"Motive"의 두 판 사이의 차이

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geometry roughly= cohomology<br><br>example<br><br>circle S^1<br><br>Betti cohomolgy (singular cohomology)<br><br>$H^0(S^1,Z)=Z$<br><br>$H^1(S^1,Z)=Z$<br><br>$\mathbb{G}_m = \mathbb{C}^{x} = \mathbb{C}/{0}$same homotopy class as$S^1$<br><br>Betti cohomology is same<br><br>$H^0(\mathbb{G}_m,Z)=Z$<br><br>$H^1(\mathbb{G}_m,\mathbb{Z})=Z$, this is dual to$H_1(\mathbb{G}_m,Z)$we can call the generator as$\gamma_0^{\vee}$where$\gamma_0$is the homology generator.<br><br>de Rham cohomology<br><br>$H^0_{dR}(\mathbb{G}_m)=\mathbb{C}$<br><br>$H^1_{dR}(\mathbb{G}_m)=\mathbb{C}\frac{dz}{z}$<br><br>De Rham isomorphism<br><br>$H^1(\mathbb{G}_m,Z) \times H^1_{dR}(\mathbb{G}_m) \to \mathbb{C}$is a perfect pairing<br><br>$(\gamma,\omega) \to \int_{\gamma}\omega$<br><br>i.e.$H^1_{dR}(\mathbb{G}_m) = ^1(\mathbb{G}_m,Z)\otimes_{\mathbb{Z}}\mathbb{C}$<br><br>question. under this isomorphism,$\frac{dz}{z} = c\times \gamma_0^{\vee}$ what is c?<br><br>$c = \int_{\gamma_0}\frac{dz}{z} = 2\pi i$<br><br>Etale cohomology<br><br>exponential map :$\mathbb{C}\to \mathbb{C}^{*}$<br><br>$H^1(\mathbb{G}_m,\mathbb{Z}) = Hom(\pi_1(\mathbb{C}^{*}),\mathb{Z})$<br><br>Let l be a prime.<br><br>$H^1_{et}(\mathbb{G}_m,\mathbb{Q}_{l})$is a 1-dimensional$\mathbb{Q}_{l}$vector space on which$Gal(\bar{\mathbb{Q}}/\mathbb{Q})$acts.<br><br>We get a character called the cyclotomic character.<br><br>general picture<br><br>Let k be a field$(Q,F_q,C,\cdots)$<br><br>from (separable finit type k-schemes) to category of motives<br><br>Betti cohomology Vec over Q (Hodeg structure)<br>de Rham cohomology Vec over k if char k = 0 (graded vector space)<br>if$l\neq$char(k) etale cohomology vec over$\mathbb{Q}_l$(Galois representation)<br>crystalline cohomology Vec over$\mathbb{Q}_p$<br><br>(category of motives) can do linear algebra<br><br>$\mathbb{Q}$-linear\otimes$-category<br><br>bigger picture obtained when we compare cohomologies<br><br>Betti <-> de Rham , Hodge theory<br><br>crystalline(de Rham) <-> etale, p-adic Hodge theory<br><br>What we like in linear algebra :<br><br>1 dimension<br><br>2$f : V\to V$, characteristic polynomial<br><br>something we don't know :<br><br>$X over (k = \bar{k}), char(k)\neq 0$<br><br>for all l prime to characteristic,$dim_{\mathbb{Q}_l H^i_{et}(X,\mathbb{Q}_l)$<br><br>We don't know how to show that these numbers are independent of$l$.<br><br>we know that if$X$over$k$is smooth and proper,<br><br>$k=\bar{\mathbb{F}_q}$, then we know that these numbers are independent of l (Deligne-Weil II, trace formula for etale cohomology)<br><br>X smooth, alternating sum of dimension,\sum(-1)^i dim_{\mathbb{Q}_l H^i_{et}(X,\mathbb{Q}_l) is independent of l. (intersection theory of cycles)<br><br>ex : elliptic curve<br><br>E : y^2=x^3-Ax-B,\Delta\neq0 , A,B in\mathbb{Q}<br><br>over complex numbers, let\alpha,\betagenerators H_1<br><br>H^0(E,\Omega^1_E) =\mathbb{C}\cdot\frac{dx}{2y}<br><br>\omega_{\alpha}=\int_{\alpha}\frac{dx}{2y},\omega_{\beta}=\int_{\beta}\frac{dx}{2y}\in\mathbb{C}<br><br>These are linearly independent over real numbers so we get a lattice\Lambda=\mathbb{Z}\omega_{\alpha}+\mathbb{Z}\omega_{\beta}\subset\mathbb{C}<br><br>\intE(C)\to\mathbb{C}/\Lambdais an isomorphism<br><br>inverse map : Weierstrass\wp-function<br><br>abelian varieties form a\mathbb{Z}-linear category. So take a tensor with\mathbb{Q}<br><br>(abelian varieties)\otimes\mathbb{Q}  = (category of ab. varieties up to isogeny) . these are\mathbb{Q}-linear<br><br>this is inside the category of motives.
 
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[http://en.wikipedia.org/wiki/Motive_%28algebraic_geometry%29 http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)]
 
[http://en.wikipedia.org/wiki/Motive_%28algebraic_geometry%29 http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)]
  

2011년 11월 10일 (목) 08:31 판

http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)

 

 

Feynman motive

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