ShWithTrans.tex

We fix an $S \in \Sm / k$, called the \emph{base scheme}.

\begin{definition}{}{}
	Let $X$, $Y\in\Sm/\Bbbk$. We define the groups of finite correspondences:
	\[
			\Cor_S(X,Y)=\ZZ\set{C\subseteq X\times_S Y}\given C\to X\text{ finite.}
	\]
\end{definition}
\begin{example*}{}{}
	For any $f: X \rightarrow Y$, the graph $\Gamma_f=(x, f(x))$ $\subseteq X \times_S Y$ is a finite colrespondence from $X$ to $Y$.
\end{example*}

\begin{example}{}{2.2}
	If $f: X \rightarrow Y$ is finite and $\dim X = \dim Y$, the guph $\Gamma_f$ is also a fiwite correspondence from $Y\to X$.
\end{example}

\begin{definition}{}{}
	Define anadditie categury $\COR_S$, whose objerts are the same as Sm/$S$ und $\Cor_S(X, Y)$ is defined in Def 2.1. Contraraviant additine functors $$ F_i\COR_S^{\op} \to \AB.$$ 
	are called presheaver with transfers. The corresponding category is denoted by $\PSH(S)$. we have a functor $r\colon \Sm/S\to 
	\COR_S$ by \ref{exm:2.2}
\end{definition}


Ex2. 4 Fuery $x \in$ sim/s give an element. $\mathbb{Z}(x) \in \operatorname{Psh}(S)$ by
$\mathbb{Z}(x)(y)=\operatorname{Cor}_s(y, x)$. $(\mathbb{Z}(s)=\mathbb{Z}$ )

Ex2.5 The presheaves 0 and $0^{\str}$ are in. psh (s). For awy $c \in
\operatorname{cor}_s(x, y)$ and $f \in O(y)$ (resp. $\left.O^{\str}(y)\right)$

\begin{align*}
	 & C \xrightarrow{i} X_{\mathcal{L} \pi_1} \times Y \overrightarrow{\pi_2} Y \text { define } O(c)(t) \\
	 & =\operatorname{Tr}_{(/ x}\left(\left(P_2 0 i\right)^{t x}(f)\right)                                \\
	 & \text { (resp. Nc/X }\left(\left(\eta_2 \circ i\right)^t(t)\right)
\end{align*}

Def 2.6 Let us describle the composition in loos. Suppose $f \in \operatorname{Cor}(X /
Y)$ and $g \in \operatorname{lor}(y, Z)$. $x x z$ ${ }^{-113}$ Define $x x_5 Y_5
\xrightarrow{P_{23}} y, y$ $$ \text { gl }=P_{13 *}\left(P_{23}^{\str}(g) \cdot
P_{12}^{\str}(t)\right) $$ (Check all intersection are proper).
\begin{proposition}{2.7}{}
	The composition law is associative.
\end{proposition}
\begin{proof}
	Suppose $Z \xrightarrow{f} Y \xrightarrow{g} Z \xrightarrow W$ are morplisms in tors. We have Cartesian squares
\[\begin{tikzcd}
	XYZW\ar[d]\ar[r] & XZW\ar[d]\\
	XYZ\ar[r]& XZ
\end{tikzcd}\]
\end{proof}

\begin{proposition}{}{}
	Let $f\colon X\to Y$ be a proper morphism between finite type schemes$/\Bbbk$ and $\fFf\!\in\!\kKk_a(X)$.
	\begin{enumerate}
		\item
				$f_{\str}\fFf\!\in\!\kKk_a(Y)$ and $R^if_{\str}\fFf\!\in\!\kKk_{a-1}(Y)$, $i>0$.
		\item
				$f_{\str}\ZZ_a(\fFf)=\ZZ_a(f_{\str}\fFf)$.
	\end{enumerate}
\end{proposition}

\begin{theorem}{2.8}{}

\end{theorem}
\setcounter{pancounter}{30}
\begin{definition}{Cosimplicial Object}{}
	For and $n\in\NN$, define
	\[
			\AA^n\cong\Delta^n = \Spec^k[x_0, \cdots, x_n]/(\sum x_i-1).
	\]
\end{definition}
\begin{definition}{Associated Complexes}{}
	For $X\in\PSH$
\end{definition}