\[
  c(E) - c(F) - c\bigl(\underline{\mathcal{O}}_{X}(E)/f\,\underline{\mathcal{O}}_{X}(F)\bigr),
\]

\[
  \mathcal{D} + N = E_{0}, \qquad \mathcal{D} \cap N = \mathcal{L},
  \qquad \struck{\Phi + N = E}, \qquad \Phi \cap N = \Phi_{N} .
\]

\begin{equation*}
  \int(F) \;=\; K_{1}\!\left( c(F) \cdot \left(1 - \tfrac{1}{2}K\right) \right)
  \qquad (K \in \mathfrak{P}) \tag{1}
\end{equation*}

\[
  \left.
  \begin{array}{l}
    \text{si } x \in I(X),\ \ill{} \\[2pt]
    \gamma^{i}(x) = 0 \quad \text{pour } i \geqslant 2
  \end{array}
  \right\}
  \qquad F^{i}_{\gamma}(K^{\bullet}(X)) = 0 \ \text{si } i \geqslant 2 .
\]

\[
  K^{\bullet}(X) \simeq \mathbb{Z}^{\pi_{0}(X)} \times K^{(1)}(X),
  \qquad
  K^{(1)}(X) = \mathrm{Pic}(X) \simeq \prod_{i \in \pi_{0}(X)} \mathrm{Pic}(X_{i}),
\]

\[
  \mathrm{cl}(E) = \bigl[\mathrm{rg}(E),\, c^{1}(E)\bigr]
  \qquad \text{où } c^{1}(E) = \det(E),
\]

\[
  \Bigl[\ \text{i.e.}\quad
  \mathrm{cl}(E) = \sum_{i} \mathrm{rg}_{i}(E)\cdot 1_{i} + \bigl[\mathrm{cl}(\det E) - 1\bigr]
  = \sum_{i} \bigl(\mathrm{rg}_{i}(E) - 1\bigr) 1_{i} + \mathrm{cl}(\det E)\ \Bigr]
\]

\[
  \mathbb{Z}^{k^{(0)}(X)} \times \mathrm{Pic}'(X)
  \;=\; \mathbb{Z}^{k^{(0)}(X)} \times \prod_{i \in \pi_{0}^{(1)}(X)} K_{0}(X_{i})
\]

\[
  \mathrm{lg}(1) = (\mu_{i}) \in \mathbb{Z}
  \qquad \text{seul invariant de la situation}
\]

\[
  K^{\bullet}(X) \xrightarrow{\ \mathrm{rg}\ } \mathbb{Z},
  \qquad
  K_{\bullet}(X) \xrightarrow{\ \mathrm{lg}\ } \mathbb{Z}
\]

\[
  K^{\bullet}(X) = \mathbb{Z} \times K^{(1)}
  \xrightarrow[\ = \mathrm{id}\ ]{\ c^{\bullet} = \mathrm{ch}^{\bullet}\ }
  B^{\bullet}(X) \simeq \mathbb{Z} \times K^{(1)} ,
\]

\[
  K_{\bullet}(X) = \mathbb{Z}^{K} \times K_{(0)}
  \ \struck{\ill{}} \ = \ B_{\bullet}(X) \simeq \mathbb{Z}^{K} \times K_{(0)} .
\]

\[
  \mathbb{Z}^{K} \longrightarrow K_{(0)},
  \qquad \struck{\ill{}} \qquad
  \xi \rightsquigarrow \bigl(\uncertain{\lambda_{i}}\, u_{i}(\xi)\bigr)_{i \in K} .
\]

\[
  1_{\bullet} = \sum \mu_{i}\, 1_{i} + \delta_{\varepsilon} .
\]

\[
  u(n, \xi) \;=\; \sum n\,\mu_{i}\, 1_{i}
  \;+\; \Bigl( n\,\delta_{\varepsilon} + \sum \mu_{i}\, u_{i}(\xi) \Bigr),
  \qquad n \in \mathbb{Z},\ \xi \in K^{(1)} .
\]

\[
  X \xrightarrow{\ i\ } X \times \mathbb{P}^{r},
  \qquad F \otimes L .
\]

\[
  \underline{K}(X \times \mathbb{P}^{r})
  = \underline{K}(X) \otimes \underline{K}(\mathbb{P}^{r}) \big/ \bigl(\xi = 1 - L^{-1}\bigr) .
\]

