"Motive"의 두 판 사이의 차이

2021년 2월 17일 (수) 01:40 기준 최신판

introduction

Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology.
Feynman motive

encyclopedia

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

expositions

Rios, Michael. “On a Final Theory of Mathematics and Physics.” arXiv:1502.04794 [hep-Th, Physics:physics], February 16, 2015. http://arxiv.org/abs/1502.04794.
Rej, Abhijnan, and Matilde Marcolli. “Motives: An Introductory Survey for Physicists.” arXiv:0907.4046 [hep-Th, Physics:math-Ph], July 23, 2009. http://arxiv.org/abs/0907.4046.

articles

Dupont, Clément. “Odd Zeta Motive and Linear Forms in Odd Zeta Values.” arXiv:1601.00950 [math], January 5, 2016. http://arxiv.org/abs/1601.00950.
Totaro, Burt. “Adjoint Functors on the Derived Category of Motives.” arXiv:1502.05079 [math], February 17, 2015. http://arxiv.org/abs/1502.05079.

메타데이터

위키데이터

ID : Q840594

Spacy 패턴 목록

[{'LEMMA': 'motive'}]
[{'LEMMA': 'Motif'}]

@@ 1번째 줄: / 1번째 줄: @@
-geometry roughly= cohomology
+==introduction==
+* Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology.
+* Feynman motive
-example
+==encyclopedia==
+* [http://en.wikipedia.org/wiki/Motive_%28algebraic_geometry%29 http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)]
-circle S^1
-Betti cohomolgy (singular cohomology)
+==expositions==
+* Rios, Michael. “On a Final Theory of Mathematics and Physics.” arXiv:1502.04794 [hep-Th, Physics:physics], February 16, 2015. http://arxiv.org/abs/1502.04794.
+* Rej, Abhijnan, and Matilde Marcolli. “Motives: An Introductory Survey for Physicists.” arXiv:0907.4046 [hep-Th, Physics:math-Ph], July 23, 2009. http://arxiv.org/abs/0907.4046.
-H^0(S^1,Z)=Z
+==articles==
+* Dupont, Clément. “Odd Zeta Motive and Linear Forms in Odd Zeta Values.” arXiv:1601.00950 [math], January 5, 2016. http://arxiv.org/abs/1601.00950.
+* Totaro, Burt. “Adjoint Functors on the Derived Category of Motives.” arXiv:1502.05079 [math], February 17, 2015. http://arxiv.org/abs/1502.05079.
-H^1(S^1,Z)=Z
+[[분류:개인노트]]
+[[분류:math and physics]]
+[[분류:math]]
+[[분류:migrate]]
-\mathbb{G}_m = \mathbb{C}^{x} = \mathbb{C}/{0} same homotopy class as S^1
+==메타데이터==
+===위키데이터===
-Betti cohomology is same
+* ID :  [https://www.wikidata.org/wiki/Q840594 Q840594]
+===Spacy 패턴 목록===
-H^0(\mathbb{G}_m,Z)=Z
+* [{'LEMMA': 'motive'}]
+* [{'LEMMA': 'Motif'}]
-H^1(\mathbb{G}_m,\mathbb{Z})=Z , this is dual to H_1(\mathbb{G}_m,Z) we can call the generator as <math>\gamma_0^{\vee}</math> where <math>\gamma_0</math> is the homology generator.
-de Rham cohomology
-H^0_{dR}(\mathbb{G}_m)=\mathbb{C}
-H^1_{dR}(\mathbb{G}_m)=\mathbb{C}\frac{dz}{z}
-De Rham isomorphism
-H^1(\mathbb{G}_m,Z) \times H^1_{dR}(\mathbb{G}_m) \to \mathbb{C} is a perfect pairing
-(\gamma,\omega) \to \int_{\gamma}\omega
-i.e. H^1_{dR}(\mathbb{G}_m) = ^1(\mathbb{G}_m,Z)\otimes_{\mathbb{Z}}\mathbb{C}
-question. under this isomorphism, \frac{dz}{z} = c\times <math>\gamma_0^{\vee}</math>  what is c?
-c = \int_{\gamma_0}\frac{dz}{z} = 2\pi i
-Etale cohomology
-exponential map : \mathbb{C}\to \mathbb{C}^{*}
-H^1(\mathbb{G}_m,\mathbb{Z}) = Hom(\pi_1(\mathbb{C}^{*}),\mathb{Z})
-Let l be a prime.
-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.
-We get a character called the cyclotomic character.
-general picture
-k field (Q,F_q,C,\cdots)
-from (separable finit type k-schemes) to category of motives
-* Betti cohomology Vec over Q (Hodeg structure)
-* de Rham cohomology Vec over k if char k = 0 (graded vector space)
-* if l\neq char(k) etal cohomology vec over \mathbb{Q}_l (Galois representation)
-* crystalline cohomology Vec over \mathbb{Q}_p
-(category of motives) can do linear algebra
-\mathbb{Q}-linear \otimes-category
-bigger picture obtained when we compare cohomologies
-Betti <-> de Rham , Hodge theory
-crystalline(de Rham) <-> etale, p-adic Hodge theory
-What we like in linear algebra :
-dimension
-f : V\to V, characteristic polynomial
-something we don't know :
-X over (k = \bar{k}), char(k)\neq 0
-for all l prime to characteristic, dim_{\mathbb{Q}_l H^i_{et}(X,\mathbb{Q}_l)
-We don't know how to show that these numbers are independent of l.
-we know that if X over k is smooth and proper,
-k=\bar{\mathbb{F}_q}, then we know that these numbers are independent of l (Deligne-Weil II, trace formula for etale cohomology)
-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)
-ex : elliptic curve
-E : y^2=x^3-Ax-B, \Delta\neq 0 , A,B in \mathbb{Q}
-over complex numbers, let \alpha, \beta generators H_1
-H^0(E,\Omega^1_E) = \mathbb{C}\cdot \frac{dx}{2y}
-\omega_{\alpha}=\int_{\alpha}\frac{dx}{2y}, \omega_{\beta}=\int_{\beta}\frac{dx}{2y} \in \mathbb{C}
-These are linearly independent over real numbers so we get a lattice \Lambda=\mathbb{Z}\omega_{\alpha}+\mathbb{Z}\omega_{\beta}\subset \mathbb{C}
-\int E(C)\to \mathbb{C}/\Lambda is an isomorphism
-inverse map : Weierstrass \wp-function
-abelian varieties form a \mathbb{Z}-linear category. So take a tensor with \mathbb{Q}
-(abelian varieties)\otimes \mathbb{Q}  = (category of ab. varieties up to isogeny) . these are \mathbb{Q}-linear
-this is inside the category of motives.
-[http://en.wikipedia.org/wiki/Motive_%28algebraic_geometry%29 http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)]

"Motive"의 두 판 사이의 차이

2021년 2월 17일 (수) 01:40 기준 최신판

목차

introduction

encyclopedia

expositions

articles

메타데이터

위키데이터

Spacy 패턴 목록

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