathcal A)\) with isomorphisms \(\Phi X_n\cong Z_n\) compatible with the transition maps, unique up to unique isomorphism.

**Proof.** The first assertion follows from the formula for morphisms of projective systems. For the second, choose \(X_n\) with \(\Phi X_n\cong Z_n\) by Theorem 2.3; the transition maps \(Z_n\to Z_{n+1}\) are the images of unique maps \(X_n\to X_{n+1}\), and uniqueness follows from full faithfulness. \(\square\)

## 3. The filtered transitivity complex

**Construction 3.1.** Let \(A\to B\xrightarrow{g}C\) be ring maps of a topos \(T\). Put \(W=B\times C\), let \(\gamma=(\mathrm{id},g)\colon B\to W\), an injective ring map, and let \(\mathrm{pr}\colon W\to C\) be the second projection; \(C\) is a direct summand of the \(W\)-module \(W\), so it is a flat \(W\)-algebra. Let \(P\) and \(Q\) be the standard resolutions of \(B\) and of \(W\) over \(A\), and \(\overline Q=Q\otimes_PB\). In the proof of Theorem 4.1 of *The cotangent complex*, in the sheaf form of its Section 10, \(Q\) is degreewise flat over \(P\) (on sections it is degreewise polynomial over \(P\), as \(\gamma\) is injective), \(\overline Q\) is a polynomial resolution of \(W\) over \(B\), and the normalized degreewise differential sequences (4.6) of \(A\to B\xrightarrow{\gamma}W\) form a short exact sequence of complexes of \(W\)-modules with flat terms. Tensoring it over \(W\) with \(C\) gives a short exact sequence of complexes of flat \(C\)-modules
\[
\begin{aligned}&0\to N(\Omega_{P/A}\otimes_PC)\\&\quad\to N(\Omega_{Q/A}\otimes_QC)\\&\quad\to N(\Omega_{\overline Q/B}\otimes_{\overline Q}C)\to0.\end{aligned}
\tag{3.1}
\]
The *filtered transitivity complex* \(L_{C/B/A}\in CF^{[0,1]}(C)\) is the middle term of (3.1), with \(F^1L_{C/B/A}\) the first term.

**Proposition 3.2.** Let \(A\to B\to C\) be ring maps of a topos \(T\), with \(W\) and \(\mathrm{pr}\) as in Construction 3.1.

1. \(L_{C/B/A}\) vanishes in positive degrees and has flat graded pieces; it is filtered K-flat. Its first step is \(F^1L_{C/B/A}=L_{B/A}\otimes_BC\), which computes \(L_{B/A}\otimes^{\mathbf L}_BC\).
2. The functoriality maps (2.6) of *The cotangent complex* for \(\mathrm{pr}\) are quasi-isomorphisms \(L_{C/B/A}=L_{W/A}\otimes_WC\to L_{C/A}\) and, through the identification of \(N(\Omega_{\overline Q/B}\otimes_{\overline Q}W)\) with \(L_{W/B}\) of its Lemma 2.1, \(\operatorname{gr}^0L_{C/B/A}\to L_{C/B}\). Under these and (1), the triangle (0.1) of \(L_{C/B/A}\) is the transitivity triangle (4.2) of Theorem 4.1 there, whose first two maps are functoriality maps.
3. A commutative diagram of ring maps from \(A\to B\to C\) to \(A'\to B'\to C'\) induces a filtered map \(L_{C/B/A}\to L_{C'/B'/A'}\), linear over \(C\to C'\), compatible with composition, with (1) and with (2). For a morphism of topoi \(w\colon T'\to T\), \(w^{-1}L_{C/B/A}=L_{w^{-1}C/w^{-1}B/w^{-1}A}\) as filtered complexes, compatibly with (1) and (2).

**Proof.** (1) The terms of (3.1) are flat and vanish in positive degrees, and an extension of flat modules is flat [Tag 03EY]; bounded above complexes of flat modules are K-flat [Tag 06YQ], so the graded pieces are K-flat with flat terms (Definition 2.1 of the lesson on filtered tensor products). Normalization commutes with \(-\otimes_BC\), and \(L_{B/A}=N(\Omega_{P/A}\otimes_PB)\) is a bounded above complex of flat \(B\)-modules, so \(L_{B/A}\otimes_BC\) computes the derived tensor product.

(2) The middle term of (3.1) is \(N(\Omega_{Q/A}\otimes_QW)\otimes_WC=L_{W/A}\otimes_WC\), and the third is \(N(\Omega_{\overline Q/B}\otimes_{\overline Q}W)\otimes_WC\). For an arbitrary \(g\), the proof of Theorem 4.1 there constructs (4.2) as the triangle of (3.1) [Tag 0152], through the isomorphisms (4.4) over \(A\) and over \(B\), whose components are the functoriality maps of the projections; so the two maps are quasi-isomorphisms, and the triangle of \(L_{C/B/A}\) is (4.2). When \(g\) is injective, that proof uses instead the sequence (4.6) of \(A\to B\to C\). The map of triples \((\mathrm{id},\mathrm{id},\mathrm{pr})\) from \(A\to B\to W\) to \(A\to B\to C\) induces a map from (3.1) to that sequence, which is the identity on the first terms and the functoriality maps of \(\mathrm{pr}\) on the others; so the two sequences give the same triangle under these identifications.