\[
  i_{!}(x) = p^{!}(x) \otimes q^{!}(\xi^{r}) = x \otimes \xi^{r},
\]

\[
  \widetilde{C}(i_{!}(x)) = \widetilde{C}(x \otimes \xi^{r})
  = \widetilde{C}(x) * \widetilde{C}(\xi^{r}) .
\]

\[
  \xi^{r} = \sum_{i=0}^{r} \binom{r}{i} (-1)^{i} L^{-i},
  \qquad
  \widetilde{C}(\xi^{r}) = \prod_{i=0}^{r} (1 - i\eta)^{(-1)^{i} \binom{r}{i}} .
\]

\[
  \widetilde{C}(i_{!}(x))
  = \prod_{i=0}^{r} \Bigl[\, \widetilde{C}(x) * (1 - i\eta) \,\Bigr]^{(-1)^{i} \binom{r}{i}}
\]

\[
  \struck{= \prod_{i=0}^{r} \Bigl[ (1 - i\xi)^{\varepsilon(x)} \sum_{i=0}^{\infty} C^{i}(x) \Bigr]}
\]

\[
  \widetilde{C}(x) * (1 - i\eta)
  = \widetilde{C}(x) * \bigl[1,\ 1 - i\eta\bigr] - \widetilde{C}(x)
  = \Bigl[\ \varepsilon(x),\ 1 + \cdots + \sum_{j=0}^{k} C^{j}(x)\,(-i\eta)^{k-j}\ \Bigr]
\]

\[
  \widetilde{C}(\xi^{r}) = 1 + (-1)^{r+1}(r-1)!\ \eta^{r},
\]

\[
  \widetilde{C}(i_{!}(x)) = \widetilde{C}(x) * \Bigl(1 + (-1)^{r+1}(r-1)!\ \eta^{r}\Bigr)
\]

\[
  = \Bigl(1 + (-1)^{r+1}(r-1)!\ \eta^{r}\Bigr)^{\varepsilon(x)}
    \Bigl( C(x) * \bigl(1 + (-1)^{r+1}(r-1)!\ \eta^{r}\bigr) \Bigr)
\]

\[
  = 1 + \sum_{i=1}^{\infty} Q_{i}\bigl(c^{1}(x), \ldots, c^{i}(x);\ 0, \ldots,
    \uncertain{\alpha^{(i/r)}}, 0, \ldots, 0\bigr)
\]

\[
  Q_{i}(\cdots) =
  \begin{cases}
    0 & \text{si } i < r, \\[4pt]
    (-1)^{r-1}(r-1)!\ Q^{(r)}_{i-r}\bigl(c^{1}(x), \ldots, c^{i}(x)\bigr) &
  \end{cases}
\]

\[
  \widetilde{C}(i_{!}(x)) = 1 + (-1)^{r-1}(r-1)!\ \eta^{r}
  \Bigl[\ \varepsilon(x) + \sum_{i=0}^{\infty} Q^{(r)}_{i}\bigl(c^{1}(x), \ldots, c^{i}(x)\bigr) \Bigr]
\]

\[
  (i_{*})_{r}\Bigl[\ \eta \sum_{0}^{\infty} x^{i} \Bigr]
  = 1 + (-1)^{r-1}(r-1)!\ \eta^{r}
    \Bigl[\ 1 + \sum_{i=0}^{\infty} Q^{(r)}_{i}\bigl(x^{i}, \ldots, x^{i}\bigr) \Bigr]
\]

\[
  Q^{(r)}_{1}(c^{1}) - \struck{\ill{}}
  = -\ \frac{(r+1-1)!}{(r-1)!\,(1-1)!}\ c^{1} = -\,r\,c^{1} .
\]

\[
  \widetilde{C}_{X}(Y) = \bigl(1 - \struck{c^{1}(x)}\ x'\bigr)^{-1} = \sum x'^{\,n} ,
\]

\[
  \widetilde{C}_{X \times \mathbb{P}^{r}}\bigl(i_{!}(\underline{\mathcal{O}}_{Y})\bigr)
  = \widetilde{C}_{X \times \mathbb{P}^{r}}(Y \times \mathbb{P}^{r})
  = 1 + (-1)^{r-1}(r-1)! \sum_{i=1}^{\infty}
      Q^{(r)}_{i}\bigl(x',\, (x')^{2}, \ldots, (x')^{i}\bigr)
\]

