Chern classes of perfect complexes

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

For a map of rings $A\to B$ of a topos, every perfect complex $E$ of $B$-modules carries its Atiyah class $\operatorname{at}(E)\colon E\to E\otimes^{\mathbf L}_BL_{B/A}[1]$, an endomorphism of $E$ with coefficients in $L_{B/A}[1]$. Its L-series and characteristic series, with the shift isomorphism $LS^n(L_{B/A}[1])\cong L\Lambda^n(L_{B/A})[n]$, give classes $c^n(E)\in H^n(T,L\Lambda^nL_{B/A}),$ the Chern classes of $E$ with values in the cotangent complex. Their images in $H^n(T,\Omega^n_{B/A})$ are Chern classes in Hodge cohomology. The lesson proves:

the total Chern class $c_t(E)=\sum c^n(E)t^n$ is multiplicative on filtrations, gradings and distinguished triangles, so it defines a homomorphism from the Grothendieck group of perfect complexes to the group of series with constant term $1$;
it is compatible with base change along squares of rings and along morphisms of ringed topoi;
$c^n(E)=0$ for $n>r$ when $E$ is a module in degree $0$ that is locally a direct summand of a free module of rank $r$;
when $B$ is a local ring, perfect complexes have an integral rank, additive on triangles, and the rank and the total Chern class together define a homomorphism on the Grothendieck group;
the Hodge-cohomology classes $c'_t(E)$ satisfy the same formulas, are computed from the extension of principal parts, and for an invertible module $c'_t(E)=1+c'^1(E)t$, with $c'^1(E)$ the negative of the class of the extension of $B$ by $\Omega_{B/A}$ whose local splittings are the local connections on $E$; on a smooth scheme this is $c'^1(L)=-c_1^{\mathrm{Hodge}}(L)$, the first Chern class defined through $\mathrm d\log$.

Prerequisites. The Atiyah class of filtered complexes (Definition 1.1, Proposition 1.2, Theorem 3.1, Corollary 3.2, Definition 3.3, Proposition 3.4, Definition 4.1, Proposition 4.2); L-series of endomorphisms of perfect complexes (Propositions 1.2, 1.3, 3.2, 3.6, Theorem 3.3, Corollaries 3.4, 3.7, Definition 3.5); The Atiyah class of a module (Lemma 1.2, Proposition 3.2); Lemma 3.7 of Deformations of modules and of their morphisms (boundaries through local lifts); Traces and cup products of perfect filtered complexes (Definitions 1.1, 2.1, Theorem 2.3); Filtered and graded powers of complexes (Lemma 1.3, Theorem 2.6); Filtered derived categories (Proposition 2.2). From the AI-integrated Stacks project: invertible modules and the first Chern class in Hodge cohomology [Tag 09NU, Tag 0FLE]; locally ringed topoi [Stacks, Tag 04ES, Tag 04EU]; strictly perfect complexes [Tag 08FL], and morphisms out of them are locally maps of complexes [Tag 08FR]; the third vertex of a distinguished triangle [Tag 014A]; triangles of short exact sequences [Stacks, Tag 0152] and termwise split replacements [Tag 014M]; the cotangent complex of a morphism of ringed topoi [Tag 08SU].

Conventions

As in the lesson on the Atiyah class of filtered complexes: $T$ is a topos, $A\to B$ a map of rings of $T$, $L=L_{B/A}$, $\Omega=\Omega_{B/A}$, $\alpha\colon L\to\Omega$ the augmentation, tensor products derived, and $\operatorname{at}(E)\colon E\to E\otimes L[1]$ the Atiyah class of $E\in D^-(B)$. An object of $D(B)$ has bounded perfect amplitude if it has perfect amplitude in some interval, as in the lesson on L-series; such objects lie in $D^-(B)$. For $X\in D(B)$ and $n\in\mathbf Z$, $H^n(T,X)=\operatorname{Hom}_{D(B)}(B,X[n])$. Since $L$ is concentrated in degrees $\leq0$, $L[1]\in D^{\leq0}(B)$, and the rings $\operatorname{Hom}(B,LS^\bullet(L[1]))$ and $\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])=$ $\prod_nH^n(T,L\Lambda^nL)$ of the lesson on L-series are commutative (Proposition 1.2 (2) there); $\sigma^S$ is the ring isomorphism between them of Proposition 1.3 there.

