GALOIS THEORY MICHAELMAS 2010 (M.W.F. 11AM, MR3
... was a field as well as being a vector space over R. Rings which are at the same time a vector space over a field K are called K-algebras (see subsection ix), which constitute a natural category in which to build the theory of equations, or more generally algebraic geometry, over the field K. Naturally, ...
... was a field as well as being a vector space over R. Rings which are at the same time a vector space over a field K are called K-algebras (see subsection ix), which constitute a natural category in which to build the theory of equations, or more generally algebraic geometry, over the field K. Naturally, ...
The Structure of Abelian Pro-Lie Groups - Mathematik@TU
... and the vector spaces dual of a topological vector space are naturally isomorphic. The compact open topology on the dual of a weakly complete vector space is the finest locally convex topology on the vector space dual. Since the character group of a weakly complete vector group is a vector space and ...
... and the vector spaces dual of a topological vector space are naturally isomorphic. The compact open topology on the dual of a weakly complete vector space is the finest locally convex topology on the vector space dual. Since the character group of a weakly complete vector group is a vector space and ...
notes
... Sets of points valued in a p-adic field The td-spaces that we are interested in are all related to algebraic groups over nonarchimedean local field. A local nonarchimedean field is a field F that is complete with respect to a discrete valuation ord F : F × → Z and whose residue field is finite. If R ...
... Sets of points valued in a p-adic field The td-spaces that we are interested in are all related to algebraic groups over nonarchimedean local field. A local nonarchimedean field is a field F that is complete with respect to a discrete valuation ord F : F × → Z and whose residue field is finite. If R ...
ON THE REPRESENTABILITY OF ACTIONS IN A SEMI
... contravariant functor from V to the category Set of sets. Our concern in this paper is with the representability of this functor; that is, with the existence of an object [X] of V and a natural isomorphism Act(G, X) ∼ = V G, [X] . We first need an alternative description of Act(G, X) in terms of ...
... contravariant functor from V to the category Set of sets. Our concern in this paper is with the representability of this functor; that is, with the existence of an object [X] of V and a natural isomorphism Act(G, X) ∼ = V G, [X] . We first need an alternative description of Act(G, X) in terms of ...
What are operator spaces? - Universität des Saarlandes
... Let X be a matricially normed space and X0 ⊂ X a linear subspace. Then Mn (X0 ) ⊂ Mn (X), and X0 together with the restriction of the operator space norm again is a matricially normed space. The embedding X0 ,→ X is completely isometric. If X is an operator space and X0 ⊂ X is a closed subspace, the ...
... Let X be a matricially normed space and X0 ⊂ X a linear subspace. Then Mn (X0 ) ⊂ Mn (X), and X0 together with the restriction of the operator space norm again is a matricially normed space. The embedding X0 ,→ X is completely isometric. If X is an operator space and X0 ⊂ X is a closed subspace, the ...
Derived Algebraic Geometry XI: Descent
... applications to the theory of cohomological Brauer groups). The first few sections of this paper are devoted to developing some general tools for proving these types of descent theorems. The basic observation (which we explain in §3) is that if F is a functor defined on the category of commutative r ...
... applications to the theory of cohomological Brauer groups). The first few sections of this paper are devoted to developing some general tools for proving these types of descent theorems. The basic observation (which we explain in §3) is that if F is a functor defined on the category of commutative r ...
On Brauer Groups of Lubin
... residue field κ has characteristic different from 2, then there exist atomic E-algebras which are homotopy commutative; such algebras are never Azumaya. p2q Not every Azumaya algebra over E is Morita equivalent to an atomic E-algebra. For example, the Lubin-Tate spectrum E itself is not Morita equiv ...
... residue field κ has characteristic different from 2, then there exist atomic E-algebras which are homotopy commutative; such algebras are never Azumaya. p2q Not every Azumaya algebra over E is Morita equivalent to an atomic E-algebra. For example, the Lubin-Tate spectrum E itself is not Morita equiv ...
Lectures on Groups and Their Connections to Geometry Anatole
... Definition 0.8. A set X together with an associative binary operation ⋆ is called a monoid. Remark. We must emphasise that apart from associativity, no other hypotheses whatsoever are placed on ⋆. In particular, we do not assume that ⋆ is commutative—that is, that a ⋆ b = b ⋆ a—one may easily check ...