\[
  = 1 + (-1)^{r-1}(r-1)!\ \eta^{r} \sum_{i=1}^{\infty}
      Q^{(r)}_{i}\bigl(x',\, (x')^{2}, \ldots, (x')^{i}\bigr)
\]

\[
  A(\mathcal{X}, G)
\]

\[
  u^{*} : A(\mathcal{X}, G') \longrightarrow A(\mathcal{X}, G)
\]

\[
  u_{*} : A(\mathcal{X}, G) \longrightarrow A(\mathcal{X}, G') .
\]

\[
  u^{*} u_{*}(F) \simeq F^{(G'/G)} \quad \ill{}
\]

\[
  u^{*}(F' \otimes G') \simeq u^{*}(F') \otimes u^{*}(G')
\]

\[
  u_{*}\bigl(F \otimes u^{*}(G')\bigr) \simeq u_{*}(F) \otimes G' .
\]

\[
  F \in \mathrm{Ob}\ A(\mathcal{X}, G), \qquad G \in \mathrm{Ob}\ A(\mathcal{X}, H),
\]

\[
  F \boxtimes G = \mathrm{pr}_{1}^{*}(F) \otimes \mathrm{pr}_{2}^{*}(G)
  \ \in\ \mathrm{Ob}\ A(\mathcal{X}, H \times G) .
\]

\[
  u^{*}(F \boxtimes G) \simeq u^{*}(F) \boxtimes u^{*}(G),
\]

\[
  F \otimes G \ \rightsquigarrow \ = (\mathrm{diag}_{G})^{*}(F \boxtimes G) .
\]

\[
  u_{pq} : \mathfrak{S}_{p} \times \mathfrak{S}_{q} \longrightarrow \mathfrak{S}_{p+q}
\]

\[
  F * G = u_{pq\,*}\bigl(F \boxtimes G\bigr) \ \in\ \mathrm{Ob}\ A(\mathcal{X}, p+q) .
\]

\[
  T^{n}(F + G) \simeq \coprod_{p+q=n} T^{p}(F) * T^{q}(G),
\]

\[
  T^{m}\bigl(T^{n}(F)\bigr) \simeq w_{m,n}^{*}\bigl(T^{mn}(F)\bigr),
\]

\[
  T^{m}(F \otimes G) \simeq T^{m}(F) \otimes T^{m}(G)
\]

\[
  \bigl[\ w_{m,n} : \mathfrak{S}_{m} \longrightarrow \mathfrak{S}_{\ill{}}
  \ \ill{} \text{convenable} \ \bigr]
\]

\[
  u_{pq\,*}\bigl(F_{p_{1} p_{2} \ldots p_{r}}\bigr)
  \qquad \Bigl(\sum p_{i} = n,\ \ill{},\ p_{i} \geqslant 0,\ r \in \mathbb{N}\Bigr)
\]

\[
  \lambda^{i}(X_{s}) = 0 \ \text{si } i > \varepsilon(s),
  \qquad \lambda^{\varepsilon(s)}(X_{s}) \ \text{inversible} .
\]

\[
  B = \mathbb{Z}\Bigl[\bigl(\lambda^{i}(X_{s})\bigr)_{s \in S,\ 1 \leqslant i \leqslant \varepsilon(s)}\Bigr]
      \Bigl[\bigl(\lambda^{\varepsilon(s)}(X_{s})\bigr)^{-1}\Bigr] .
\]

\[
  CB = \mathbb{Z}\Bigl[\bigl(C^{i}_{s}\bigr)_{\struck{\ill{}}}\Bigr],
  \qquad \ill{} \quad
  \widetilde{C}(X_{s}) = \Bigl[\varepsilon(s),\ 1 + C^{1}_{s} \cdots \ill{}\Bigr] .
\]

\[
  c_{B}^{i}(\xi^{i}) \;=\; C^{i}_{B}(P) ,
\]

\[
  \varphi_{B}\psi_{B}(\xi^{i}) \;=\; \varphi_{B}\bigl(C^{i}_{B}(P)\bigr)
  \;=\; c^{i}_{B}(P) \;\equiv\; \struck{c^{i}_{B}}\ \struck{dans}\
  \gamma^{i}\bigl(P - \varepsilon(P)\bigr) \pmod{B^{(i+1)}} ,
\]