1. Chern classes with values in the cotangent complex

Definition 1.1. Let $E\in D(B)$ have bounded perfect amplitude. The L-series and the total Chern class of $E$ relative to $A\to B$ are $\begin{aligned}&L_t(E)=L_t(E,B/A)\\ &\quad=\sigma^S\bigl(L_t(\operatorname{at}(E))\bigr),\\ &c_t(E)=c_t(E,B/A)\\ &\quad=\sigma^S\bigl(c_t(\operatorname{at}(E))\bigr)=L_t(E)^{-1},\end{aligned}$ in $\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])$, where $L_t(\operatorname{at}(E))$ and $c_t(\operatorname{at}(E))$ are the series of Definition 3.5 of the lesson on L-series for $M=L[1]$. The coefficient of $t^n$ in $c_t(E)$ is the $n$-th Chern class $c^n(E)=c^n(E,B/A)\in H^n(T,L\Lambda^nL)$; $c^0(E)=1$.

Proposition 1.2. Let $E,E'\in D(B)$ have bounded perfect amplitude.

If $E\cong E'$, then $L_t(E)=L_t(E')$.
$L_t(E[1])=c_t(E)$; hence $L_t(E[2])=L_t(E)$.
If $E\in D^{\leq0}(B)$, then $c_t(\operatorname{at}(E))=$ $\sum_n(-1)^n\operatorname{Tr}(\psi^nL\Lambda^n\operatorname{at}(E))t^n$, with $\psi^n=\psi^n_{E,L[1]}$.

Proof. (1) For an isomorphism $u\colon E\to E'$, $\operatorname{at}(E')=(u\otimes1)\circ\operatorname{at}(E)\circ u^{-1}$ (Proposition 3.4 of the lesson on the Atiyah class of filtered complexes), and Proposition 3.2 (2) of the lesson on L-series, in the form of Proposition 3.6 there, applies. (2) $\operatorname{at}(E[1])=\operatorname{at}(E)[1]$ (Proposition 3.4 of the lesson on the Atiyah class of filtered complexes), and $L_t(\operatorname{at}(E)[1])=c_t(\operatorname{at}(E))$ (Proposition 3.6 of the lesson on L-series). (3) Corollary 3.7 of the lesson on L-series, $E$ and $L[1]$ being in $D^{\leq0}(B)$. $\square$

Theorem 1.3 (multiplicativity).

Let $E\in DF(B)$ have bounded perfect amplitude. Then $L_t(E)=\prod_pL_t(\operatorname{gr}^pE)$ and $c_t(E)=\prod_pc_t(\operatorname{gr}^pE)$.
Let $E\in DG(B)$ have bounded perfect amplitude, with components $E_i$. Then $L_t(\bigoplus_iE_i)=\prod_iL_t(E_i)$, and likewise for $c_t$.
For a distinguished triangle $E'\to E\to E''\to E'[1]$ of objects of bounded perfect amplitude, $L_t(E)=L_t(E')L_t(E'')$ and $c_t(E)=c_t(E')c_t(E'')$.

Proof. (1) By Proposition 3.4 of the lesson on the Atiyah class of filtered complexes, $\operatorname{at}^F(E)\colon E\to E\otimes L[1]$ is a morphism of $DF(B)$, $L[1]$ in filtration $[0,0]$, with underlying morphism $\operatorname{at}(E)$ and graded pieces $\operatorname{at}(\operatorname{gr}^pE)$. Theorem 3.3 (1) and (2) of the lesson on L-series, in the form of Proposition 3.6 there, give $L_t(\operatorname{at}(E))=$ $L_t(\bigoplus_p\operatorname{at}(\operatorname{gr}^pE))=$ $\prod_pL_t(\operatorname{at}(\operatorname{gr}^pE))$, and $\sigma^S$ is a ring isomorphism. The ring is commutative, so the statement for $c_t$ follows by taking inverses. (2) $\operatorname{at}(\bigoplus_iE_i)=\bigoplus_i\operatorname{at}(E_i)$ is a morphism of $DG(B)$, and Theorem 3.3 (2) of the lesson on L-series applies. (3) By [Tag 014M] and [Tag 0152] the triangle is isomorphic to the triangle of a short exact sequence $0\to X'\to X\to X''\to0$ of complexes. The complex $X$ filtered by $F^1X=X'$, $F^0X=X$, $F^2X=0$ is an object of $DF(B)$ with $\operatorname{gr}^1=X'$ and $\operatorname{gr}^0=X''$, of bounded perfect amplitude; (1) and Proposition 1.2 (1) give the statement. $\square$

Corollary 1.4 (Grothendieck group). Let $K(B)$ be the Grothendieck group of the objects of $D(B)$ of bounded perfect amplitude: the free abelian group on their isomorphism classes, modulo $[E]=[E']+[E'']$ for every distinguished triangle $E'\to E\to E''\to E'[1]$. Then $[E]\mapsto c_t(E)$ is a homomorphism from $K(B)$ to the multiplicative group of series with constant term $1$ in $\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])$, and $[E]\mapsto L_t(E)$ is the composite with inversion. The trace of the identity, $[E]\mapsto\operatorname{Tr}(1_E)\in\Gamma(T,B)$, is additive on triangles (Theorem 2.3 of the lesson on traces, for the filtered object of the proof of Theorem 1.3 (3)), so $[E]\mapsto(\operatorname{Tr}(1_E),c_t(E))$ is a homomorphism to $\Gamma(T,B)\times\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])^+$.

