Arithmetic fundamental groups and moduli of curves
... direction, by H. Nakamura, F. Pop, A. Tamagawa, and S. Mochizuki, and others. For example, the conjecture is true for curves over p-adic elds with nonabelian fundamental groups. An exposition on these researches in Japanese is available, and its English translation is to appear [25]. 2.3. Arithmeti ...
... direction, by H. Nakamura, F. Pop, A. Tamagawa, and S. Mochizuki, and others. For example, the conjecture is true for curves over p-adic elds with nonabelian fundamental groups. An exposition on these researches in Japanese is available, and its English translation is to appear [25]. 2.3. Arithmeti ...
18. Fibre products of schemes The main result of this section is
... S. Indeed if Z maps to Xij and Y over S, it maps to Xi and Y over S. But then Z maps to Fi , by the universal property of the fibre product. It is clear that the image of Z lands in Fij , so that Fij is the fibre product. But then there are natural isomorphisms φij : Fij −→ Fji such that φik = φjk ◦ ...
... S. Indeed if Z maps to Xij and Y over S, it maps to Xi and Y over S. But then Z maps to Fi , by the universal property of the fibre product. It is clear that the image of Z lands in Fij , so that Fij is the fibre product. But then there are natural isomorphisms φij : Fij −→ Fji such that φik = φjk ◦ ...
SUBDIVISIONS OF SMALL CATEGORIES Let A be a
... category with a single object, BG is called the classsifying space of the group G. The space BG is often written as K(G, 1) and called an Eilenberg-Mac Lane space. It is characterized (up to homotopy type) as a connected space with π1 (K(G, 1)) = G and with all higher homotopy groups πq (K(G, 1)) = ...
... category with a single object, BG is called the classsifying space of the group G. The space BG is often written as K(G, 1) and called an Eilenberg-Mac Lane space. It is characterized (up to homotopy type) as a connected space with π1 (K(G, 1)) = G and with all higher homotopy groups πq (K(G, 1)) = ...
Orbifolds and their cohomology.
... that an orbifold chart contains more data than simply the topological quotient Ũ /G. In particular, an orbifold “remembers” where the Gactions in each of its charts have isotropy. Example 1.0.1. Let Zn act on C by multiplication by nth roots of unity. Then there is an orbifold X = [C/Zn ] with a si ...
... that an orbifold chart contains more data than simply the topological quotient Ũ /G. In particular, an orbifold “remembers” where the Gactions in each of its charts have isotropy. Example 1.0.1. Let Zn act on C by multiplication by nth roots of unity. Then there is an orbifold X = [C/Zn ] with a si ...
To translate algebraic sentences
... “The sum of a number squared and fifteen is equal to twenty-four.” 3) 8 + (x + 3) ...
... “The sum of a number squared and fifteen is equal to twenty-four.” 3) 8 + (x + 3) ...
WHAT IS A GLOBAL FIELD? A global field K is either • a finite
... G(AK )1 acts unitarily on L1 (G(AK )1 /G(K)). This representation is supposed to encode deep informations concerning the global field K, e.g, the Galois group Gal(K/K), motives over K and their Hasse-Weil zeta functions. 3. Varieties over global fields Let K = κ(C) be a field as in Weil’s Rosetta St ...
... G(AK )1 acts unitarily on L1 (G(AK )1 /G(K)). This representation is supposed to encode deep informations concerning the global field K, e.g, the Galois group Gal(K/K), motives over K and their Hasse-Weil zeta functions. 3. Varieties over global fields Let K = κ(C) be a field as in Weil’s Rosetta St ...
Computing Galois groups by specialisation
... a homomorphism. The image φ(G0 ) consists of those elements of Hom(Y, µ4 ) whose kernel contains (1 + i). Lemma 7. Let H be a subgroup of Hom(Y, µ4 ). A set of generators for φ−1 H is given by • the inverse images of a set of generators for H ∩ φ(G0 ), together with • the inverse image of any elemen ...
... a homomorphism. The image φ(G0 ) consists of those elements of Hom(Y, µ4 ) whose kernel contains (1 + i). Lemma 7. Let H be a subgroup of Hom(Y, µ4 ). A set of generators for φ−1 H is given by • the inverse images of a set of generators for H ∩ φ(G0 ), together with • the inverse image of any elemen ...
Trigonometric sums
... due to the fact that F , and hence Fˇ is totally ramified. Theorem 3.2. The cohomology of Gm with coefficient in F (ψχ −1 ) satisfies the following: (i) If χ is non-trivial, then H c∗ → H ∗ is an isomorphism. (ii) H ci = 0 for i 6= 1 and dim H c1 = 1. (iii) F acts on H c1 via multiplication by τ(χ, ...