\[
  \gamma^{i}\bigl(P - \varepsilon(P)\bigr) \;-\; (-1)^{i-1}(i-1)!\,P
  \;\in\; B^{(i+1)} ,
\]

\[
  \varphi_{B}\psi_{B} \;=\; (-1)^{i-1}(i-1)! ,
\]

\[
  CK \;=\; \mathbb{Z}[\xi] \big/ \bigl(\text{idéal engendré par } \ill{}\bigr),
  \qquad
  (1+\xi)^{\ill{}} \;=\; 1 + (-1)^{n-1}(n-1)!\,\xi^{n} + \cdots
\]

\[
  B \;=\; \mathbb{Z}\bigl[\,(\lambda^{i}(X_{s}))_{s \in S,\; 1 \leqslant i \leqslant \varepsilon(s)},\;
  (\lambda^{d}(Y_{t}))_{t \in T,\; d \geqslant 1}\,\bigr] ,
\]

\[
  CB \;=\; \mathbb{Z}\bigl[\,(\xi^{i}_{s})_{s \in S,\; 1 \leqslant i \leqslant \varepsilon(s)},\;
  (\eta^{d}_{t})_{t \in T,\; d \geqslant 1}\,\bigr] .
\]

\[
  \gamma^{i}\bigl(X_{s} - \varepsilon(X_{s})\bigr), \qquad
  \gamma^{d}\bigl(Y_{t} - \varepsilon(Y_{t})\bigr) ,
\]

\[
  x \;\longmapsto\; \widetilde{c}(x) \;=\;
  \bigl[\,\varepsilon(x),\; 1 + c^{1}(x)\,t + \cdots \,\bigr]
\]

\[
  \widetilde{c}(x+y) \;=\; \widetilde{c}(x)\,\widetilde{c}(y)
\]

\[
  \widetilde{c}(xy) \;=\; \widetilde{c}(x) \ast \widetilde{c}(y)
\]

\[
  \widetilde{c}(xy) \;=\; Q_{i}\bigl(c^{1}(x), \ldots, c^{i}(x);\;
  c^{1}(y), \ldots, c^{i}(y)\bigr) \qquad \text{soit} \ \ill{}
\]

\[
  \gamma^{i}(xy) \;=\; Q_{i}\bigl(\gamma^{1}(x), \ldots, \gamma^{i}(x);\;
  \gamma^{1}(y), \ldots, \gamma^{i}(y)\bigr) .
\]

\[
  \gamma^{i}\bigl(x - \varepsilon(x)\bigr) \;=\; 0 \qquad \text{pour } i > n ,
\]

\[
  (i-1)!\;x^{i} \;=\; 0 \qquad \text{pour } i > n ,
\]

\[
  D^{i}_{t}\bigl(\lambda_{t}(X)\,\lambda_{t}(Y)\bigr)\big|_{t=1} \;=\; 0
  \qquad \text{pour } 0 \leqslant i \leqslant m .
\]

\[
  \varepsilon : K \longrightarrow \mathbb{Z} ,
\]

\[
  \begin{cases}
    \varepsilon(E_{s}) = n_{s} \geqslant 0, \\[2pt]
    \lambda^{i} E_{s} = 0 & \text{si } i > n_{s}, \\[2pt]
    \lambda^{n_{s}} E_{s} \text{ inversible},
  \end{cases}
\]

\[
  K \;\xrightarrow{\ \text{dét}\ }\; K^{*}
\]

\[
  \text{dét}(E_{s}) \;=\; \lambda^{n_{s}}(E_{s}) .
\]

\[
  c : K \longrightarrow \widetilde{A}
\]

\[
  c^{i}(E_{s}) \;=\; 0 \quad \text{pour } i > n_{s} ,
  \qquad
  c^{1}(E) \;=\; c^{1}(\text{dét } E) .
\]

\[
  c^{i}(x) \qquad i \geqslant 1
\]

\[
  \begin{cases}
    c^{i}(1) = 0 & i \geqslant 1, \\[3pt]
    c^{n}(x+y) = \displaystyle\sum_{i+j=n} c^{i}(x)\,c^{j}(y)
      & x, y \in K,\ n \geqslant 1, \\[8pt]
    c^{n}(x \ast y) = Q_{n}\bigl(c^{1}(x), \ldots, c^{n}(x);\;
      c^{1}(y), \ldots, c^{n}(y)\bigr)
      & x, y \in I(K),\ n \geqslant 1, \\[5pt]
    c^{n}(E_{s}) = 0 & n > n_{s},
  \end{cases}
\]