Proof. Theorem 1.3 (3) and Proposition 1.2 (1). $\square$

Proposition 1.5 (base change).

Let $(A\to B)\to(A'\to B')$ be a commutative square of rings of $T$, $c\colon L_{B/A}\otimes B'\to L_{B'/A'}$ the functoriality map, and let $\overline c\colon\operatorname{Hom}(B,L\Lambda^\bullet(L_{B/A})[1])\to$ $\operatorname{Hom}(B',L\Lambda^\bullet(L_{B'/A'})[1])$ be extension of scalars to $B'$ followed by $L\Lambda^n(c)[n]$. For $E$ of bounded perfect amplitude and an isomorphism $E\otimes^{\mathbf L}_BB'\cong E'$, $L_t(E',B'/A')=\overline c\,L_t(E,B/A)$ and $c_t(E',B'/A')=\overline c\,c_t(E,B/A)$.
Let $f\colon T'\to T$ be a morphism of topoi. Then $L_t(f^{-1}E,f^{-1}B/f^{-1}A)=$ $f^{-1}L_t(E,B/A)$, through $f^{-1}L_{B/A}=L_{f^{-1}B/f^{-1}A}$.
Let $g\colon X'\to X$ and $h\colon Y'\to Y$ be morphisms of ringed topoi with $f\circ g=h\circ f'$ for morphisms of ringed topoi $f\colon X\to Y$, $f'\colon X'\to Y'$, and write $L_{X/Y}=L_{\mathcal O_X/f^{-1}\mathcal O_Y}$ [Tag 08SU], $L_t(E,X/Y)=L_t(E,\mathcal O_X/f^{-1}\mathcal O_Y)$. For $E$ of bounded perfect amplitude on $X$, $g^*L_t(E,X/Y)=L_t(Lg^*E,X'/Y')$, where $g^*$ is the inverse image followed by the functoriality map $Lg^*L_{X/Y}\to L_{X'/Y'}$.

Proof. (1) Let $u\colon E\otimes B'\to E'$ be the isomorphism. By Proposition 1.2 (2) of the lesson on the Atiyah class of filtered complexes (with Proposition 3.4 there), $\operatorname{at}(E')=(u\otimes c[1])\circ(\operatorname{at}(E)\otimes B')\circ u^{-1}$. By Proposition 3.2 of the lesson on L-series (invariance, naturality in $M$ for $c[1]$, and base change along the morphism of ringed topoi $(T,B')\to(T,B)$), $L_t(\operatorname{at}(E'))$ is the image of $L_t(\operatorname{at}(E))$ under base change followed by $LS^n(c[1])$. Finally $\sigma^S$ is natural in the coefficient object. (2) Proposition 1.2 (3) of the lesson on the Atiyah class of filtered complexes and Proposition 3.2 (3) of the lesson on L-series. (3) $Lg^*E=\mathcal O_{X'}\otimes_{g^{-1}\mathcal O_X}g^{-1}E$: apply (2) to the underlying morphism of topoi of $g$, then (1) on $X'$ to the square of rings $(g^{-1}f^{-1}\mathcal O_Y\to g^{-1}\mathcal O_X)\to$ $(f'^{-1}\mathcal O_{Y'}\to\mathcal O_{X'})$, whose functoriality map, composed with the identification of (2), is the functoriality map $Lg^*L_{X/Y}\to L_{X'/Y'}$. $\square$

Proposition 1.6 (rank). Let $E$ be a $B$-module, placed in degree $0$, such that every object of $T$ has a covering on whose members $E$ is a direct summand of $B^r$. Then $E$ has perfect amplitude in $[0,0]$, and $c^n(E)=0$ for $n>r$.