(3) Standard resolutions are functorial for maps of ring pairs (proof of Theorem 2.2 there), and \(\gamma\) and \(\mathrm{pr}\) are natural in the triple. Inverse images under morphisms of topoi are exact and preserve the polynomial constructions, products, differentials, tensor products and normalization (Section 10 there). \(\square\)

**Definition 3.3.** Let \(f\colon X\to Y\) be a morphism of diagrams of schemes of type \(u\), and \(g\colon Y\to Z\) one of type \(v\) (Conventions of [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md)). The *filtered transitivity complex* of \(X\to Y\to Z\) is
\[
L_{X/Y/Z}=L_{\mathcal O_X/f^{-1}\mathcal O_Y/(gf)^{-1}\mathcal O_Z},
\]
for the ring maps \((gf)^{-1}\mathcal O_Z=\) \(f^{-1}g^{-1}\mathcal O_Z\to\) \(f^{-1}\mathcal O_Y\to\) \(\mathcal O_X\) of \(\operatorname{Top}(X_{\mathrm{Zar}})\).

**Proposition 3.4.** In Definition 3.3:

1. \(L_{X/Y/Z}\in CF^{[0,1]}(\mathcal O_X)\) is filtered K-flat and vanishes in positive degrees, and \(F^1L_{X/Y/Z}=f^*L_{Y/Z}\), which computes \(Lf^*L_{Y/Z}\).
2. The functoriality maps are quasi-isomorphisms \(L_{X/Y/Z}\to L_{X/Z}\) and \(\operatorname{gr}^0L_{X/Y/Z}\to L_{X/Y}\). Under them, the triangle (0.1) of \(L_{X/Y/Z}\) is the transitivity triangle (1.1) of the lesson on deformations of diagrams; its connecting map is the Kodaira–Spencer class \(K(X/Y/Z)\).
3. For every object \(i\) of the index category of \(X\), \(i^*L_{X/Y/Z}=L_{X_i/Y_{u(i)}/Z_{vu(i)}}\). The construction is natural for maps of triples of rings of \(\operatorname{Top}(X_{\mathrm{Zar}})\).

**Proof.** Since \(f^{-1}\) preserves the polynomial constructions, \(L_{f^{-1}\mathcal O_Y/f^{-1}g^{-1}\mathcal O_Z}=f^{-1}L_{Y/Z}\), and \(F^1L_{X/Y/Z}=\) \(f^{-1}L_{Y/Z}\otimes_{f^{-1}\mathcal O_Y}\mathcal O_X=\) \(f^*L_{Y/Z}\); with Proposition 3.2 this gives (1) and (2), the triangle (1.1) being the sheaf form of (4.2) for these ring maps (Proposition 1.3 there). (3) The restriction \(i^*\) is the inverse image of a morphism of topoi (Conventions there), and Proposition 3.2 (3) applies. \(\square\)

## 4. The equivariant filtered transitivity triangle

In this section \(G\) is a category scheme over \(S\), and \(X\xrightarrow{f}Y\xrightarrow{g}Z\) are morphisms of \(G\)-schemes such that \(f\) and \(G\), and \(g\) and \(G\), are Tor independent. The notation is that of [*Equivariant cotangent complexes*](equivariant-cotangent-complexes.md): \(\varphi=N(G,f)\), \(\varepsilon\) the comparison morphisms, \(\operatorname{ner}\) the nerve functor of its Proposition 1.5, \(f_B^*\) the pullback of its Lemma 3.1, \(t_{[}L^G\) the equivariant cotangent complexes. Let
\[
\mathbf L=L_{N(G,X)/N(G,Y)/N(G,Z)}
\]
be the filtered transitivity complex of the morphisms of simplicial schemes \(N(G,X)\to N(G,Y)\to N(G,Z)\) (Definition 3.3). It is filtered K-flat, so the filtered complex \(\varepsilon^*\mathbf L\), filtered by the \(\varepsilon^*F^p\mathbf L\), represents the filtered derived pullback \(L\varepsilon^*\mathbf L\) on \(\operatorname{Top}(N(G,X)_{\mathrm{fppf}})\), with \(F^1\varepsilon^*\mathbf L=\varepsilon^*F^1\mathbf L\) and \(\operatorname{gr}^0\varepsilon^*\mathbf L=\varepsilon^*\operatorname{gr}^0\mathbf L\) (Proposition 4.1 of the lesson on filtered tensor products). Since \(\varepsilon^*\) of the short exact sequence of \(\mathbf L\) is that of \(\varepsilon^*\mathbf L\), the triangle (0.1) of \(\varepsilon^*\mathbf L\) is \(L\varepsilon^*\) of the transitivity triangle of the nerves (Proposition 3.4 (2)). Put
\[
\Lambda_n=t_{[n}Lf_B^*t_{[n}L^G_{Y/Z},
\]
the terms of the projective system \(Lf_B^*t_{[}L^G_{Y/Z}\) of Proposition 3.3 of the lesson on equivariant cotangent complexes. By its part (1), \(\operatorname{ner}(\Lambda_n)\cong t_{[n}\varepsilon^*F^1\mathbf L\), compatibly with the transition maps; since \(\operatorname{ner}\) is exact and fully faithful on \(D^b\), the transition map induces an isomorphism \(t_{[n+1}\Lambda_n\cong\Lambda_{n+1}\).

