Motivic interpretation of Milnor K
... First, we will briefly review the definition of mixed K-groups from [ZerCy] and [RS00]. 2.1 Let k be a field, and X a smooth quasi-projective varieties over k. We use the notation CH0 (X) for the group of zero-cycles on X modulo rational equivalence. If G is a group scheme defined over k and A is k- ...
... First, we will briefly review the definition of mixed K-groups from [ZerCy] and [RS00]. 2.1 Let k be a field, and X a smooth quasi-projective varieties over k. We use the notation CH0 (X) for the group of zero-cycles on X modulo rational equivalence. If G is a group scheme defined over k and A is k- ...
The Kazhdan-Lusztig polynomial of a matroid
... one does not recover the classical Kazhdan-Lusztig polynomials for the Coxeter group Sn from the braid matroid. Polo [Pol99] has shown that any polynomial with non-negative coefficients and constant term 1 appears as a Kazhdan-Lusztig polynomial associated to some symmetric group, while Kazhdan-Lus ...
... one does not recover the classical Kazhdan-Lusztig polynomials for the Coxeter group Sn from the braid matroid. Polo [Pol99] has shown that any polynomial with non-negative coefficients and constant term 1 appears as a Kazhdan-Lusztig polynomial associated to some symmetric group, while Kazhdan-Lus ...
Tannaka Duality for Geometric Stacks
... Remark 4.6. If (S, OS ) is the underlying topos of a complex analytic space (or complex-analytic orbifold) and X is any algebraic stack of finite type over C, then HomC (S, X) ' Hom(S, X an ). To prove this, we note that equality holds when X is a scheme or algebraic space, essentially by the defini ...
... Remark 4.6. If (S, OS ) is the underlying topos of a complex analytic space (or complex-analytic orbifold) and X is any algebraic stack of finite type over C, then HomC (S, X) ' Hom(S, X an ). To prove this, we note that equality holds when X is a scheme or algebraic space, essentially by the defini ...
MILNOR K-THEORY OF LOCAL RINGS WITH FINITE RESIDUE
... site of all schemes. Assume F is continuous, i.e. that it commutes with filtering inverse limits of affine schemes, see Definition 5, and has some kind a transfer for finite étale extensions of local rings with infinite residue fields – for an explanation see Section 1. In Theorem 7 we prove: Theor ...
... site of all schemes. Assume F is continuous, i.e. that it commutes with filtering inverse limits of affine schemes, see Definition 5, and has some kind a transfer for finite étale extensions of local rings with infinite residue fields – for an explanation see Section 1. In Theorem 7 we prove: Theor ...
ABELIAN VARIETIES A canonical reference for the subject is
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
Derived splinters in positive characteristic
... This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities, as suggested by the work of Kovács. Our ...
... This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities, as suggested by the work of Kovács. Our ...
Closed locally path-connected subspaces of finite
... not homeomorphic to closed subspaces of finite-dimensional topological groups. First we state a Key Lemma treating locally continuum-connected subspaces of finite-dimensional topological groups. By a continuum we understand a connected compact Hausdorff space. We shall say that two points x, y of a top ...
... not homeomorphic to closed subspaces of finite-dimensional topological groups. First we state a Key Lemma treating locally continuum-connected subspaces of finite-dimensional topological groups. By a continuum we understand a connected compact Hausdorff space. We shall say that two points x, y of a top ...
A very brief introduction to étale homotopy
... Representing a covering by a morphism U → X is convenient because this carries over to étale coverings. One would like to apply this construction to a Grothendieck topology on a scheme X, for example, to the small étale site Xét of X. The obvious problem is that open étale “subsets” are rarely c ...
... Representing a covering by a morphism U → X is convenient because this carries over to étale coverings. One would like to apply this construction to a Grothendieck topology on a scheme X, for example, to the small étale site Xét of X. The obvious problem is that open étale “subsets” are rarely c ...
The Essential Dimension of Finite Group Schemes Corso di Laurea Magistrale in Matematica
... definition for L/K. In fact consider our previous example of k( t)/k(t). Take as Kn = k(t2n+1 ) and Ln = k(tn+1/2T ). Then L/K is surely defined over Kn for each n but not on their intersection n Kn = k. This thesis will not try to follow this direct approach, which proves itself inconvenient when t ...
... definition for L/K. In fact consider our previous example of k( t)/k(t). Take as Kn = k(t2n+1 ) and Ln = k(tn+1/2T ). Then L/K is surely defined over Kn for each n but not on their intersection n Kn = k. This thesis will not try to follow this direct approach, which proves itself inconvenient when t ...
Problem Set #1 - University of Chicago Math
... III. Which, if any, among the relations “is finer than”, “is coarser than”, and “is comparable to” forms an equivalence relations among the class of topologies? For those which do not form an equivalence relation, which of the three axioms of an equivalence relation do they satisfy? IV. Complete the ...