Proof. $E$ is locally a direct summand of a finite free module, so it is flat and has perfect amplitude in $[0,0]$, and $L\Lambda^nE=\Lambda^nE$, which vanishes for $n>r$ because it vanishes locally, being a direct summand of $\Lambda^nB^r=0$. By Proposition 1.2 (3) the coefficient of $t^n$ in $c_t(\operatorname{at}(E))$ is a trace over the zero object, which is $0$, and $\sigma^S$ preserves degrees. $\square$

In the rest of this section, $B$ is a local ring: for every object $V$ of $T$ and $f\in B(V)$, $V$ has a covering on whose members $f$ or $1-f$ is invertible, and $B(V)$ is the zero ring only if $V$ is initial. These are the conditions of a locally ringed topos [Stacks, Tags 04ES, 04EU]. The second is stated there as: the subsheaf of the final object over which $0=1$ in $B$ is initial. The two forms agree, because a sheaf with a morphism to the initial sheaf is itself initial: the initial sheaf has no sections over an object unless the empty family covers it, and then one. Over an initial object every sheaf has exactly one section.

A finite free complex over an object $U$ is a bounded complex $S$ of $B|_U$-modules whose terms are finite free, $S^i\cong B^{r_i}|_U$; it is strictly perfect [Stacks, Tag 08FL], a strictly perfect complex being a bounded complex whose terms are direct summands of finite free modules. The Euler number of a finite free complex is $\chi(S)=\sum_i(-1)^ir_i$. By the following lemma it is well defined when $U$ is not initial.

Lemma 1.7. Let $U$ be an object of $T$ that is not initial.

Every covering of $U$ has a member that is not initial, and $B(U)\neq0$.
If $B^m|_U\cong B^n|_U$ as $B|_U$-modules, then $m=n$.
Every direct summand of $B^n|_U$ is locally free: $U$ has a covering on whose members it is isomorphic to some $B^k$.
If $S$ is an acyclic finite free complex over $U$, then $\chi(S)=0$.

Proof. (1) If every member $U_j$ of a covering were initial, then $B(U)\to\prod_jB(U_j)$ would be injective, with target the zero ring, so $B(U)=0$ and $U$ would be initial.

(2) Morphisms $B^m|_U\to B^n|_U$ are $n\times m$ matrices over $B(U)$, so the isomorphism and its inverse give $B(U)^m\cong B(U)^n$. Tensoring with the residue field of a maximal ideal of the nonzero ring $B(U)$ gives vector spaces of dimensions $m$ and $n$, so $m=n$.

(3) Let $P$ be a direct summand of $F=B^n|_U$, with complement $Q$, and $e\in M_n(B(U))$ the projection onto $P$ along $Q$. We argue by induction on $n$, the case $n=0$ being clear. Let $f_1,\ldots,f_n$ be the standard basis. On the members of a covering, $e_{11}$ or $1-e_{11}$ is invertible. Where $e_{11}$ is invertible, $v=e(f_1)\in P$ has invertible first coordinate, so $v,f_2,\ldots,f_n$ is a basis of $F$, its matrix having determinant $e_{11}$. Let $C$ be the span of $f_2,\ldots,f_n$. Since $Bv\subseteq P$ and $F=Bv\oplus C$, $P=Bv\oplus(P\cap C)$. Moreover $P\cap C\cong P/Bv$ is a direct summand of $F/Bv=P/Bv\oplus Q\cong C\cong B^{n-1}$, so it is locally free by induction, and so is $P$. Where $1-e_{11}$ is invertible, the same argument for $Q$, the image of $1-e$, gives $w=(1-e)(f_1)$ with $F=Bw\oplus C$ and $Bw\subseteq Q$. Then $P$ maps isomorphically onto a direct summand of $F/Bw=P\oplus Q/Bw\cong B^{n-1}$, and it is locally free by induction.

(4) Let $S$ have terms in degrees $[a,b]$, with $S^i\cong B^{r_i}$. If $a=b$, then $S^a=0$ and $r_a=0$ by (2). Let $a<b$. The differential $d\colon S^{b-1}\to S^b$ is an epimorphism, so on a covering the basis of $S^b$ lifts, which gives $s\colon S^b\to S^{b-1}$ with $ds=1$. Then $S^{b-1}=K\oplus s(S^b)$, with $K=\ker d$ a direct summand of $B^{r_{b-1}}$. By (3), on a further covering $K\cong B^k$. The complex with terms $S^i$ for $i<b-1$ and $K$ in degree $b-1$ is an acyclic finite free complex in degrees $[a,b-1]$. On a non-initial member of the covering, which exists by (1), $r_{b-1}=k+r_b$ by (2), and by induction $\sum_{i<b-1}(-1)^ir_i+(-1)^{b-1}k=0$. Hence $\chi(S)=0$. $\square$

