GROUP ALGEBRAS. We will associate a certain algebra to a
... Therefore the representations U2 and U3 are isomorphic. We have C[G] = U0 ⊕ U1 ⊕ U2 ⊕ U3 with dim U0 = dim U2 = 1 and corresponding representations are nonisomorphic. And dim U2 ∼ = dim U3 = 2 and the corresponding representations are isomorphic. We have completely classified all irreducible represe ...
... Therefore the representations U2 and U3 are isomorphic. We have C[G] = U0 ⊕ U1 ⊕ U2 ⊕ U3 with dim U0 = dim U2 = 1 and corresponding representations are nonisomorphic. And dim U2 ∼ = dim U3 = 2 and the corresponding representations are isomorphic. We have completely classified all irreducible represe ...
A Brief Summary of the Statements of Class Field Theory
... If K is archimedean, then θ : K × → Gal(K ab /K) is surjective and its kernel is the connected component of the identity in K × . For the rest of Section 1.3, we assume that K is nonarchimedean. Then θ is injective: The choice of a uniformizer π ∈ O lets us write K × = O× π Z ' O× × Z, and O× is alr ...
... If K is archimedean, then θ : K × → Gal(K ab /K) is surjective and its kernel is the connected component of the identity in K × . For the rest of Section 1.3, we assume that K is nonarchimedean. Then θ is injective: The choice of a uniformizer π ∈ O lets us write K × = O× π Z ' O× × Z, and O× is alr ...
Cyclic Homology Theory, Part II
... 1. The algebra slr (k) is reductive, (glr (A)⊗n )Sn is an slr (k)-module, and we can consider the projection on the component corresponding to the trivial representation K (glr (A)⊗n )Sn ։ ((glr (A)⊗n )Sn )slr (k) . The kernel K has trivial homology, so the projection is a quasi-isomorphism. 2. Th ...
... 1. The algebra slr (k) is reductive, (glr (A)⊗n )Sn is an slr (k)-module, and we can consider the projection on the component corresponding to the trivial representation K (glr (A)⊗n )Sn ։ ((glr (A)⊗n )Sn )slr (k) . The kernel K has trivial homology, so the projection is a quasi-isomorphism. 2. Th ...
Homology With Local Coefficients
... We shalldeal onlywithpropertiesofsystemswhichare invariantunderisomorphisms. In each case theproofofinvarianceis trivialand willbe omitted. It was proved in ?2 that the collection {Fx} is a systemof local groups. It is simpleif and only if it is abelian. In some instances a system {G1, will consist ...
... We shalldeal onlywithpropertiesofsystemswhichare invariantunderisomorphisms. In each case theproofofinvarianceis trivialand willbe omitted. It was proved in ?2 that the collection {Fx} is a systemof local groups. It is simpleif and only if it is abelian. In some instances a system {G1, will consist ...
here
... (1) The Hodge and Tate conjectures for abelian varieties: Raskind outlined some of the known results for divisors on and endomorphisms of abelian varieties and Milne gave some indications of one of the basic methods of proof used by Tate, Zarhin and Faltings. Milne then explained his approach for pr ...
... (1) The Hodge and Tate conjectures for abelian varieties: Raskind outlined some of the known results for divisors on and endomorphisms of abelian varieties and Milne gave some indications of one of the basic methods of proof used by Tate, Zarhin and Faltings. Milne then explained his approach for pr ...
A very brief introduction to étale homotopy
... If we allow all the n-fold intersections to be disjoint unions of contractible sets, we arrive at the notion of a good covering. If U → X is a good covering, then the topological realisation |π0 (U )| of its Čech nerve is weakly equivalent to X. Representing a covering by a morphism U → X is conven ...
... If we allow all the n-fold intersections to be disjoint unions of contractible sets, we arrive at the notion of a good covering. If U → X is a good covering, then the topological realisation |π0 (U )| of its Čech nerve is weakly equivalent to X. Representing a covering by a morphism U → X is conven ...
dmodules ja
... all f ∈ H 0 U A -module is a sheaf on X which is quasi-coherent as an -module and has the structure of a module over . A -module is coherent if it is locally finitely generated over . We write -Mod and Mod- for the categories of left and right -modules, respectively. The full subcatego ...
... all f ∈ H 0 U A -module is a sheaf on X which is quasi-coherent as an -module and has the structure of a module over . A -module is coherent if it is locally finitely generated over . We write -Mod and Mod- for the categories of left and right -modules, respectively. The full subcatego ...
BASIC DEFINITIONS IN CATEGORY THEORY MATH 250B 1
... A covariant functor F : C → Set said to be representable by A ∈ C if one has an isomorphism of functors: F ∼ = HomC (A, •). Similarly, a contravariant functor G : C → Set is said to be representable by A ∈ C if one has an isomorphism of functors: G∼ = HomC (•, A). It is more-or-less standard to call ...
... A covariant functor F : C → Set said to be representable by A ∈ C if one has an isomorphism of functors: F ∼ = HomC (A, •). Similarly, a contravariant functor G : C → Set is said to be representable by A ∈ C if one has an isomorphism of functors: G∼ = HomC (•, A). It is more-or-less standard to call ...
Profinite Groups - Universiteit Leiden
... image is dense in G. Find a group G for which f is not injective. (b) What is the profinite completion of the additive group of Z? Exercise 1.7. Let p be a prime number. (a) Show that there is a group G whose profinite completion is isomorphic to the additive group Zp . Can you find such a G that is ...
... image is dense in G. Find a group G for which f is not injective. (b) What is the profinite completion of the additive group of Z? Exercise 1.7. Let p be a prime number. (a) Show that there is a group G whose profinite completion is isomorphic to the additive group Zp . Can you find such a G that is ...