**Theorem 4.1 (the equivariant filtered transitivity triangle).**

1. For every integer \(n\) there is an object \(t_{[n}L^G_{X/Y/Z}\) of \(D^bF^{[0,1]}(B[G,X]^{\mathrm o})\) with an isomorphism
\[
\begin{aligned}&\operatorname{ner}\bigl(t_{[n}L^G_{X/Y/Z}\bigr)\\&\quad \cong t_{[n}\varepsilon^*\mathbf L\end{aligned} \tag{4.1}
\]
in \(DF(\operatorname{Top}(N(G,X)_{\mathrm{fppf}}))\), and maps \(t_{[n}L^G_{X/Y/Z}\to t_{[n+1}L^G_{X/Y/Z}\) whose nerves are the transition maps. The objects, the isomorphisms and the maps are unique up to unique isomorphism. The projective system \(t_{[}L^G_{X/Y/Z}\) is the *equivariant filtered transitivity triangle* of \(X\to Y\to Z\).
2. There are unique isomorphisms from the underlying object of \(t_{[n}L^G_{X/Y/Z}\) to \(t_{[n}L^G_{X/Z}\) and from \(\operatorname{gr}^0t_{[n}L^G_{X/Y/Z}\) to \(t_{[n}L^G_{X/Y}\), compatible with (4.1), with the isomorphisms (2.1) of the lesson on equivariant cotangent complexes and with the quasi-isomorphisms of Proposition 3.4 (2); they are compatible with the transition maps.
3. There are unique maps \(\psi_n\colon\Lambda_n\to F^1t_{[n}L^G_{X/Y/Z}\) and \(\theta_n\colon F^1t_{[n}L^G_{X/Y/Z}\to t_{[n+1}\Lambda_n\) whose nerves are the maps \(\psi_n\), \(\theta_n\) of Lemma 1.2 (2) for \(\varepsilon^*\mathbf L\). Through \(t_{[n+1}\Lambda_n\cong\Lambda_{n+1}\), they are inverse isomorphisms of projective systems between \(Lf_B^*t_{[}L^G_{Y/Z}\) and \((F^1t_{[n}L^G_{X/Y/Z})_n\).
4. For every \(n\), the triangle (0.1) of \(t_{[n}L^G_{X/Y/Z}\), which under (2) reads
\[
\begin{aligned}&F^1t_{[n}L^G_{X/Y/Z}\to t_{[n}L^G_{X/Z}\\&\quad\to t_{[n}L^G_{X/Y}\xrightarrow{\ \partial_n\ }F^1t_{[n}L^G_{X/Y/Z}[1],\end{aligned}
\]
is distinguished in \(D^b(B[G,X]^{\mathrm o})\), and these triangles form a projective system. The three maps of Proposition 3.3 (2) of the lesson on equivariant cotangent complexes are the first map composed with \(\psi_n\), the second map, and
\[
K_{G,n}=\theta_n[1]\circ\partial_n.
\]
So the equivariant Kodaira–Spencer class \(K_G(X/Y/Z)\) is the morphism of projective systems \((\partial_n)_n\), followed by the inverse \((\theta_n[1])_n\) of the isomorphism \((\psi_n[1])_n\).

**Proof.** (1) By Lemma 1.2 (2), \(\operatorname{gr}^0t_{[n}\varepsilon^*\mathbf L=\tau_{\geq n}\varepsilon^*\operatorname{gr}^0\mathbf L\) and \(\operatorname{gr}^1t_{[n}\varepsilon^*\mathbf L=F^1\tau_{\geq n}\varepsilon^*\mathbf L\). The first is \(t_{[n}L\varepsilon^*L_{N(G,X)/N(G,Y)}\) (Proposition 3.4 (2)), which is \(\operatorname{ner}(t_{[n}L^G_{X/Y})\) by (2.1) there. The second has cohomology only in degrees \([n,0]\). By Lemma 1.2 (1) and (3), its \(H^q\) is \(H^q(\varepsilon^*F^1\mathbf L)\) for \(q\geq n+1\), and its \(H^n\) is the cokernel of the connecting map \(H^{n-1}(\varepsilon^*\operatorname{gr}^0\mathbf L)\to H^n(\varepsilon^*F^1\mathbf L)\), a morphism of modules of \(\operatorname{Top}(N(G,X)_{\mathrm{fppf}})\). The complexes \(\varepsilon^*F^1\mathbf L\) and \(\varepsilon^*\operatorname{gr}^0\mathbf L\) compute \(L\varepsilon^*L\varphi^*L_{N(G,Y)/N(G,Z)}\) and \(L\varepsilon^*L_{N(G,X)/N(G,Y)}\), whose cohomology modules have invertible structure maps along every \(\alpha\) with \(\alpha(0)=0\), by Propositions 3.3 (1) and 2.3 (1) there, as in the proof of Proposition 2.3 (2) there. The inverse images along the levels of the fppf topoi are exact, so the cokernel has invertible structure maps along these \(\alpha\) as well. Hence both graded pieces lie in the essential image of \(\operatorname{ner}\) on \(D^b\) (Proposition 1.5 (3) there), and Theorem 2.3 and Corollary 2.4, with \(\Phi=\operatorname{ner}\) and \(D'=D^b(B[G,X]^{\mathrm o})\), give the objects, the isomorphisms and the maps.

(2) \(\operatorname{ner}\) commutes with the underlying objects and with \(\operatorname{gr}^0\). The underlying object of \(t_{[n}\varepsilon^*\mathbf L\) is \(\tau_{\geq n}\varepsilon^*\mathbf L\cong t_{[n}L\varepsilon^*L_{N(G,X)/N(G,Z)}\), which is \(\operatorname{ner}(t_{[n}L^G_{X/Z})\) by (2.1) there, and \(\operatorname{gr}^0\) was treated in (1). Since \(\operatorname{ner}\) is fully faithful on \(D^b\), these isomorphisms come from unique isomorphisms, compatible with the transition maps.