\[
  \gamma^{n}(E_{s} - n_{s}) \;=\; 0 \qquad \text{pour } n > n_{s} ,
\]

\[
  \varphi_{K} : CK \longrightarrow GK, \qquad
  c^{i}(x) \;\rightsquigarrow\; \gamma^{i}\bigl(x - \varepsilon(x)\bigr)
  \bmod K^{(i+1)} ,
\]

\[
  \chi : K \longrightarrow \widetilde{A}
\]

\[
  \psi : \mathrm{Gr}\,K \longrightarrow \struck{A}\ G(\widetilde{A}) \simeq A\,!
\]

\[
  \psi : \mathrm{Gr}\,K \longrightarrow \ill{} ,
  \qquad
  K \longrightarrow CK\,!
\]

\[
  D_{\mathrm{parf},\Phi}(X), \quad K_{\Phi}(X), \qquad
  D_{\mathrm{parf},\Phi}(X) = \varinjlim_{Z \in \Phi} D_{\mathrm{parf},Z}(X),
  \quad
  K_{\Phi}(X) = \varinjlim_{Z \in \Phi} K_{Z}(X) .
\]

\[
  R\,i_{*} : D_{\mathrm{parf}}(\ill{}) \longrightarrow D_{Z}(X),
  \qquad
  i_{!} : K_{\ill{}}(\ill{}) \longrightarrow K'_{Z}(X) .
\]

\[
  \varinjlim_{n} D_{\mathrm{parf}}(i_{n}) \longrightarrow D_{Z}(X) ,
\]

\[
  \struck{A}\ A_{\Phi}(X)
\]

\[
  c^{i}_{\Phi} : K_{\Phi}(X) \longrightarrow A_{\Phi}(X) \qquad [\,i > 0\,]
\]

\[
  c^{n}_{\Phi}(x+y) \;=\; c^{n}_{\Phi}(x) + c^{n}_{\Phi}(y)
  \;+\; \sum_{\substack{p+q=n \\ p,\,q > 0}} c^{p}_{\Phi}(x)\,c^{q}_{\Phi}(y)
  \qquad \bigl(x, y \in K_{\Phi}(X)\bigr)
\]

\[
  K^{\Phi}_{\ill{}}(X/S) \;=\; K^{\Phi}_{\ill{}}(f),
  \qquad
  A^{\Phi}_{\ill{}}(X/S) \;=\; A^{\Phi}_{\ill{}}(f),
\]

\[
  \operatorname{ch}^{X,\Phi}_{\ill{}} : K_{\ill{}}(X/S) \longrightarrow
  A_{\ill{}}(X,S),
  \qquad
  K^{\Phi}_{\ill{}}(X/S), \quad A^{\Phi}_{\ill{}}(X/S) \ \text{covariants} ,
\]

\[
  \operatorname{ch}^{Y/S}_{\ill{}}\bigl(g_{\ill{}}(x)\bigr)
  \;=\; g_{*}\bigl(\operatorname{ch}^{X/S}_{\ill{}}(x)\bigr) ,
\]

\[
  \bigl[\,X/S \ \text{parfait} \ \ill{}\,\bigr]
\]

\[
  K^{\bullet}(X) \longrightarrow K_{\ill{}}(X/\Sigma) ,
\]

\[
  \varinjlim_{n} \struck{D^{\bullet}(i_{n})} \ill{}
\]

\[
  \operatorname{ch}\bigl(i_{!}\,x\bigr)\,\operatorname{Todd}(X'/S)
  \qquad X'/S
\]

\[
  \tau_{h}(Y)\,\varphi_{Y}\bigl(f_{!}(\xi)\bigr)
  \;=\; f_{*}\bigl(\tau_{h}(X)\,\varphi_{X}(\xi)\bigr)
  \tag{1}
\]

\[
  c^{d}_{Y}\bigl(f_{!}(\xi)\bigr) \;=\; f_{*}\bigl[\,c^{\ill{}}(\ill{})\,\bigr]
\]

