ETALE COHOMOLOGY AND THE WEIL CONJECTURES Sommaire 1.
... We will be particularly interested in the set X(0) of closed point (also called the atomization of X). We would like to “count” these points. As there are usually infinitely many we need a notion of “size” for x ∈ X(0) . This can be conveniently done for schemes of finite type over Z. Lemma 2.3.2. L ...
... We will be particularly interested in the set X(0) of closed point (also called the atomization of X). We would like to “count” these points. As there are usually infinitely many we need a notion of “size” for x ∈ X(0) . This can be conveniently done for schemes of finite type over Z. Lemma 2.3.2. L ...
Our Number Theory Textbook
... This chapter starts off with an introduction into some basic operations on integers as well as definitions that will be used throughout the entire book. The operations used in this chapter center around the definitions of divides, greatest common divisor (gcd), and relatively prime. The chapter star ...
... This chapter starts off with an introduction into some basic operations on integers as well as definitions that will be used throughout the entire book. The operations used in this chapter center around the definitions of divides, greatest common divisor (gcd), and relatively prime. The chapter star ...
Class Field Theory - Purdue Math
... more explicitly, let π be a generator of the unique maximal ideal of Ap . Then every x ∈ K ∗ can be uniquely written as uπ n , where u is a unit in Ap and n is an integer. We then define ordp (x) = n (and set ordp (0) = ∞). This valuation extends uniquely to K ∗ , and it induces a nonarchimedean abs ...
... more explicitly, let π be a generator of the unique maximal ideal of Ap . Then every x ∈ K ∗ can be uniquely written as uπ n , where u is a unit in Ap and n is an integer. We then define ordp (x) = n (and set ordp (0) = ∞). This valuation extends uniquely to K ∗ , and it induces a nonarchimedean abs ...
Some structure theorems for algebraic groups
... algebraic groups of a specific geometric nature, such as smooth, connected, affine, proper... A first result in this direction asserts that every algebraic group G has a largest connected normal subgroup scheme G0 , the quotient G/G0 is finite and étale, and the formation of G0 commutes with field ...
... algebraic groups of a specific geometric nature, such as smooth, connected, affine, proper... A first result in this direction asserts that every algebraic group G has a largest connected normal subgroup scheme G0 , the quotient G/G0 is finite and étale, and the formation of G0 commutes with field ...
Chapter IV. Quotients by group schemes. When we work with group
... a (left) action of the group G(T ) on the set X(T ). (ii) Let an action of G on X be given. A morphism q: X → Y in C is said to be G-invariant if q ◦ ρ = q ◦ prX : G × X → Y . By the Yoneda lemma this is equivalent to the requirement that for every T ∈ C, if x1 , x2 ∈ X(T ) are two points in the sam ...
... a (left) action of the group G(T ) on the set X(T ). (ii) Let an action of G on X be given. A morphism q: X → Y in C is said to be G-invariant if q ◦ ρ = q ◦ prX : G × X → Y . By the Yoneda lemma this is equivalent to the requirement that for every T ∈ C, if x1 , x2 ∈ X(T ) are two points in the sam ...
On fusion categories - Annals of Mathematics
... Acknowledgments. The research of P.E. was partially supported by the NSF grant DMS-9988796, and was done in part for the Clay Mathematics Institute. The research of D.N. was supported by the NSF grant DMS-0200202. The research of V.O. was supported by the NSF grant DMS-0098830. We are grateful to A. ...
... Acknowledgments. The research of P.E. was partially supported by the NSF grant DMS-9988796, and was done in part for the Clay Mathematics Institute. The research of D.N. was supported by the NSF grant DMS-0200202. The research of V.O. was supported by the NSF grant DMS-0098830. We are grateful to A. ...
Conjugacy and cocycle conjugacy of automorphisms of O2 are not
... Theorem 1.1. The relations of conjugacy and cocycle conjugacy of automorphisms of O2 are complete analytic sets when regarded as subsets of Aut(O2 ) × Aut(O2 ), and in particular, are not Borel. Furthermore if C is any class of countable structures such that the corresponding isomorphism relation ∼ ...
... Theorem 1.1. The relations of conjugacy and cocycle conjugacy of automorphisms of O2 are complete analytic sets when regarded as subsets of Aut(O2 ) × Aut(O2 ), and in particular, are not Borel. Furthermore if C is any class of countable structures such that the corresponding isomorphism relation ∼ ...
Modern Algebra: An Introduction, Sixth Edition
... all symmetry types was not easy. Like the problem in two dimensions, however, it was also settled in the nineteenth century: in contrast to the 17 different, symmetry types of plane-filling designs, there are 230 different types of symmetry for figures that fill threedimensional space. Again, groups ...