(3) The sources and targets of the maps \(\psi_n\colon\tau_{\geq n}\varepsilon^*F^1\mathbf L\to F^1\tau_{\geq n}\varepsilon^*\mathbf L\) and \(\theta_n\colon F^1\tau_{\geq n}\varepsilon^*\mathbf L\to\tau_{\geq n+1}\varepsilon^*F^1\mathbf L\) of Lemma 1.2 (2) are the nerves of \(\Lambda_n\), of \(F^1t_{[n}L^G_{X/Y/Z}\) (by (4.1)) and of \(t_{[n+1}\Lambda_n\). As \(\operatorname{ner}\) is fully faithful on \(D^b\), these maps come from unique maps, and the relations of Lemma 1.2 (2) hold after applying \(\operatorname{ner}\), hence hold.

(4) The triangle of an object of \(DF^{[0,1]}\) is distinguished and natural (Conventions). Its image under \(\operatorname{ner}\) is the triangle of \(t_{[n}\varepsilon^*\mathbf L=\tau_{\geq n}\varepsilon^*\mathbf L\). The maps of Proposition 3.3 (2) there are the unique maps whose nerves are \(t_{[n}\) of the maps of \(L\varepsilon^*\) of the transitivity triangle of the nerves, that is, of the triangle of \(\varepsilon^*\mathbf L\). By Lemma 1.3 for \(\varepsilon^*\mathbf L\), these are the nerves of the stated composites; as \(\operatorname{ner}\) is faithful on \(D^b\), the maps agree. \(\square\)

## 5. Descent along a torsor and the filtered Atiyah triangle

In this section \(G\) is a group scheme over \(S\), flat and locally of finite presentation, \(f\colon X\to Y\) is a \(G_Y\)-torsor and \(Y\to S\) a morphism of schemes. The notation is that of [*Torsors, co-Lie complexes and the Atiyah class*](torsors-co-lie-complexes-and-the-atiyah-class.md): \(T\) is the big fppf topos of \(S\), \(a\colon N(G,X)\to Y\) the morphism to the constant simplicial scheme, \(\varphi=N(G,f)\), \(f^{G*}\) and \(f^G_*\) the functors of its Construction 2.1, and \(\varepsilon\) the comparison morphisms. For an interval \([c,d]\), \(D^bF^{[c,d]}_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) is the full subcategory of \(D^bF^{[c,d]}(Y_{\mathrm{Zar}})\) of the objects whose graded pieces lie in \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\).

**Theorem 5.1 (filtered descent along a torsor).** Let \([c,d]\) be an interval.

1. \(f^{G*}\) induces an equivalence \(DF(T/Y)\to DF(B[G,X]^{\mathrm o})\), with quasi-inverse \(f^G_*\), and
\[
\begin{aligned}&a^*=\operatorname{ner}\circ f^{G*}\colon D^bF^{[c,d]}(T/Y)\\&\quad\longrightarrow D^bF^{[c,d]}\bigl(\operatorname{Top}(N(G,X)_{\mathrm{fppf}})\bigr)\end{aligned}
\]
is fully faithful, with essential image the objects whose graded pieces have cartesian cohomology modules.
2. The functor
\[
\begin{aligned}&a^*\colon D^bF^{[c,d]}_{\mathrm{qc}}(Y_{\mathrm{Zar}})\\&\quad\longrightarrow D^bF^{[c,d]}\bigl(\operatorname{Top}(N(G,X)_{\mathrm{Zar}})\bigr)\end{aligned}
\]
is fully faithful, with essential image the objects whose graded pieces are quasi-cartesian with termwise quasi-coherent cohomology.
3. The functors \(a^*\) of (1) and (2) are fully faithful on projective systems, and a projective system of objects of their essential images is isomorphic, compatibly with the transition maps, to the image of a projective system, unique up to unique isomorphism.

**Proof.** (1) \(f^{G*}\) and \(f^G_*\) are exact quasi-inverse equivalences of abelian categories (Proposition 2.2 (3) there), so they induce quasi-inverse equivalences of the categories of filtered complexes and, as they preserve filtered quasi-isomorphisms, of the filtered derived categories. The functor \(a^*=\operatorname{ner}\circ f^{G*}\) on modules is exact (Construction 2.1 (a) there), and on \(D^b(T/Y)\) it is fully faithful with essential image the objects with cartesian cohomology modules (Theorem 2.4 there); Theorem 2.3 applies with \(D'=D^b(T/Y)\).

(2) Every \(a_n\) is flat, so \(a^*\) is exact (proof of Theorem 2.5 there). Theorem 2.5 there and Theorem 2.3 with \(D'=D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), which is stable under shifts, give the assertion.

(3) Corollary 2.4. \(\square\)

**Theorem 5.2 (the filtered Atiyah triangle).** Let \(\mathbf L=L_{N(G,X)/N(G,Y)/N(G)}\) be the filtered transitivity complex of \(N(G,X)\xrightarrow{\varphi}N(G,Y)\to N(G)\) (Definition 3.3).

