(Covariant) Chow Motives and Voevodsky Motives Review
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Only in setting of Chow motives, that is, the fixed adquate equivalance relationship is rational equivalance.
Correspondence and the category of (covariant) effective motives
For varieties \(X\) and \(Y\) smooth projective over \(k\), \(\mathrm{Corr}^0(X,Y)\) is defined as \(\mathrm{CH}^{\dim X}(X\times Y)\otimes\mathbb{Q}\) (Here \(\mathbb{Q}\) is the ring of coefficients, we can also choose other rings). The category of effective pure motives is modelled on the set of pairs \((X,p)\) where \(p\in\mathrm{Corr}^0{X,X}\) is a projector (which cuts out a direct summand of the object \((X,id_X)\)), with morphism between objects defined as \[ \mathrm{Hom}((X,p),(Y,q))=q\circ\mathrm{Corr}^0(X,Y)\circ p\in\mathrm{Corr}^0(X,Y). \] Additionally, we have the projective bundle formula, in short which implies (TSP 02TY) for an affine bundle over a variety \[ p:E=\mathrm{Spec}_X(\mathrm{Sym}^*(\mathcal{E}))\rightarrow X \] we have \[ p^*:\mathrm{CH}^i(X)\rightarrow\mathrm{CH}^i(E) \] is an isomorphism.
We now give a decomposition of the motive of \(\mathbb{P}^1\). Note that \[ \mathrm{Corr}^0(\mathbb{P}^1,\mathbb{P}^1)=\mathrm{CH}^1(\mathbb{P}^1\times\mathbb{P}^1)\otimes\mathbb{Q} \] has two special elements, namely \[ \begin{aligned} c_0=[0\times\mathbb{P}^1]\\ c_1=[\mathbb{P}^1\times 0]. \end{aligned} \] By simple calculation we have \[ \begin{aligned} c_0\circ c_0&=c_0\\ c_2\circ c_2&=c_2\\ c_0\circ c_2&=0\\ c_2\circ c_0&=0\\ \end{aligned} \] furthermore \[ id_{\mathbb{P}^1}=[\Delta]=c_0+c_2 \] so $$