... all symmetry types was not easy. Like the problem in two dimensions, however, it was also settled in the nineteenth century: in contrast to the 17 different, symmetry types of plane-filling designs, there are 230 different types of symmetry for figures that fill threedimensional space. Again, groups ...
ABSTRACT APPROACH TO FINITE RAMSEY
... After that we prove Theorem 5.3 and Corollary 5.4 that show that the local pigeonhole principle, under mild conditions, implies the Ramsey property for Ramsey domains over normed composition spaces. Section 6: We show two results allowing us to propagate the pigeonhole principle. In the first one, Pr ...
... After that we prove Theorem 5.3 and Corollary 5.4 that show that the local pigeonhole principle, under mild conditions, implies the Ramsey property for Ramsey domains over normed composition spaces. Section 6: We show two results allowing us to propagate the pigeonhole principle. In the first one, Pr ...
AN ALGEBRAIC APPROACH TO SUBFRAME LOGICS. MODAL
... in between T0 and T1 (see, e.g., [1]). In a recent paper [2], it was shown that the modal logic of all T0 -spaces is wK4T0 = wK4 + p ∧ 3(q ∧ 3p) → 3p ∨ 3(q ∧ 3q), thus providing a useful modal logic strictly in between wK4 and K4. In fact, there are continuum many logics between wK4 and K4. It is re ...
... in between T0 and T1 (see, e.g., [1]). In a recent paper [2], it was shown that the modal logic of all T0 -spaces is wK4T0 = wK4 + p ∧ 3(q ∧ 3p) → 3p ∨ 3(q ∧ 3q), thus providing a useful modal logic strictly in between wK4 and K4. In fact, there are continuum many logics between wK4 and K4. It is re ...
Problems in the classification theory of non-associative
... numbers to quaternions, associativity still holds in H. This is no longer the case for O. For example, ((i, 0)(j, 0))(0, i) = (k, 0)(0, i) = (0, ik) = −(0, j), but (i, 0)((j, 0)(0, i)) = (i, 0)(0, ij) = (i, 0)(0, k) = (0, ki) = (0, j). However, the octonions still satisfy the weaker condition of alt ...
... numbers to quaternions, associativity still holds in H. This is no longer the case for O. For example, ((i, 0)(j, 0))(0, i) = (k, 0)(0, i) = (0, ik) = −(0, j), but (i, 0)((j, 0)(0, i)) = (i, 0)(0, ij) = (i, 0)(0, k) = (0, ki) = (0, j). However, the octonions still satisfy the weaker condition of alt ...
The Brauer group of a field - Mathematisch Instituut Leiden
... algebras over a field k are division rings for which a ring isomorphism between the center and k is given and the underlying k-vector space is finite-dimensional, as are the n × nmatrix rings over these division rings for n ∈ Z>0 . The ring of quaternions H, introduced by William Hamilton (1805–1865), ...
... algebras over a field k are division rings for which a ring isomorphism between the center and k is given and the underlying k-vector space is finite-dimensional, as are the n × nmatrix rings over these division rings for n ∈ Z>0 . The ring of quaternions H, introduced by William Hamilton (1805–1865), ...
Representations of GL_2(A_Q^\infty)
... Q ). The reason why this should seem like a generalization (and we’ll see later is literally a generalization) is that the congruence subgroups Γ ⊆ SL2 (Z) are precisely subgroups of SL2 (Z) of the form U ∩ SL2 (Z) with U ⊆ GL2 (A∞ Q ) compact open. So, without further delay let us define such a mod ...
... Q ). The reason why this should seem like a generalization (and we’ll see later is literally a generalization) is that the congruence subgroups Γ ⊆ SL2 (Z) are precisely subgroups of SL2 (Z) of the form U ∩ SL2 (Z) with U ⊆ GL2 (A∞ Q ) compact open. So, without further delay let us define such a mod ...
Introduction to representation theory
... mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics and quantum field theory. Representation theory was born in 1896 in the work of the German mathematician F. G. Frobenius. This work was triggered by a let ...
... mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics and quantum field theory. Representation theory was born in 1896 in the work of the German mathematician F. G. Frobenius. This work was triggered by a let ...
Lubin-Tate Formal Groups and Local Class Field
... is called their ramified and unramified parts. When K is a local field and O its ring of integers, the ring O contains a unique maximal ideal m that is also its only prime ideal. In an unramified extension, this ideal remains prime, so unramified extensions occur in bijection with extensions of the ...
... is called their ramified and unramified parts. When K is a local field and O its ring of integers, the ring O contains a unique maximal ideal m that is also its only prime ideal. In an unramified extension, this ideal remains prime, so unramified extensions occur in bijection with extensions of the ...