1. For every integer \(n\) there is an object \(\ell_n(X/Y/S)\) of \(D^bF^{[0,1]}_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) with an isomorphism \(a^*\ell_n(X/Y/S)\cong t_{[n}\mathbf L\) in \(DF(\operatorname{Top}(N(G,X)_{\mathrm{Zar}}))\), and maps \(\ell_n(X/Y/S)\to\ell_{n+1}(X/Y/S)\) whose images under \(a^*\) are the transition maps, all unique up to unique isomorphism. The projective system \(\ell_{X/Y/S}=(\ell_n(X/Y/S))_n\) is the *filtered Atiyah triangle* of the torsor.
2. There are unique isomorphisms from the underlying object of \(\ell_n(X/Y/S)\) to \(\ell_n(X/S)\) and from \(\operatorname{gr}^0\ell_n(X/Y/S)\) to \(t_{[n}\ell_{X/Y}\), which is \(\ell_{X/Y}\) for \(n\le-1\), and unique maps \(\psi_n\colon t_{[n}L_{Y/S}\to F^1\ell_n(X/Y/S)\) and \(\theta_n\colon F^1\ell_n(X/Y/S)\to t_{[n+1}L_{Y/S}\), whose images under \(a^*\) are, through the isomorphisms of Theorem 3.3 (1), (2) and Lemma 3.1 (3) of the lesson on torsors and of Proposition 3.4 (2), the corresponding identifications and the maps \(\psi_n\), \(\theta_n\) of Lemma 1.2 (2) for \(\mathbf L\). The maps \(\psi_n\) and \(\theta_n\) are inverse isomorphisms of projective systems, and \(\psi_n\) is an isomorphism for \(n\le-1\).
3. For \(n\le-1\), under the isomorphisms of (2), the triangle (0.1) of \(\ell_n(X/Y/S)\) is the Atiyah triangle (3.1) of the lesson on torsors: its maps are \(\beta_n\), \(\gamma_n\) and \(\mathrm{at}_n\).

**Proof.** (1) By Lemma 1.2 (2), \(\operatorname{gr}^0t_{[n}\mathbf L=\tau_{\geq n}\operatorname{gr}^0\mathbf L\), and \(\operatorname{gr}^0\mathbf L\) is quasi-isomorphic to \(L_{N(G,X)/N(G,Y)}\) (Proposition 3.4 (2)), which is bounded and quasi-cartesian with termwise quasi-coherent cohomology (Lemma 3.1 (2), (4) there); so is its truncation (Lemma 3.1 (5) there). The other graded piece \(K=F^1\tau_{\geq n}\mathbf L\) is bounded. By Lemma 1.2 (1) and (3), \(H^q(K)=H^q(F^1\mathbf L)\) for \(q\geq n+1\), and \(H^n(K)\) is the cokernel of the connecting map \(H^{n-1}(\operatorname{gr}^0\mathbf L)\to H^n(F^1\mathbf L)\). Here \(F^1\mathbf L=\varphi^*L_{N(G,Y)/N(G)}\cong a^*L_{Y/S}\) (Lemma 3.1 (3) there) and \(\operatorname{gr}^0\mathbf L\) are quasi-cartesian with termwise quasi-coherent cohomology. The faces of \(N(G,X)\) are flat (Proposition 1.3 (2) there), so pullback along a face is exact, and the cohomology modules of these two complexes have invertible structure maps along the faces. Pullbacks are right exact, so the same holds for the cokernel \(H^n(K)\), and all cohomology modules of \(K\) are termwise quasi-coherent with invertible structure maps along the faces. Hence the structure maps of \(K\) along the faces are isomorphisms, and \(K\) is quasi-cartesian (Lemma 3.1 (5) there). Theorem 5.1 (2) and (3) give the objects and the maps.

(2) The underlying object of \(t_{[n}\mathbf L\) is \(\tau_{\geq n}\mathbf L\cong t_{[n}L_{N(G,X)/N(G)}\) (Proposition 3.4 (2)), which is \(a^*\ell_n(X/S)\) by Theorem 3.3 (2) there; \(\operatorname{gr}^0t_{[n}\mathbf L=\) \(\tau_{\geq n}\operatorname{gr}^0\mathbf L\cong\) \(t_{[n}L_{N(G,X)/N(G,Y)}\cong\) \(a^*t_{[n}\ell_{X/Y}\) by Theorem 3.3 (1) there, \(a^*\) being exact. The sources and targets of \(\psi_n\) and \(\theta_n\) for \(\mathbf L\) are \(\tau_{\geq n}F^1\mathbf L\cong a^*t_{[n}L_{Y/S}\), \(F^1t_{[n}\mathbf L\) and \(\tau_{\geq n+1}F^1\mathbf L\cong a^*t_{[n+1}L_{Y/S}\). By the full faithfulness of \(a^*\) on \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) (Theorem 2.5 there), all these isomorphisms and maps come from unique ones, and the relations of Lemma 1.2 (2) hold. For \(n\le-1\), \(H^{n-1}(\operatorname{gr}^0\mathbf L)=0\), as \(L_{N(G,X)/N(G,Y)}\) lies in \(D^{[-1,0]}\) (Lemma 3.1 (2) there); so \(\psi_n\) is a quasi-isomorphism (Lemma 1.2 (3)), and its descent is an isomorphism.