Lie Algebra Cohomology
... with Hom(B, C) deleted as before. The nth homology of this is Ext n (B, C). Again, even though directions are reversed (since Hom(·, C) is contravariant), different projective resolutions give isomorphic homology, and everything in sight is well defined. ...
... with Hom(B, C) deleted as before. The nth homology of this is Ext n (B, C). Again, even though directions are reversed (since Hom(·, C) is contravariant), different projective resolutions give isomorphic homology, and everything in sight is well defined. ...
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... The question of whether or not X and X φ are birationally isomorphic over k is delicate in general. Birational isomorphism over K = k(X)G is more accessible because it has a natural interpretation in terms of Galois cohomology. In this section we will to show that in many cases X and X φ are, indeed ...
... The question of whether or not X and X φ are birationally isomorphic over k is delicate in general. Birational isomorphism over K = k(X)G is more accessible because it has a natural interpretation in terms of Galois cohomology. In this section we will to show that in many cases X and X φ are, indeed ...
introduction to algebraic topology and algebraic geometry
... These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tool ...
... These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tool ...
Part C4: Tensor product
... (0) (unity) R ⊗ M ∼ = M. (1) (commutative) M ⊗ N ∼ = N!⊗ M (2) (distributive) N ⊗ ⊕Mi ∼ = (N ⊗ Mi ) (3) (associative) (A ⊗ B) ⊗ C ∼ = A ⊗ (B ⊗ C) (4) (right exactness) M ⊗ − is right exact, i.e., a short exact sequence A " B # C gives an exact sequence ...
... (0) (unity) R ⊗ M ∼ = M. (1) (commutative) M ⊗ N ∼ = N!⊗ M (2) (distributive) N ⊗ ⊕Mi ∼ = (N ⊗ Mi ) (3) (associative) (A ⊗ B) ⊗ C ∼ = A ⊗ (B ⊗ C) (4) (right exactness) M ⊗ − is right exact, i.e., a short exact sequence A " B # C gives an exact sequence ...
ON MACKEY TOPOLOGIES IN TOPOLOGICAL ABELIAN
... composite C −→ A −→ Ci is continuous. But for any character Ci −→ T, the composite A −→ Ci −→ T is continuous and since C −→ A is weakly continuous, the composite C −→ A −→ Ci −→ T is also continuous. Thus each composite C −→ A −→ Ci is weakly continuous and, by Glicksberg’s condition, is continuous ...
... composite C −→ A −→ Ci is continuous. But for any character Ci −→ T, the composite A −→ Ci −→ T is continuous and since C −→ A is weakly continuous, the composite C −→ A −→ Ci −→ T is also continuous. Thus each composite C −→ A −→ Ci is weakly continuous and, by Glicksberg’s condition, is continuous ...
October 17, 2014 p-DIVISIBLE GROUPS Let`s set some conventions
... Definition. A connected p-divisible group is a G such that G = G0 . An étale p-divisible group is a G such that G = Gét . Note that a connected p-divisible group is one such that each Gpn is connected, and similarly for étale p-divisible groups. ...
... Definition. A connected p-divisible group is a G such that G = G0 . An étale p-divisible group is a G such that G = Gét . Note that a connected p-divisible group is one such that each Gpn is connected, and similarly for étale p-divisible groups. ...
Endomorphisms The endomorphism ring of the abelian group Z/nZ
... Given two groups (G, *) and (H, ), a group isomorphism from (G, *) to (H, ) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function such that for all u and v in G it holds that ...
... Given two groups (G, *) and (H, ), a group isomorphism from (G, *) to (H, ) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function such that for all u and v in G it holds that ...
1_Modules_Basics
... right action of R op on M by defining mr := rm which makes M a right Rop-module. Furthermore, if R is commutative then R op = R and in general we have R op @ R . Since we mainly deal with left R-modules, unless otherwise specified, by an R-module we mean a left Rmodule. Let M be an R-module. The fol ...
... right action of R op on M by defining mr := rm which makes M a right Rop-module. Furthermore, if R is commutative then R op = R and in general we have R op @ R . Since we mainly deal with left R-modules, unless otherwise specified, by an R-module we mean a left Rmodule. Let M be an R-module. The fol ...
Intersection homology
... The only allowable one-cycles in X are linear combinations of the ones shown in green in the diagram. (All other 1-cycles pass through the singular point of X and are therefore not allowable.) The green cycles are both boundaries of faces and hence they are trivial in homology, giving IH1 (X) = 0. ...
... The only allowable one-cycles in X are linear combinations of the ones shown in green in the diagram. (All other 1-cycles pass through the singular point of X and are therefore not allowable.) The green cycles are both boundaries of faces and hence they are trivial in homology, giving IH1 (X) = 0. ...
The plus construction, Bousfield localization, and derived completion Tyler Lawson June 28, 2009
... Definition 9. The nonunital O-algebra operad O+ ⊂ O has O+ (k) = O(k) for all k > 0 and O+ (0) = ∗. Definition 10. The O-module operad MO (1) ⊂ O has MO (1) = O(1) and MO (k) = ∗ if k 6= 1. The symmetric spectrum O(1) is an associative R-algebra, and an algebra over MO is precisely a module over O(1 ...
... Definition 9. The nonunital O-algebra operad O+ ⊂ O has O+ (k) = O(k) for all k > 0 and O+ (0) = ∗. Definition 10. The O-module operad MO (1) ⊂ O has MO (1) = O(1) and MO (k) = ∗ if k 6= 1. The symmetric spectrum O(1) is an associative R-algebra, and an algebra over MO is precisely a module over O(1 ...