... III. Which, if any, among the relations “is finer than”, “is coarser than”, and “is comparable to” forms an equivalence relations among the class of topologies? For those which do not form an equivalence relation, which of the three axioms of an equivalence relation do they satisfy? IV. Complete the ...
THE GEOMETRY OF TORIC VARIETIES
... algebraic variety. The second circumstance is closely related to the first, consisting in the fact that varieties "locally" are frequently toric in structure, or toroidal. As a trivial example, a smooth variety is locally isomorphic to affine space A". Toroidal varieties are interesting in that one ...
... algebraic variety. The second circumstance is closely related to the first, consisting in the fact that varieties "locally" are frequently toric in structure, or toroidal. As a trivial example, a smooth variety is locally isomorphic to affine space A". Toroidal varieties are interesting in that one ...
The derived category of sheaves and the Poincare-Verdier duality
... If we set F n : F T n so that F n pX q : F pX rns q. Using the axiom TR2 we deduce that for every distinguished triangle pX, Y, Z, f, g, hq we obtain a long exact sequence in B q f rns n p1q grns n p1q hrns n 1 Ñ F npX q p1ÝÑ F pY q ÝÑ F pZ q ÝÑ F pX q Ñ (1.5) In particu ...
... If we set F n : F T n so that F n pX q : F pX rns q. Using the axiom TR2 we deduce that for every distinguished triangle pX, Y, Z, f, g, hq we obtain a long exact sequence in B q f rns n p1q grns n p1q hrns n 1 Ñ F npX q p1ÝÑ F pY q ÝÑ F pZ q ÝÑ F pX q Ñ (1.5) In particu ...
The congruence subgroup problem
... A k-group T is a torus if it is connected and can be conjugated into diagonal matrices in GL(n, C). It is again a basic result that if G is a reductive k-group, G contains a central k-torus T such that G/T has no connected abelian normal subgroups. Information on T and G/T separately can be pieced t ...
... A k-group T is a torus if it is connected and can be conjugated into diagonal matrices in GL(n, C). It is again a basic result that if G is a reductive k-group, G contains a central k-torus T such that G/T has no connected abelian normal subgroups. Information on T and G/T separately can be pieced t ...
GEOMETRIC PROOFS OF SOME RESULTS OF MORITA
... Cg → Mg . It is well known that H 2 (Cg , Q) has basis λ and ω. The class ω is often denoted by −ψ in the physics literature.1 One has the universal abelian variety J → Ag . This has (orbifold) fundamental group isomorphic to Spg (Z)⋉HZ , where HZ denotes the first homology group of the reference ab ...
... Cg → Mg . It is well known that H 2 (Cg , Q) has basis λ and ω. The class ω is often denoted by −ψ in the physics literature.1 One has the universal abelian variety J → Ag . This has (orbifold) fundamental group isomorphic to Spg (Z)⋉HZ , where HZ denotes the first homology group of the reference ab ...
on the shape of torus-like continua and compact connected
... Thus one can apply the results of [3] to compute the Cech cohomology of Y over any integral domain R. In the next section we will show that if Ii is a collection of compact connected Lie groups, then a continuum X may be Tl-like without having the shape of a compact connected topological group. 2. C ...
... Thus one can apply the results of [3] to compute the Cech cohomology of Y over any integral domain R. In the next section we will show that if Ii is a collection of compact connected Lie groups, then a continuum X may be Tl-like without having the shape of a compact connected topological group. 2. C ...
Connes–Karoubi long exact sequence for Fréchet sheaves
... Lemma 2.3. Let (X, OX ) denote the formal scheme obtained by completing an integral noetherian scheme X ′ of finite type over C along a closed primary integral subscheme X, as in example (a) above. Then, for each open subset U of X, the ring OX (U ) is an ultrametric Banach algebra, i.e., (X, OX ) d ...
... Lemma 2.3. Let (X, OX ) denote the formal scheme obtained by completing an integral noetherian scheme X ′ of finite type over C along a closed primary integral subscheme X, as in example (a) above. Then, for each open subset U of X, the ring OX (U ) is an ultrametric Banach algebra, i.e., (X, OX ) d ...
de Rham cohomology
... between the de Rham groups and topology was first proved by Georges de Rham himself in the 1931. In these notes we give a proof of de Rham’s Theorem, that states there exists an isomorphism between de Rham and singular cohomology groups of a smooth manifold. In the first chapter we recall some notio ...
... between the de Rham groups and topology was first proved by Georges de Rham himself in the 1931. In these notes we give a proof of de Rham’s Theorem, that states there exists an isomorphism between de Rham and singular cohomology groups of a smooth manifold. In the first chapter we recall some notio ...
on h1 of finite dimensional algebras
... usual Hochschild complex of cochains computing the Hochschild cohomology H i (Λ, X), see [10, 18, 19]. At zero-degree we have H 0 (Λ, X) = HomΛe (Λ, X) = X Λ where X Λ = {x ∈ X|λx = xλ ∀λ ∈ Λ}. Indeed, to ϕ ∈ HomΛe (Λ, X) we associate ϕ(1) which belongs to X Λ . In particular if the Λ-bimodule X is ...