Proposition 1.8 (the integral rank). Let $E\in D(B)$ have bounded perfect amplitude. There is a unique global section $\operatorname{rg}(E)$ of the constant sheaf $\mathbf Z$ on $T$ such that $\operatorname{rg}(E)|_U=\chi(S)$ for every object $U$ and every finite free complex $S$ over $U$ with $S\cong E|_U$. It depends only on the isomorphism class of $E$, and for every distinguished triangle $E'\to E\to E''\to E'[1]$ of objects of bounded perfect amplitude, $\operatorname{rg}(E)=\operatorname{rg}(E')+\operatorname{rg}(E'').$

Proof. By definition of bounded perfect amplitude, the final object has a covering on whose members $E$ is isomorphic to a strictly perfect complex. By Lemma 1.7 (3), applied to its finitely many terms, there is a finer covering $\{U_j\}$ with finite free complexes $S_j\cong E|_{U_j}$. Let $S$ over $U$ and $S'$ over $U'$ be two such representatives, and $W=U\times U'$. On $W$, the isomorphism $S|_W\cong E|_W\cong S'|_W$ of $D(B|_W)$ is given, on the members $W_k$ of a covering, by maps of complexes $\varphi_k\colon S|_{W_k}\to S'|_{W_k}$ [Stacks, Tag 08FR]. These are quasi-isomorphisms, so their cones are acyclic finite free complexes, with Euler numbers $\chi(S')-\chi(S)$ because the cone of $\varphi_k$ has the terms $S^{i+1}\oplus S'^i$. By Lemma 1.7 (4), $\chi(S)=\chi(S')$ on every non-initial $W_k$. On an initial object all sections of $\mathbf Z$ agree. So the sections $\chi(S_j)$ of $\mathbf Z$ over the $U_j$ agree on the overlaps and glue to a unique global section $\operatorname{rg}(E)$, with the stated property for every representative. A representative of $E$ is one of every object isomorphic to $E$.

Let $E'\to E\to E''\to E'[1]$ be a distinguished triangle. On the members of a covering of the final object, $E'$ and $E$ have finite free representatives $S'$ and $S$, and by [Stacks, Tag 08FR] the morphism $E'\to E$ is given by a map of complexes $\varphi\colon S'\to S$. The triangle $S'\to S\to\operatorname{cone}(\varphi)\to S'[1]$ is distinguished, so $E''$ is isomorphic there to $\operatorname{cone}(\varphi)$ [Tag 014A], a finite free complex with $\chi(\operatorname{cone}(\varphi))=\chi(S)-\chi(S')$. Hence $\operatorname{rg}(E'')=\operatorname{rg}(E)-\operatorname{rg}(E')$ on every member, and so globally. $\square$

Corollary 1.9. If $B$ is a local ring, then $\begin{aligned}[E]&\mapsto(\operatorname{rg}(E),c_t(E))\in\Gamma(T,\mathbf Z)\\&\qquad\times\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])^+\end{aligned}$ is a homomorphism of groups on the Grothendieck group $K(B)$ of Corollary 1.4, where $\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])^+$ is the multiplicative group of series with constant term $1$.

Proof. Proposition 1.8 and Corollary 1.4. $\square$

2. Chern classes in Hodge cohomology

Lemma 2.1. Let $V$ be a simplicial module and $n\geq1$. Then $\pi_0(\Lambda^nV)=\Lambda^n(\pi_0V)$ and $\pi_0(S^nV)=S^n(\pi_0V)$. Consequently, for a $B$-module $Q$ placed in degree $0$, $H^0(L\Lambda^nQ)=\Lambda^nQ$ and the truncation $L\Lambda^nQ\to H^0(L\Lambda^nQ)$ is a morphism $L\Lambda^nQ\to\Lambda^nQ$, compatible with the products.

Proof. $\pi_0$ of a simplicial module $W$ is the cokernel of $d_0-d_1\colon W_1\to W_0$. Let $N=(d_0-d_1)(V_1)\subseteq V_0$, so $\pi_0V=V_0/N$. By Lemma 1.3 of the lesson on filtered and graded powers, $\Lambda^nV_0\to\Lambda^n(V_0/N)$ is surjective with kernel $N\wedge\Lambda^{n-1}V_0$. For $w=v_1\wedge\cdots\wedge v_n$ with $v_i\in V_1$, $\begin{aligned}&d_0w-d_1w\\ &\quad=\sum_kd_1v_1\wedge\cdots\wedge d_1v_{k-1}\\ &\qquad\wedge(d_0-d_1)v_k\\ &\qquad\wedge d_0v_{k+1}\wedge\cdots\wedge d_0v_n,\end{aligned}$ which lies in $N\wedge\Lambda^{n-1}V_0$. Conversely, for $v\in V_1$ and $m_2,\dots,m_n\in V_0$, the element $w=v\wedge s_0m_2\wedge\cdots\wedge s_0m_n$ satisfies $d_0w-d_1w=$ $(d_0-d_1)v\wedge m_2\wedge\cdots\wedge m_n$, since $d_0s_0=d_1s_0=1$. So the image of $d_0-d_1$ on $\Lambda^nV_1$ is $N\wedge\Lambda^{n-1}V_0$, which gives the first statement; the argument for $S^n$ is the same. For $Q$, $L\Lambda^nQ$ is represented by $N\Lambda^n(V)$ for a simplicial resolution $V\to Q$ by flat modules (Theorem 2.6 of the lesson on filtered and graded powers), whose $H^0$ is $\pi_0\Lambda^n(V)=\Lambda^nQ$; the complex lies in degrees $\leq0$, so the truncation is a morphism to $H^0$. The products of $\Lambda(V)$ induce on $\pi_0$ the products of $\Lambda(Q)$. $\square$