(3) Let \(n\le-1\), and write \(A=F^1\mathbf L\), \(B=\mathbf L\), \(C=\operatorname{gr}^0\mathbf L\), with the maps \(u\), \(v\), \(\partial\) of the triangle of \(\mathbf L\), which is the transitivity triangle of \(N(G,X)\to N(G,Y)\to N(G)\) (Proposition 3.4 (2)). As \(C\in D^{\geq-1}\subset D^{\geq n}\), Lemma 3.2 of the lesson on torsors makes from it the distinguished triangle
\[
t_{[n}A\xrightarrow{t_{[n}(u)}t_{[n}B\xrightarrow{v'}C\xrightarrow{w'}t_{[n}A[1],
\]
with \(v'\) the factorization of \(v\) and \(w'=(A\to t_{[n}A)[1]\circ\partial\). By Lemma 1.3, with the isomorphisms \(\psi_n\) and \(C\to\tau_{\geq n}C\), the triangle of \(\tau_{\geq n}\mathbf L\) is carried to this one: \((F^1\tau_{\geq n}\mathbf L\to\tau_{\geq n}\mathbf L)\circ\psi_n=t_{[n}(u)\), \(t_{[n}(v)=(C\to\tau_{\geq n}C)\circ v'\), by naturality and the uniqueness of \(v'\), and
\[
\begin{aligned}&\partial_{\tau_{\geq n}\mathbf L}\circ(C\to\tau_{\geq n}C)\\&\quad=(F^1\mathbf L\to F^1\tau_{\geq n}\mathbf L)[1]\circ\partial\\&\quad=\psi_n[1]\circ w'\end{aligned}
\]
by (1.1) and the definition of \(\psi_n\).
The triangle of \(\ell_n(X/Y/S)\) is carried by \(a^*\) to the triangle of \(\tau_{\geq n}\mathbf L\). By Theorem 3.3 (3) of the lesson on torsors, \(\beta_n\), \(\gamma_n\) and \(\mathrm{at}_n\) are the unique maps whose images under \(a^*\) are the maps of the triangle of Lemma 3.2 there; as \(a^*\) is faithful on \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), the maps of the triangle of \(\ell_n(X/Y/S)\) correspond to them under the isomorphisms of (2). \(\square\)

**Proposition 5.3.** Let \(n\in\mathbf Z\), and write \(\ell_n=\ell_n(X/Y/S)\).

1. The unit \(\ell_n\to Ra_*a^*\ell_n\cong Ra_*t_{[n}\mathbf L\) is an isomorphism in \(DF(Y_{\mathrm{Zar}})\).
2. If \(s\colon Y\to X\) is a section of \(f\), then \(\ell_n\cong Ls^*t_{[n}L_{X/Y/S}\) in \(DF(Y_{\mathrm{Zar}})\), where \(L_{X/Y/S}\) is the filtered transitivity complex of \(X\to Y\to S\) (Definition 3.3).
3. For the equivariant filtered transitivity triangle of \(X\to Y\to S\) (Theorem 4.1, \(G\) acting trivially on \(Y\) and \(S\)), there are isomorphisms
\[
\begin{gathered}t_{[n}L^G_{X/Y/S}\cong f^{G*}t_{[n}L\varepsilon^*\ell_n,\\ \ell_n\cong R\varepsilon_*f^G_*t_{[n}L^G_{X/Y/S},\end{gathered}
\]
in \(DF(B[G,X]^{\mathrm o})\) and in \(DF(Y_{\mathrm{Zar}})\), where \(\varepsilon\colon T/Y\to Y_{\mathrm{Zar}}\) is the comparison morphism.

**Proof.** (1) By Lemma 1.4 (1), \(F_p\) of the unit is the unit of \(F_p\ell_n\), which lies in \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), being an iterated extension of graded pieces of \(\ell_n\); it is an isomorphism by Corollary 2.6 (1) of the lesson on torsors. Lemma 1.1 (3) concludes.

(2) The first vertex of \(N_0(G,X)=X\) is the identity, so \(a_0=f\), and the restriction \(0^*\) to the level \(0\) satisfies \(0^*a^*=f^*\) on modules. Both functors are exact, so this holds on filtered complexes. As \(f\) is flat, \(Lf^*=f^*\) (Lemma 1.4 (2)), and \(fs=\mathrm{id}_Y\) gives
\[
\begin{aligned}\ell_n&\cong L(fs)^*\ell_n\cong Ls^*f^*\ell_n\\&=Ls^*0^*a^*\ell_n\cong Ls^*0^*t_{[n}\mathbf L.\end{aligned}
\]
Finally \(0^*t_{[n}\mathbf L=t_{[n}0^*\mathbf L\) (Lemma 1.1 (5)), and \(0^*\mathbf L=L_{X/Y/S}\) (Proposition 3.4 (3)), as \(N_0(G,X)=X\), \(N_0(G,Y)=Y\) and \(N_0(G)=S\).