... usual Hochschild complex of cochains computing the Hochschild cohomology H i (Λ, X), see [10, 18, 19]. At zero-degree we have H 0 (Λ, X) = HomΛe (Λ, X) = X Λ where X Λ = {x ∈ X|λx = xλ ∀λ ∈ Λ}. Indeed, to ϕ ∈ HomΛe (Λ, X) we associate ϕ(1) which belongs to X Λ . In particular if the Λ-bimodule X is ...
DUALITY AND STRUCTURE OF LOCALLY COMPACT ABELIAN
... work is the beautiful theorem proved independently by Lev Pontryagin in 1931 which states that up to topological isomorphism there are only two non-discrete locally compact fields - the field of real numbers and the field of complex numbers. By the early 1930's many mathematicians were working with ...
... work is the beautiful theorem proved independently by Lev Pontryagin in 1931 which states that up to topological isomorphism there are only two non-discrete locally compact fields - the field of real numbers and the field of complex numbers. By the early 1930's many mathematicians were working with ...
Universal covering spaces and fundamental groups in
... 1.1. Relation to earlier work. The fundamental group family of §4 can be recovered from Deligne’s profinite fundamental group [D, §10]. For X Noetherian, Deligne constructs a profinite lisse sheaf P on X × X, called the profinite fundamental group [D, §10.17]. P pulled back under the diagonal map X → X ...
... 1.1. Relation to earlier work. The fundamental group family of §4 can be recovered from Deligne’s profinite fundamental group [D, §10]. For X Noetherian, Deligne constructs a profinite lisse sheaf P on X × X, called the profinite fundamental group [D, §10.17]. P pulled back under the diagonal map X → X ...
An introduction to schemes - University of Chicago Math
... in R, and is denoted by Spec(R) Classically, we have a natural identification of the maximal ideals in C[x1 , . . . , xn ] and points in Cn . These points are still here, but even for a polynomial ring over C, we’ve just added in a bunch of extra points. Why? One reason is that if we have a homomorp ...
... in R, and is denoted by Spec(R) Classically, we have a natural identification of the maximal ideals in C[x1 , . . . , xn ] and points in Cn . These points are still here, but even for a polynomial ring over C, we’ve just added in a bunch of extra points. Why? One reason is that if we have a homomorp ...
October 17, 2014 p-DIVISIBLE GROUPS Let`s set some conventions
... have good reduction, if m - char(k) we get EK [m](K) ,→ Ek (k) is injective. However, while the theorem holds for elliptic curves, it fails for even simple examples of finite flat group schemes. For example, if we let R = Zp [ζp ] and consider the map Z/pZ −→ µp of group schemes given by 1 7→ ζp , w ...
... have good reduction, if m - char(k) we get EK [m](K) ,→ Ek (k) is injective. However, while the theorem holds for elliptic curves, it fails for even simple examples of finite flat group schemes. For example, if we let R = Zp [ζp ] and consider the map Z/pZ −→ µp of group schemes given by 1 7→ ζp , w ...
Sheaf theory - Department of Mathematics
... satisfying the following local coherence condition: There are a cover U = {Ui } of U and sections s(i) ∈ F (Ui ) such that t(x) = s(i)x ∈ Fx for all x ∈ Ui . The presheaf F + has natural restriction maps given by restriction of functions and it is easy to check that F + is a sheaf. Indeed, we may ac ...
... satisfying the following local coherence condition: There are a cover U = {Ui } of U and sections s(i) ∈ F (Ui ) such that t(x) = s(i)x ∈ Fx for all x ∈ Ui . The presheaf F + has natural restriction maps given by restriction of functions and it is easy to check that F + is a sheaf. Indeed, we may ac ...
Categories and functors
... the free abelian group functor G = Z[−] : Sets → Ab. Example 22.11. Hom(A, −) and A ⊗ − are adjoint additive functors: Hom(A ⊗ B, C) ∼ = Hom(B, Hom(A, C)) The well-know fact that tensor product is right exact and Hom is left exact are special cases of the following more general theorem where the def ...
... the free abelian group functor G = Z[−] : Sets → Ab. Example 22.11. Hom(A, −) and A ⊗ − are adjoint additive functors: Hom(A ⊗ B, C) ∼ = Hom(B, Hom(A, C)) The well-know fact that tensor product is right exact and Hom is left exact are special cases of the following more general theorem where the def ...
ON MACKEY TOPOLOGIES IN TOPOLOGICAL ABELIAN
... class of topological abelian groups, called nuclear groups, on which the circle group is injective. We refer the reader to the source for the definition. The class is described there as, roughly speaking, the smallest class of groups containing both locally compact abelian groups and nuclear topologi ...
... class of topological abelian groups, called nuclear groups, on which the circle group is injective. We refer the reader to the source for the definition. The class is described there as, roughly speaking, the smallest class of groups containing both locally compact abelian groups and nuclear topologi ...