Definition 2.2. The augmentation $\alpha\colon L\to\Omega$ and Lemma 2.1 give morphisms $L\Lambda^n(L)\to L\Lambda^n(\Omega)\to\Lambda^n\Omega$, compatible with the products, hence a ring homomorphism $\begin{aligned}&h\colon\operatorname{Hom}(B,L\Lambda^\bullet(L)[1])\\ &\quad\to\prod_{n\geq0}H^n(T,\Lambda^n\Omega),\end{aligned}\tag{2.1}$ the product on the right being that of $\operatorname{Hom}(B,\Lambda^\bullet(\Omega)[1])$, cup product followed by the exterior product. For $E\in D(B)$ of bounded perfect amplitude put $c'_t(E)=h(c_t(E))=\sum_nc'^n(E)t^n$, with $c'^n(E)\in H^n(T,\Lambda^n\Omega)$, the Chern classes of $E$ in Hodge cohomology.

Proposition 2.3.

$c'_t(E)$ is the image under the analogue of $h$ for $L\Lambda^\bullet(\Omega)$ of $\sigma^S\bigl(c_t(\operatorname{at}^\Omega(E))\bigr)$, the characteristic series of the principal parts class of Definition 4.1 of the lesson on the Atiyah class of filtered complexes.
(Whitney formula) For a distinguished triangle $E'\to E\to E''\to E'[1]$ of objects of bounded perfect amplitude, $c'_t(E)=c'_t(E')c'_t(E'')$; for $E\in DF(B)$ of bounded perfect amplitude, $c'_t(E)=\prod_pc'_t(\operatorname{gr}^pE)$.
(Functoriality) In the situation of Proposition 1.5 (3), $g^*c'_t(E,X/Y)=c'_t(Lg^*E,X'/Y')$.
If $E$ is as in Proposition 1.6, then $c'^n(E)=0$ for $n>r$.

Proof. (1) By Proposition 4.2 (2) of the lesson on the Atiyah class of filtered complexes, $\operatorname{at}^\Omega(E)=(1\otimes\alpha[1])\circ\operatorname{at}(E)$, so by Proposition 3.2 (1) of the lesson on L-series, $c_t(\operatorname{at}^\Omega(E))=\alpha_*c_t(\operatorname{at}(E))$, and $\sigma^S$ is natural in the coefficient object. (2), (4) Theorem 1.3, Proposition 1.6, and the fact that $h$ is a ring homomorphism. (3) Proposition 1.5 (3), the maps of Definition 2.2 being compatible with the functoriality maps of $L$ and of $\Omega$. $\square$

Proposition 2.4 (invertible modules). Let $E$ be a $B$-module such that every object of $T$ has a covering on whose members $E$ is isomorphic to $B$, and let $E^\vee=\mathcal Hom_B(E,B)$. Then $c'_t(E)=1+c'^1(E)t$, and $c'^1(E)\in H^1(T,\Omega)=\operatorname{Ext}^1_B(B,\Omega)$ is the negative of the class of the extension $0\to\Omega\to E^\vee\otimes_BP^1_{B/A}(E)\to B\to0$ obtained from the extension of principal parts $0\to\Omega\otimes_BE\to P^1_{B/A}(E)\to E\to0$ by tensoring with $E^\vee$ and identifying $E^\vee\otimes E$ with $B$ by evaluation. The local $B$-linear splittings of this extension correspond to the local connections on $E$ (Proposition 3.2 of the lesson on the Atiyah class). If $e$ is a local basis of $E$ and $\nabla_e$ the connection with $\nabla_e(be)=\mathrm db\otimes e$, then $\nabla_{ge}=\nabla_e-g^{-1}\mathrm dg$ for a local unit $g$ of $B$, as maps $E\to\Omega\otimes E$ with $g^{-1}\mathrm dg$ acting by multiplication.