... Definition 0.8. A set X together with an associative binary operation ⋆ is called a monoid. Remark. We must emphasise that apart from associativity, no other hypotheses whatsoever are placed on ⋆. In particular, we do not assume that ⋆ is commutative—that is, that a ⋆ b = b ⋆ a—one may easily check ...
model categories of diagram spectra
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
Propiedades de regularidad homol´ogica de variedades
... that is noetherian connected N-graded algebras which, following the general notions of noncommutative geometry, we regard as analogues of homogeneous coordinate rings of certain projective varieties. The first family is that of quantum toric varieties, which are graded subalgebras of quantum tori. W ...
... that is noetherian connected N-graded algebras which, following the general notions of noncommutative geometry, we regard as analogues of homogeneous coordinate rings of certain projective varieties. The first family is that of quantum toric varieties, which are graded subalgebras of quantum tori. W ...
Linear Spaces
... Planes A plane is determined by two vectors v and w which points in different directions (linearly independent). For any scalar s and t, the vector sv + tw is called a linear combination of v and w. It is clear that all the linear combinations of v and w lie on the plane determined by the two vector ...
... Planes A plane is determined by two vectors v and w which points in different directions (linearly independent). For any scalar s and t, the vector sv + tw is called a linear combination of v and w. It is clear that all the linear combinations of v and w lie on the plane determined by the two vector ...
Vector Bundles and K-Theory
... dimension of the fiber is at least three, then the classification is of the same order of difficulty as the fundamental but still largely unsolved problem of computing the homotopy groups of spheres. In the absence of a full classification of all the different vector bundles over a given base space, ...
... dimension of the fiber is at least three, then the classification is of the same order of difficulty as the fundamental but still largely unsolved problem of computing the homotopy groups of spheres. In the absence of a full classification of all the different vector bundles over a given base space, ...
Chapter 1 I. Fibre Bundles
... A topological space together with a (continuous) action of a group G is called a G-space. We will be particularly interested in fibre bundles which come with an action of some topological group. To this end we define: Definition 1.1.4 Let G be a topological group and B a topological space. A princip ...
... A topological space together with a (continuous) action of a group G is called a G-space. We will be particularly interested in fibre bundles which come with an action of some topological group. To this end we define: Definition 1.1.4 Let G be a topological group and B a topological space. A princip ...
FROM COMMUTATIVE TO NONCOMMUTATIVE SETTINGS 1
... and H(·, f ) = e0 ∈H0 H(e0 , f ). For A, B ⊂ H we define AB = { ab ∈ H : a ∈ A, b ∈ B, t(a) = s(b) }, and if a ∈ H, we define aB = {a}B and Ba = B{a}. We say that b ∈ H left divides a ∈ H, and write b |l a, if a ∈ bH. Two elements a, b ∈ H are left coprime if, for all c ∈ H, c |l a and c |l b implie ...
... and H(·, f ) = e0 ∈H0 H(e0 , f ). For A, B ⊂ H we define AB = { ab ∈ H : a ∈ A, b ∈ B, t(a) = s(b) }, and if a ∈ H, we define aB = {a}B and Ba = B{a}. We say that b ∈ H left divides a ∈ H, and write b |l a, if a ∈ bH. Two elements a, b ∈ H are left coprime if, for all c ∈ H, c |l a and c |l b implie ...
PDF - File
... Part III about tensor product systems of Hilbert modules [BS00] may be considered as the heart of these notes. In some sense it is related to every other part. The most important product system consists of time ordered Fock modules (treated in detail in Part II) and by Section 14.1 any dilation on a ...
... Part III about tensor product systems of Hilbert modules [BS00] may be considered as the heart of these notes. In some sense it is related to every other part. The most important product system consists of time ordered Fock modules (treated in detail in Part II) and by Section 14.1 any dilation on a ...
Math 845 Notes on Lie Groups
... It is useful to recognize when a matrix g belongs to On without having to check the condition |gu| = |u| for every vector u ∈ Rn . Proposition 2.1 On For a matrix g ∈ GLn (R), the following are equivalent. ...
... It is useful to recognize when a matrix g belongs to On without having to check the condition |gu| = |u| for every vector u ∈ Rn . Proposition 2.1 On For a matrix g ∈ GLn (R), the following are equivalent. ...
PARTIAL DYNAMICAL SYSTEMS AND C∗
... representations whose range projections satisfy a given set of relations. In Proposition 4.1 we describe the spectrum associated to the relations and give a canonical partial action of the group on this spectrum. The resulting crossed product has a universal property with respect to partial represen ...
... representations whose range projections satisfy a given set of relations. In Proposition 4.1 we describe the spectrum associated to the relations and give a canonical partial action of the group on this spectrum. The resulting crossed product has a universal property with respect to partial represen ...
