Affine group schemes over symmetric monoidal categories

... [Tate 1997], for instance), they are a natural starting point for such a theory. For more on group schemes, we refer the reader to [Demazure and Gabriel 1970]. 2. Notations In this section, we introduce notation that we will maintain throughout this paper. We let (C, ⊗, 1) denote an abelian symmetri ...

... [Tate 1997], for instance), they are a natural starting point for such a theory. For more on group schemes, we refer the reader to [Demazure and Gabriel 1970]. 2. Notations In this section, we introduce notation that we will maintain throughout this paper. We let (C, ⊗, 1) denote an abelian symmetri ...

AUTOMORPHISM GROUPS AND PICARD GROUPS OF ADDITIVE

... modules N . Then Pic(C) is naturally isomorphic to AutA (C). Such construction agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as develop ...

... modules N . Then Pic(C) is naturally isomorphic to AutA (C). Such construction agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as develop ...

Distintos tipos de estructuras celulares en espacios topológicos

... spaces generalize CW-complexes and our constructions, applications and results keep the geometric intuition and the combinatorial structure of J.H.C. Whitehead’s original theory. We delve deeply into the topological and homotopical properties of CW(A)-complexes, their localizations and changes of th ...

... spaces generalize CW-complexes and our constructions, applications and results keep the geometric intuition and the combinatorial structure of J.H.C. Whitehead’s original theory. We delve deeply into the topological and homotopical properties of CW(A)-complexes, their localizations and changes of th ...

arXiv:1510.01797v3 [math.CT] 21 Apr 2016 - Mathematik, Uni

... In the case where R = k is a field, Sweedler’s finite dual coalgebra construction A• [22] of a kalgebra A provides an extension of the above construction to arbitrary algebras. The underlying vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinit ...

... In the case where R = k is a field, Sweedler’s finite dual coalgebra construction A• [22] of a kalgebra A provides an extension of the above construction to arbitrary algebras. The underlying vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinit ...

DIRECTED HOMOTOPY THEORY, II. HOMOTOPY CONSTRUCTS

... Directed homotopies are based on the standard directed interval ↑I, the directed cylinder functor, ↑I(X) = X×↑I, and its right adjoint, the directed path functor, ↑P (X) = X ↑I . Such functors, with a structure consisting of faces, degeneracy, connections and interchange, satisfy the axioms of an IP ...

... Directed homotopies are based on the standard directed interval ↑I, the directed cylinder functor, ↑I(X) = X×↑I, and its right adjoint, the directed path functor, ↑P (X) = X ↑I . Such functors, with a structure consisting of faces, degeneracy, connections and interchange, satisfy the axioms of an IP ...

Topological types of Algebraic stacks - IBS-CGP

... between the topological type h(F ) of F in a topos T and the topological type h(T /F ) of the localized topos T /F (see 2.3.30). (iii) The Barnea-Schlank model category structure in the case of pro-simplicial sets is simply the strict model category structure [15, 4.15] on the pro-category induced b ...

... between the topological type h(F ) of F in a topos T and the topological type h(T /F ) of the localized topos T /F (see 2.3.30). (iii) The Barnea-Schlank model category structure in the case of pro-simplicial sets is simply the strict model category structure [15, 4.15] on the pro-category induced b ...

A Bousfield-Kan algorithm for computing homotopy

... a simplicial set and converges to its homotopy groups; the classical Adams spectral sequence can be derived of the BKSS. Combinatorial does not imply constructive3 . In the case of the BKSS, its combinatorial nature does not imply it is constructive. For example one may naively hope to have an “Adam ...

... a simplicial set and converges to its homotopy groups; the classical Adams spectral sequence can be derived of the BKSS. Combinatorial does not imply constructive3 . In the case of the BKSS, its combinatorial nature does not imply it is constructive. For example one may naively hope to have an “Adam ...

course notes

... By abuse of notation, we will usually write χ for χ̃. Furthermore, we let χ̄ denote the complex conjugate of χ, defined by χ̄ : Z → C m 7→ χ(m). One checks immediately that χ̄ is a Dirichlet character satisfying ...