\[
  \ill{}
  \;=\; f_{*}\left[\,
  \frac{c^{q+d}\bigl(\struck{\ill{}}\ \widetilde{C}(\xi)
    \ast \lambda_{-1}\bigl(\widetilde{C}(T_{Y/X})\bigr)\bigr)}
  {(-1)^{q}\,c^{q}_{Y}(T_{Y/X})}\,\right]
  \tag{2}
\]

\[
  \varphi_{Y}\bigl(f_{!}(\xi)\bigr)
  \;=\; f_{*}\Bigl(\varphi_{X}(\xi)\,\tau_{h}(T_{Y/X})^{-1}\Bigr)
  \tag{1 bis}
\]

\[
  f_{!}\bigl(\tau_{X}(F)\bigr) \;=\; \tau_{Y}(F) .
\]

\[
  f_{!}\bigl(\tau_{X}(E)\bigr) \;=\; \sum_{i} (-1)^{i}\,\tau_{Y}\bigl(E^{(i)}\bigr) ,
\]

\[
  \varphi_{Y}\bigl(f_{!}(\tau_{X}(E))\bigr)
  \;=\; \sum_{i} (-1)^{i}\,\varphi_{Y}\bigl(E^{(i)}\bigr) .
\]

\[
  \tau_{h}(Y)\sum_{i}(-1)^{i}\,\varphi_{Y}\bigl(E^{(i)}\bigr)
  \;=\; f_{*}\bigl(\tau_{h}(X)\cdot\varphi_{X}(E)\bigr)
  \tag{1 ter}
\]

\[
  \sum_{i}(-1)^{i}\,\mathrm{rg}\,E^{(i)} \;=\; \chi(Z;\,E|Z)
\]

\[
  K_{p}\Bigl(i^{*}\bigl[\bigl(\tau_{h}(X)\cdot\varphi_{X}(E)\bigr)^{(p)}\bigr]\Bigr),
\]

\begin{align*}
  i^{*}[\;\;] &= i^{*}\bigl(\tau_{h}(X)\bigr)\,\struck{\ill{}}\cdot
                 i^{*}\bigl(\varphi_{X}(E)\bigr) \\
              &= \tau_{h}\bigl(C_{Z}(i^{*}(T_{X}))\bigr)\,
                 \varphi_{Z}\bigl(i^{*}(E)\bigr) \\
              &= \tau_{h}\bigl(C_{Z}(T_{Z})\bigr)\,\varphi_{Z}(E|Z)
               \;=\; \tau_{h}(Z)\,\varphi_{Z}(E|Z) .
\end{align*}

\[
  \varphi_{X}\bigl(E(n)\bigr) \;=\; \varphi_{X}(E)\,\struck{\ill{}}\,e^{n\xi} .
\]

\[
  \tau_{h}(X)\,\varphi_{X}\bigl(E(n)\bigr)
  \;=\; \tau_{h}(X)\,\varphi_{X}(E)\,e^{n\xi}
  \;=\; \sum_{k} n^{k}\left[\frac{\xi^{k}}{k!}\,\tau_{h}(X)\,\varphi_{X}(E)\right]
\]

\[
  f_{*}\bigl(\tau_{h}(X)\,\varphi_{X}(E(n))\bigr)
  \;=\; \sum_{k} n^{k}\, f_{*}\!\left(\frac{\xi^{k}}{k!}\,\tau_{h}(X)\,\varphi_{X}(E)\right)
\]

\[
  K(\ill{}) \;\simeq\; \mathbf{Z} + P(\ill{}),
\]

\[
  \varphi_{X}(\xi) = n + d,
  \qquad \widetilde{C}(T_{X}) = (1, -k),
\]

\[
  \tau_{h}(X) \;=\; \frac{-k}{1 - e^{k}}
  \;=\; 1 - \tfrac{1}{2}k,
\]

\[
  \varphi_{X}(\xi)\,\tau_{h}(X)
  \;=\; \bigl(1 - \tfrac{1}{2}k\bigr)(n + d)
  \;=\; \struck{\ill{}}\;\; n + \bigl(d - n\tfrac{1}{2}k\bigr)
\]

\[
  f_{*}\bigl(\varphi_{X}(\xi)\,\tau_{h}(X)\bigr)
  \;=\; n\delta + \bigl(f_{*}(d) - n f_{*}(\tfrac{1}{2}k)\bigr)
\]