QUANTUM GROUPS AND NONCOMMUTATIVE GEOMETRY
... X the algebra of functions O(X) from X to the base field (of coefficients). The dream of geometry is that this construction is bijective, i.e. that two different spaces are mapped to two different function algebras and that each algebra is the function algebra of some space. Actually the spaces and ...
... X the algebra of functions O(X) from X to the base field (of coefficients). The dream of geometry is that this construction is bijective, i.e. that two different spaces are mapped to two different function algebras and that each algebra is the function algebra of some space. Actually the spaces and ...
Mixed structures on fundamental groups
... Lemma 1.9: If W is algebraic, then for any X ∈ RepF (W ), the natural map H · (W, X) −→ H · (w, X) is an isomorphism. Note that the left hand side is calculated via injective resolutions in Rat(W ), while the right hand side is calculated via injective resolutions in ModU(w) . Proof: By filtering X ...
... Lemma 1.9: If W is algebraic, then for any X ∈ RepF (W ), the natural map H · (W, X) −→ H · (w, X) is an isomorphism. Note that the left hand side is calculated via injective resolutions in Rat(W ), while the right hand side is calculated via injective resolutions in ModU(w) . Proof: By filtering X ...
Random Involutions and the Distinct Prime Divisor Function
... represents the probability of an involution on Fn2 being isomorphic to F2 [Z/2]a x Fb2 , and the sum is being taken over all (a0 , b0 ) such that 2a0 + b0 = n. ...
... represents the probability of an involution on Fn2 being isomorphic to F2 [Z/2]a x Fb2 , and the sum is being taken over all (a0 , b0 ) such that 2a0 + b0 = n. ...
The Brauer group of a field - Mathematisch Instituut Leiden
... Proof. Given (a, b) ∈ A × B we have an R-bilinear map Ma,b : A × B → A ⊗R B defined by (x, y) 7→ xa ⊗ yb. Hence, Ma,b induces a unique R-linear homomorphism ma,b : A ⊗R B → A ⊗R B satisfying x ⊗ y 7→ xa ⊗ yb. Thus, for any pair (c, d) ∈ A × B we have a unique R-linear homomorphism mc,d : A ⊗R B → A ⊗ ...
... Proof. Given (a, b) ∈ A × B we have an R-bilinear map Ma,b : A × B → A ⊗R B defined by (x, y) 7→ xa ⊗ yb. Hence, Ma,b induces a unique R-linear homomorphism ma,b : A ⊗R B → A ⊗R B satisfying x ⊗ y 7→ xa ⊗ yb. Thus, for any pair (c, d) ∈ A × B we have a unique R-linear homomorphism mc,d : A ⊗R B → A ⊗ ...
COMPLEXES OF INJECTIVE kG-MODULES 1. Introduction Let k be
... under this equivalence to the left derived tensor product over C ∗ (BG; k) coming from the fact that the latter is E∞ , or “commutative up to all higher homotopies” (see Theorem 7.9 and the remarks after Theorem 4.1). If G is not a p-group, then there is more than one simple kG-module, and the only ...
... under this equivalence to the left derived tensor product over C ∗ (BG; k) coming from the fact that the latter is E∞ , or “commutative up to all higher homotopies” (see Theorem 7.9 and the remarks after Theorem 4.1). If G is not a p-group, then there is more than one simple kG-module, and the only ...
The periodic table of n-categories for low
... The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories Eugenia Cheng and Nick Gurski Abstract. We examine the periodic table of weak n-categories for the lowdimensional cases. It is widely understood that degenerate categories give rise to monoids, ...
... The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories Eugenia Cheng and Nick Gurski Abstract. We examine the periodic table of weak n-categories for the lowdimensional cases. It is widely understood that degenerate categories give rise to monoids, ...
Transition exercise on Eisenstein series 1.
... because GQ is transitive on these lines, and PQ is the stabilizer of the line {(0 ∗)}. Next, each line in Q2 meets Z2 in a free rank-one Z-module generated by a primitive vector (x, y), meaning that gcd(x, y) = 1. Call such a Z-module a primitive Z-line in Q2 . The collection of lines in Q2 is thus ...
... because GQ is transitive on these lines, and PQ is the stabilizer of the line {(0 ∗)}. Next, each line in Q2 meets Z2 in a free rank-one Z-module generated by a primitive vector (x, y), meaning that gcd(x, y) = 1. Call such a Z-module a primitive Z-line in Q2 . The collection of lines in Q2 is thus ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.