Proof. $E$ is as in Proposition 1.6 with $r=1$, so $\Lambda^nE=0$ for $n\geq2$. Since $\Omega[1]\in D^{\leq0}(B)$, Corollary 3.7 of the lesson on L-series applies to $\operatorname{at}^\Omega(E)\colon E\to E\otimes\Omega[1]$ and gives $c_t(\operatorname{at}^\Omega(E))=1-\operatorname{Tr}(\psi^1\circ\operatorname{at}^\Omega(E))t$, with $\psi^1$ the identity. By Proposition 2.3 (1), $c'_t(E)=1+c'^1(E)t$ with $c'^1(E)=-\operatorname{Tr}(\operatorname{at}^\Omega(E))$, because $\sigma^S$ is the identity in degree $1$ (Exercise 4.4 of the lesson on Koszul complexes) and the truncation $L\Lambda^1\Omega=\Omega\to\Lambda^1\Omega$ is the identity. The module $E$ is flat and in degree $0$, so $\kappa_E$ is the symmetry $\Omega\otimes E\to E\otimes\Omega$, without sign, and $\operatorname{at}^\Omega(E)$ is the class of the extension of principal parts of $E$ (Definition 4.1 of the lesson on the Atiyah class of filtered complexes, with $F=E$). With the dual $E^\vee$, the coevaluation $\eta$ and the evaluation $\varepsilon\colon E^\vee\otimes E\to B$, an isomorphism, the trace of Definition 1.1 of the lesson on traces is $\operatorname{Tr}(g)=(\varepsilon\otimes1)(1\otimes g)(\sigma\otimes1)(\eta\otimes1)$; here $\varepsilon\sigma\eta=\operatorname{Tr}(1_E)=1$, checked locally where $E\cong B$, so $\sigma\eta=\varepsilon^{-1}$ and $\operatorname{Tr}(g)=(\varepsilon\otimes1)(1\otimes g)\varepsilon^{-1}$. Applied to $g=\operatorname{at}^\Omega(E)$ this is the class of the twisted extension. For the last assertion, $\nabla_{ge}-\nabla_e$ is $B$-linear, $\nabla_e(e)=0$, and $\nabla_{ge}(e)=\nabla_{ge}(g^{-1}\cdot ge)=$ $\mathrm d(g^{-1})\otimes ge=-g^{-1}\mathrm dg\otimes e$. $\square$

Corollary 2.5 (invertible modules on a smooth scheme). Let $Y\to S$ be a smooth morphism of schemes, $T$ the Zariski topos of $Y$, $A=f^{-1}\mathcal O_S\to B=\mathcal O_Y$, and $L$ an invertible $\mathcal O_Y$-module. Then $c'^1(L)=-c_1^{\mathrm{Hodge}}(L)$ in $H^1(Y,\Omega_{Y/S})$, where $c_1^{\mathrm{Hodge}}(L)$ is the image of the class of $L$ in $H^1(Y,\mathcal O_Y^*)$ [Tag 09NU] under $\mathrm d\log\colon g\mapsto g^{-1}\mathrm dg$ [Tag 0FLE].

Proof. Choose an open covering by the $U_i$, with bases $s_i$ of $L$ over $U_i$, and put $f_{i_0i_1}=s_{i_1}s_{i_0}^{-1}$; then $c_1^{\mathrm{Hodge}}(L)$ is the image of the Čech class of $(\mathrm d\log f_{i_0i_1})$ [Tag 0FLE]. By Proposition 2.4, $-c'^1(L)$ is the class of the twisted extension $0\to\Omega\to L^\vee\otimes P^1_{B/A}(L)\to\mathcal O_Y\to0$, that is, the boundary of $1\in H^0(Y,\mathcal O_Y)$ in $H^1(Y,\Omega)$. Write $\nabla_i=\nabla_{s_i}$ and $\sigma_i=\mathrm d^1-\iota\nabla_i$ for the corresponding section of the extension of principal parts (Proposition 3.2 of the lesson on the Atiyah class). For a local basis $s$ of $L$ over $U_i$, the element $s^\vee\otimes\sigma_i(s)$, with $s^\vee$ the dual basis, does not depend on $s$, as $\sigma_i$ is $\mathcal O_Y$-linear; it lifts $1$ over $U_i$. By Proposition 2.4, $\nabla_{i_1}=\nabla_{i_0}-\mathrm d\log f_{i_0i_1}$ over $U_{i_0i_1}$. So the difference of the lifts over $U_{i_0i_1}$ is $\begin{aligned}&s^\vee\otimes(\sigma_{i_1}-\sigma_{i_0})(s)\\&\quad=s^\vee\otimes\iota(\mathrm d\log f_{i_0i_1}\otimes s),\end{aligned}$ which is $\mathrm d\log f_{i_0i_1}$ under $L^\vee\otimes\Omega\otimes L\cong\Omega$. By Lemma 3.7 of the lesson on deformations of modules, the boundary of $1$ is the image of the Čech class of $(\mathrm d\log f_{i_0i_1})$, which is $c_1^{\mathrm{Hodge}}(L)$. $\square$

3. Exercises

Exercise 3.1. Show that $c_t(B^r)=1$ and $c_t(E\oplus B^r)=c_t(E)$.