(3) On big sites \(\operatorname{ner}\circ f^{G*}=a^*\), which is exact, and \(\varepsilon\circ a=a\circ\varepsilon\) as morphisms of ringed topoi from \(\operatorname{Top}(N(G,X)_{\mathrm{fppf}})\) to \(Y_{\mathrm{Zar}}\) (proof of Theorem 2.5 of the lesson on torsors). By Lemma 1.1 (5), Lemma 1.4 (2) and (3), and Theorem 5.2 (1),
\[
\begin{aligned}&\operatorname{ner}(f^{G*}t_{[n}L\varepsilon^*\ell_n)=t_{[n}a^*L\varepsilon^*\ell_n\\&\quad\cong t_{[n}L\varepsilon^*a^*\ell_n\cong t_{[n}L\varepsilon^*t_{[n}\mathbf L\\&\quad\cong t_{[n}L\varepsilon^*\mathbf L=t_{[n}\varepsilon^*\mathbf L,\end{aligned}
\]
which is \(\operatorname{ner}(t_{[n}L^G_{X/Y/S})\) by (4.1). The graded pieces of \(\ell_n\) lie in \(D^{\le0}\), since those of \(a^*\ell_n\cong t_{[n}\mathbf L\) do and \(a^*\) is exact and faithful on modules (\(a_0=f\) is faithfully flat); so \(L\varepsilon^*\ell_n\in D^{\le0}F\), and the graded pieces of \(t_{[n}L\varepsilon^*\ell_n\) have cohomology in \([n,0]\) (Lemma 1.2 (1)). Theorem 2.3 for \(\operatorname{ner}\) gives the first isomorphism. For the second, \(f^G_*t_{[n}L^G_{X/Y/S}\cong t_{[n}L\varepsilon^*\ell_n\), and the map \(\ell_n\to R\varepsilon_*t_{[n}L\varepsilon^*\ell_n\), the unit followed by the truncation, has \(F_p\) equal to the map of Proposition 2.1 of the lesson on big fppf sites for \(F_p\ell_n\) (Lemma 1.1 (2), Lemma 1.4 (1)). It is an isomorphism: \(F_p\ell_n\) lies in \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) and has no cohomology in degrees \(<n\), because the graded pieces of \(\ell_n\) have none, by the same faithfulness argument and Lemma 1.2 (1). Lemma 1.1 (3) concludes. \(\square\)

## 6. Exercises

**6.1.** In the category of abelian groups, let \(L\) be the complex \(\mathbf Z\xrightarrow{\mathrm{id}}\mathbf Z\) in degrees \(-1\) and \(0\), with \(F^0L=L\), \(F^1L\) the subcomplex \(\mathbf Z\) in degree \(0\), and \(F^2L=0\). Compute the graded pieces of \(\tau_{\geq0}L\) and of \(\tau_{\geq-1}L\), the maps \(\psi_{-1}\), \(\psi_0\) and \(\theta_{-1}\) of Lemma 1.2, and compare \(F^1t_{[0}L\) with \(t_{[0}F^1L\).

**6.2.** Let \(\mathcal A\) be an abelian category.

1. Show that every distinguished triangle of \(D(\mathcal A)\) is isomorphic to the triangle (0.1) of an object of \(DF^{[0,1]}(\mathcal A)\).
2. Show that two objects of \(DF^{[0,1]}(\mathcal A)\) whose triangles are isomorphic are isomorphic.

**6.3.** Let \(k\) be a field, \(\Lambda=k[x]/(x^2)\), and \(\Phi\colon\operatorname{Mod}(k)\to\operatorname{Mod}(\Lambda)\) the functor giving a vector space the action \(x=0\). Let \(D'\subset D(k)\) be the strictly full subcategory of the objects isomorphic to complexes concentrated in degree \(0\). Show that \(\Phi\) is exact and fully faithful on \(D'\), but that the filtered object \(Z=\Lambda\), in degree \(0\), with \(F^1Z=x\Lambda\), whose graded pieces are isomorphic to \(\Phi(k)\), is not isomorphic to \(\Phi(X)\) for any \(X\in DF(k)\). Which hypothesis of Theorem 2.3 fails?

## 7. Complete solutions

**7.1.** Here \(\operatorname{gr}^0L=L/F^1L\) is \(\mathbf Z\) in degree \(-1\), and \(F^1L\) is \(\mathbf Z\) in degree \(0\). The truncation \(\tau_{\geq0}L\) is \(\operatorname{Coker}(\mathrm{id})=0\) in degree \(0\), so \(\tau_{\geq0}L=0\) and all its graded pieces vanish; in particular \(\operatorname{gr}^0\tau_{\geq0}L=0=\tau_{\geq0}\operatorname{gr}^0L\), as Lemma 1.2 (2) says, while \(F^1t_{[0}L=0\) and \(t_{[0}F^1L=\mathbf Z[0]\) are not isomorphic. The map \(\psi_0\colon\mathbf Z[0]\to0\) is surjective on \(H^0\) with kernel \(H^0(F^1L)=\mathbf Z\); this is the image of the connecting map \(H^{-1}(\operatorname{gr}^0L)=\mathbf Z\to H^0(F^1L)=\mathbf Z\), which is the identity: lift \(1\in\operatorname{gr}^0L^{-1}\) to \(1\in L^{-1}\) and apply \(d\). Next, \(\tau_{\geq-1}L=L\) with the same filtration, so \(\psi_{-1}\colon\) \(\tau_{\geq-1}F^1L=\) \(\mathbf Z[0]\to\) \(F^1\tau_{\geq-1}L=\) \(\mathbf Z[0]\) is the identity, a quasi-isomorphism, in accordance with \(H^{-2}(\operatorname{gr}^0L)=0\). The map \(\theta_{-1}\colon F^1\tau_{\geq-1}L=\mathbf Z[0]\to\tau_{\geq0}F^1L=\mathbf Z[0]\) is the projection onto \(\operatorname{Coker}(0\to\mathbf Z)\), the identity. Then \(\theta_{-1}\psi_{-1}=\mathrm{id}\) is the transition map \(\tau_{\geq-1}F^1L\to\tau_{\geq0}F^1L\), and \(\psi_0\theta_{-1}=0\) is the transition map \(F^1\tau_{\geq-1}L\to F^1\tau_{\geq0}L=0\). The discrepancy at \(n=0\) is absorbed by the shift of index in the isomorphism of projective systems.