\begin{align*}
  \varphi_{X}\bigl(f_{!}(\xi)\bigr)
    &= \tau_{h}(Y)^{-1} f_{*}(\cdot\;\ill{})
     = \bigl(1 + \tfrac{1}{2}k'\bigr)
       \Bigl(n\delta + \bigl(f_{*}(d) - n f_{*}(\tfrac{1}{2}k)\bigr)\Bigr) \\
    &= n\delta + \Bigl[n\delta\tfrac{1}{2}k' + f_{*}(d)
       - n f_{*}(\tfrac{1}{2}k)\Bigr]
\end{align*}

\[
  f_{!}(\xi) \;=\; \Bigl(n\delta,\;\;
    n\delta\tfrac{1}{2}k_{y} + f_{*}(d)
    - n f_{*}\bigl(\tfrac{1}{2}k_{x}\bigr)\Bigr)
  \qquad (f : X \to Y)
\]

\[
  \tfrac{1}{2}\,\struck{\ill{}}\bigl(\delta k_{Y} - f_{*}(k_{X})\bigr) .
\]

\[
  c(T_{Y}) \;=\; 1 - k_{Y}^{1} + \ill{}
\]

\[
  \tau_{h}(Y) \;=\; 1 - \tfrac{1}{2}k_{Y}^{1} + \struck{\ill{}}\;\ill{},
\]

\[
  \tau_{h}(Y)^{-1} \;=\; 1 + \tfrac{1}{2}k_{Y}^{1} + \struck{\ill{}}
  + \tfrac{1}{4}k_{Y}^{1\,2}
  \;=\; 1 + \tfrac{1}{2}k_{Y}^{1}
  + \tfrac{1}{12}\bigl[\,k_{Y}^{2}\;\ill{}\;k_{Y}^{1\,2}\,\bigr]
\]

\[
  f_{*}(1) \;=\; \delta f(X) \;=\; \delta\,\struck{\ill{}}\;C
\]

\[
  \varphi_{Y}\bigl(f_{!}(\xi)\bigr)
  \;=\; \Bigl[\,1 + \tfrac{1}{2}k_{Y} + \struck{\ill{}}\,\Bigr]
        \Bigl[\,\struck{\ill{}}\;\; n\delta C
          + \bigl(f_{*}(d) - \tfrac{1}{2}n\,f_{*}(k_{x})\bigr)\Bigr]
\]

\[
  \varphi_{Y}\bigl(f_{!}(\xi)\bigr) \;=\; n\delta C
  + \Bigl[\,f_{*}(d) - \tfrac{1}{2}n\,f_{*}(k_{x})
    + \tfrac{1}{2}n\delta\,(C\cdot k_{Y})\,\Bigr]
\]

\begin{align*}
  c_{Y}\bigl(f_{!}(\xi)\bigr)
    &= \exp\bigl(n\delta C - [\;\;]\bigr) \\
    &= 1 + n\delta C - [\;\;] + \tfrac{1}{2}n^{2}\delta^{2}C^{2}
\end{align*}

\[
  c_{Y}\bigl(f_{!}(\xi)\bigr) \;=\; 1 + n\delta C
  + \Bigl[\,\tfrac{1}{2}n^{2}\delta^{2}C^{2}
    - \tfrac{1}{2}n\delta\,C k_{Y} - f_{*}(d)
    + \tfrac{1}{2}n\,f_{*}(k_{X})\,\Bigr]
\]

\[
  c_{Y}\bigl(f_{!}(\xi)\bigr) \;=\; 1 + C
  + \Bigl[\,\tfrac{1}{2}\bigl(C^{2} - C k_{Y}\bigr) - f_{*}(d)
    + \tfrac{1}{2}f_{*}(k_{X})\,\Bigr]
\]

\[
  c^{2}(\mathcal{O}_{C}) \;=\; \tfrac{1}{2}\bigl[\,C^{2} - C k_{Y}
    + f_{*}(k_{\widetilde{C}})\,\bigr]
\]

\[
  K_{2}\bigl(c^{2}(\mathcal{O}_{C})\bigr)
  \;=\; \tfrac{1}{2}\bigl[\,C^{2} - C k_{Y} + 2g - 2\,\bigr]
  \;=\; \tfrac{1}{2}\bigl(C^{2} - C k_{Y}\bigr) + g - 1
\]