Exercise 3.2. Let $T$ be the punctual topos. Show that $c_t(E)=1$ for every $E$ of bounded perfect amplitude, although $\operatorname{at}(E)$ need not be $0$ (Exercise 5.1 of the lesson on the Atiyah class of filtered complexes).

Exercise 3.3. Let $T$ be the Zariski topos of the projective line over a field $k$, $A=k$, $B=\mathcal O$, and $E=\mathcal O(1)$, with bases $e_0$ on $U_0=\operatorname{Spec}k[x]$ and $e_1=xe_0$ on $U_1=\operatorname{Spec}k[y]$, $y=1/x$. Show that $E$ has no global connection, and deduce that $c'^1(E)\ne0$.

Exercise 3.4. Let $E$ be of bounded perfect amplitude. Show that $c_t(E[1])=L_t(E)$ and $c_t(E\oplus E[1])=1$.

4. Complete solutions

Solution 3.1. $\operatorname{at}(B^r)=0$ (Proposition 1.2 (4) of the lesson on the Atiyah class of filtered complexes, with $N=A^r$), so every $\varphi^n\circ LS^n(\operatorname{at}(B^r))$ is $0$ for $n\geq1$ and $L_t(B^r)=1$. By Theorem 1.3 (2), $c_t(E\oplus B^r)=c_t(E)c_t(B^r)=c_t(E)$.

Solution 3.2. On the punctual topos $H^n(T,X)=H^n(X)$, the cohomology of the complex in degree $n$. Since $L$ is concentrated in degrees $\leq0$, so is $L\Lambda^nL$ (Theorem 2.6 of the lesson on filtered and graded powers), and $H^n(T,L\Lambda^nL)=0$ for $n\geq1$. So every $c^n(E)$ with $n\geq1$ vanishes and $c_t(E)=1$. In Exercise 5.1 there, $\operatorname{at}(k)\ne0$ for $B=k[x]$.

Solution 3.3. A connection is determined by $\nabla(e_0)=a\,\mathrm dx\otimes e_0$ on $U_0$ and $\nabla(e_1)=b\,\mathrm dy\otimes e_1$ on $U_1$, with $a\in k[x]$, $b\in k[y]$. On $U_0\cap U_1$, $\nabla(e_1)=\nabla(xe_0)=$ $\mathrm dx\otimes e_0+x\,a\,\mathrm dx\otimes e_0=$ $(x^{-1}+a)\,\mathrm dx\otimes e_1$, and $\mathrm dy=-x^{-2}\mathrm dx$, so $x^{-1}+a(x)=-x^{-2}b(x^{-1})$, that is $x+x^2a(x)=-b(x^{-1})$ in $k[x,x^{-1}]$. The left side is a polynomial in $x$ without constant term and the right side a polynomial in $x^{-1}$, so both are constant, hence $0$; but $x+x^2a(x)\ne0$. So $E$ has no global connection, the extension of principal parts does not split (Proposition 3.2 of the lesson on the Atiyah class), and neither does its twist by $E^\vee$, whose splittings are the same. Its class in $H^1(T,\Omega)$ is therefore nonzero, and $c'^1(E)\ne0$ by Proposition 2.4.

Solution 3.4. Proposition 1.2 (2) gives $L_t(E[1])=c_t(E)$, so $c_t(E[1])=L_t(E[1])^{-1}=L_t(E)$. By Theorem 1.3 (2), $c_t(E\oplus E[1])=c_t(E)L_t(E)=1$.

Sources

The Chern classes of perfect complexes with values in the cotangent complex, their multiplicativity on filtrations and triangles, their base change and the Hodge-cohomology classes are due to Illusie, in his work on the cotangent complex, where the homomorphism of Corollary 1.9 is stated with the rank of perfect complexes over a local ring, after SGA 6, and where the question whether it is a homomorphism of pre-λ-rings is left open. Chern classes in Hodge cohomology through the Atiyah class go back to Atiyah, for holomorphic vector bundles.