... due to the fact that F , and hence Fˇ is totally ramified. Theorem 3.2. The cohomology of Gm with coefficient in F (ψχ −1 ) satisfies the following: (i) If χ is non-trivial, then H c∗ → H ∗ is an isomorphism. (ii) H ci = 0 for i 6= 1 and dim H c1 = 1. (iii) F acts on H c1 via multiplication by τ(χ, ...
∗-AUTONOMOUS CATEGORIES: ONCE MORE
... obvious, observation that the category of topological spaces can be embedded fully into Chu(Set, 2) (see 2.2). The real significance—at least to me—of this observation is that putting a Chu structure on a set can be viewed as a kind of generalized topology. A reader who is not familiar with the Chu ...
... obvious, observation that the category of topological spaces can be embedded fully into Chu(Set, 2) (see 2.2). The real significance—at least to me—of this observation is that putting a Chu structure on a set can be viewed as a kind of generalized topology. A reader who is not familiar with the Chu ...
2. Basic notions of algebraic groups Now we are ready to introduce
... (ii) G0 is a closed normal subgroup of G of finite index, and the irreducible components of G are the cosets of G0 . (iii) Any closed subgroup of G of finite index contains G0 . Proof. (i) Let X and Y be two irreducible components containing e. Then the closure of XY = µ(X × Y ) is irreducible too s ...
... (ii) G0 is a closed normal subgroup of G of finite index, and the irreducible components of G are the cosets of G0 . (iii) Any closed subgroup of G of finite index contains G0 . Proof. (i) Let X and Y be two irreducible components containing e. Then the closure of XY = µ(X × Y ) is irreducible too s ...
Garrett 10-03-2011 1 We will later elaborate the ideas mentioned earlier: relations
... This issue is similar to the issue of proving that sums and products of algebraic numbers α, β (over Q, for example) are again algebraic. Specifically, do not try to explicitly find a polynomial P with rational coefficients and P (α + β) = 0, in terms of the minimal polynomials of α, β. The methodol ...
... This issue is similar to the issue of proving that sums and products of algebraic numbers α, β (over Q, for example) are again algebraic. Specifically, do not try to explicitly find a polynomial P with rational coefficients and P (α + β) = 0, in terms of the minimal polynomials of α, β. The methodol ...
Locally cartesian closed categories and type theory
... (= E) If a,beA, cel(a,b), deC[a,a,r(a)], where C[x,y,z] is a type depending on x,yeA, zeI(x,y), then a(d)[a,b,c]eC[a,b,c]. 1-1-3. Equality rules. Using the notation of § 1-1-2, we have the following equations: (f eq) Any imposed equations on function constants induce the obvious equations. (lred) If ...
... (= E) If a,beA, cel(a,b), deC[a,a,r(a)], where C[x,y,z] is a type depending on x,yeA, zeI(x,y), then a(d)[a,b,c]eC[a,b,c]. 1-1-3. Equality rules. Using the notation of § 1-1-2, we have the following equations: (f eq) Any imposed equations on function constants induce the obvious equations. (lred) If ...
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... Assume that G is a group scheme of finite type over k , acting on a quasi-separated k -scheme X , with a non-empty invariant open subscheme on which the action is free. Let E be a G-equivariant locally free sheaf of rank r on X . Then there exists a non-empty open G-invariant subscheme U of X , such ...
... Assume that G is a group scheme of finite type over k , acting on a quasi-separated k -scheme X , with a non-empty invariant open subscheme on which the action is free. Let E be a G-equivariant locally free sheaf of rank r on X . Then there exists a non-empty open G-invariant subscheme U of X , such ...
Equivariant cohomology and equivariant intersection theory
... This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certai ...
... This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certai ...
MANIFOLDS, COHOMOLOGY, AND SHEAVES
... calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be consistent with Eduardo Cattani’s lectures at this summer school, the vector space of C ∞ differential forms on a manifold M will be denoted by A∗ (M ), instead of Ω∗ (M ). 1. Differential Forms on a Manifold This se ...
... calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be consistent with Eduardo Cattani’s lectures at this summer school, the vector space of C ∞ differential forms on a manifold M will be denoted by A∗ (M ), instead of Ω∗ (M ). 1. Differential Forms on a Manifold This se ...
1.5.4 Every abelian variety is a quotient of a Jacobian
... Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety. The modular abelian varieties that we will encounter later are, by definition, exactly the quotients of the Jacobian J1 (N ) of X1 (N ) for some N . In this section we see that merely being a quotient o ...
... Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety. The modular abelian varieties that we will encounter later are, by definition, exactly the quotients of the Jacobian J1 (N ) of X1 (N ) for some N . In this section we see that merely being a quotient o ...