**7.2.** (1) A distinguished triangle of \(D(\mathcal A)\) is isomorphic to the cone triangle \((K,M,C(u),u,i,-p)\) of a morphism of complexes \(u\colon K\to M\). Let \(\operatorname{Cyl}(u)^n=M^n\oplus K^{n+1}\oplus K^n\), with \(d(m,y,z)=(dm+u(y),-dy,dz-y)\). The maps \(j\colon K\to\operatorname{Cyl}(u)\), \(z\mapsto(0,0,z)\), and \(\pi\colon\operatorname{Cyl}(u)\to M\), \((m,y,z)\mapsto m+u(z)\), are morphisms of complexes with \(\pi j=u\): indeed \(\pi(d(m,y,z))=\) \(dm+u(y)+u(dz)-u(y)=\) \(d\pi(m,y,z)\). The map \(\pi\) is a homotopy equivalence: \(\sigma(m)=(m,0,0)\) satisfies \(\pi\sigma=\mathrm{id}\), and \(h(m,y,z)=(0,-z,0)\) satisfies \(dh+hd=\mathrm{id}-\sigma\pi\). The cokernel of the injective map \(j\) is \(M\oplus K[1]\) with \(d(m,y)=(dm+u(y),-dy)\), that is \(C(u)\). So \(X=\operatorname{Cyl}(u)\), filtered by \(F^1X=j(K)\), lies in \(DF^{[0,1]}(\mathcal A)\), and its triangle \(K\to\operatorname{Cyl}(u)\to C(u)\to K[1]\) is distinguished. The maps \(\mathrm{id}_K\) and \(\pi\) commute with \(j\) and \(u\), so they extend to a morphism to the cone triangle of \(u\), which is an isomorphism [Tag 014A].

(2) Let \(X,X'\in DF^{[0,1]}(\mathcal A)\) and \((\alpha,\beta,\gamma)\) an isomorphism from the triangle of \(X\) to that of \(X'\). Lemma 2.2 for \(p=0\) gives distinguished triangles \(\iota_1F^1X\to X\to\iota_0\operatorname{gr}^0X\xrightarrow{\delta}\iota_1F^1X[1]\) in \(DF(\mathcal A)\), and likewise for \(X'\); here \(F^1X\), with filtration in \([1,1]\), is \(\iota_1F^1X\). The underlying object functor carries the cone construction of Lemma 2.2 to that of [Tag 0152], so it carries \(\delta\) to \(\partial_X\). By Lemma 2.1 (1), with \(0\le1\), the underlying object functor is bijective on morphisms \(\iota_0\operatorname{gr}^0X\to\iota_1F^1X'[1]\), and \(\iota_1(\alpha)[1]\circ\delta\) and \(\delta'\circ\iota_0(\gamma)\) have the same underlying morphism \(\alpha[1]\circ\partial_X=\partial_{X'}\circ\gamma\); so they are equal. Hence \(\iota_1(\alpha)\) and \(\iota_0(\gamma)\) extend to a morphism of distinguished triangles of \(DF(\mathcal A)\), whose middle map \(X\to X'\) is an isomorphism [Tag 014A].

**7.3.** Kernels and cokernels of \(\Lambda\)-modules killed by \(x\) are computed on the underlying vector spaces, so \(\Phi\) is exact. For vector spaces \(V,W\) in degree \(0\), \(\operatorname{Hom}_{D(k)}(V,W)=\operatorname{Hom}_k(V,W)\) and \(\operatorname{Hom}_{D(\Lambda)}(\Phi V,\Phi W)=\operatorname{Hom}_\Lambda(\Phi V,\Phi W)\), and a map between modules killed by \(x\) is \(\Lambda\)-linear if and only if it is \(k\)-linear; so \(\Phi\) is fully faithful on \(D'\). The graded pieces of \(Z\) are \(\Lambda/x\Lambda\cong k\) and \(x\Lambda\cong k\), with \(x\) acting by zero. If \(Z\cong\Phi(X)\) in \(DF(\Lambda)\), the underlying objects are isomorphic in \(D(\Lambda)\), so \(\Lambda=H^0(Z)\cong H^0(\Phi X)=\Phi(H^0X)\), on which \(x\) acts by zero, whereas \(x\neq0\) on \(\Lambda\): a contradiction. The subcategory \(D'\) is not stable under shifts. On the smallest subcategory stable under shifts containing \(D'\), \(\Phi\) is not fully faithful: \(\operatorname{Hom}_{D(k)}(k,k[1])=\operatorname{Ext}^1_k(k,k)=0\), while \(\operatorname{Hom}_{D(\Lambda)}(\Phi k,\Phi k[1])=\operatorname{Ext}^1_\Lambda(k,k)\) contains the class of \(0\tox\Lambda\to\Lambda\to\Lambda/x\Lambda\to0\), which is not split, since \(x\) does not act by zero on \(\Lambda\). By Lemma 2.1 (1) this class is the underlying morphism of the map \(\delta\colon\iota_0\operatorname{gr}^0Z\to\iota_1F^1Z[1]\) of the proof of Theorem 2.3, which is therefore not in the image of \(\Phi\).

## Sources

The filtered form of the transitivity triangle, the equivariant transitivity triangle as a projective system of filtered objects on the classifying topos, and the filtered Atiyah triangle of a torsor with its descent statements are due to Illusie, in his work on the cotangent complex, where the filtered Atiyah triangle is called the true Atiyah triangle. The reduction of statements on filtered objects to their graded pieces follows his treatment of filtered derived categories. The homological algebra used here is that of the Stacks project.