... By abuse of notation, we will usually write χ for χ̃. Furthermore, we let χ̄ denote the complex conjugate of χ, defined by χ̄ : Z → C m 7→ χ(m). One checks immediately that χ̄ is a Dirichlet character satisfying ...

Constructible Sheaves, Stalks, and Cohomology

... As we mentioned earlier, strictly Henselian local rings are for the étale topology what local rings are for the Zariski topology. As a first step toward fully realizing this analogy, we have the following result. Theorem 2.5. Let R be a strictly Henselian local ring. Any étale neighborhood of the c ...

... As we mentioned earlier, strictly Henselian local rings are for the étale topology what local rings are for the Zariski topology. As a first step toward fully realizing this analogy, we have the following result. Theorem 2.5. Let R be a strictly Henselian local ring. Any étale neighborhood of the c ...

FORMAL PLETHORIES Contents 1. Introduction 3 1.1. Outline of the

... converging conditionally to the homotopy groups of the unstable E-completion of X. The Ext group on the left is a nonlinear comonad-derived functor of homomorphisms of comodules over a comonad G, a category which is closely related to plethory comodules. Note that if E∗ is not a graded field, the pl ...

... converging conditionally to the homotopy groups of the unstable E-completion of X. The Ext group on the left is a nonlinear comonad-derived functor of homomorphisms of comodules over a comonad G, a category which is closely related to plethory comodules. Note that if E∗ is not a graded field, the pl ...

Iterated Bar Complexes of E-infinity Algebras and Homology

... E1 = Σ−n Bn (H∗ (A)) ⇒ H∗ (Σ−n Bn (A)) = HE∗ n (A), whose E1 -term reduces to the usual n-fold bar construction of a commutative algebra for any n > 1 (the homology of an En -algebra forms a commutative algebra for n > 1, an associative algebra for n = 1). In the case of the cochain algebra A = C∗ ( ...

... E1 = Σ−n Bn (H∗ (A)) ⇒ H∗ (Σ−n Bn (A)) = HE∗ n (A), whose E1 -term reduces to the usual n-fold bar construction of a commutative algebra for any n > 1 (the homology of an En -algebra forms a commutative algebra for n > 1, an associative algebra for n = 1). In the case of the cochain algebra A = C∗ ( ...

HIGHER HOMOTOPY OF GROUPS DEFINABLE IN O

... Definable groups in o-minimal structures have been studied by several authors. The class of such groups includes all algebraic groups over either a real closed field or an algebraically closed field, but is not limited to such groups. Any compact Lie group is isomorphic to a real algebraic subgroup ...

... Definable groups in o-minimal structures have been studied by several authors. The class of such groups includes all algebraic groups over either a real closed field or an algebraically closed field, but is not limited to such groups. Any compact Lie group is isomorphic to a real algebraic subgroup ...

slides

... The existence of such products for each ordered pair of objects (and also what is called a terminal object) gives rise to a monoidal structure on C which is referred to as a cartesian monoidal category. For instance the category of sets with the cartesian product is a cartesian monoidal category, so ...

... The existence of such products for each ordered pair of objects (and also what is called a terminal object) gives rise to a monoidal structure on C which is referred to as a cartesian monoidal category. For instance the category of sets with the cartesian product is a cartesian monoidal category, so ...

A course on finite flat group schemes and p

... 2.3.1. Algebraic affine group schemes. An algebraic affine group scheme over R is an affine group scheme such that its Hopf algebra is finitely generated as an R-algebra, which means that ‘we have only finitely many coordinates’. 2.3.2. Translation action. For g ∈ G(R) we have the left translation λ ...

... 2.3.1. Algebraic affine group schemes. An algebraic affine group scheme over R is an affine group scheme such that its Hopf algebra is finitely generated as an R-algebra, which means that ‘we have only finitely many coordinates’. 2.3.2. Translation action. For g ∈ G(R) we have the left translation λ ...

A geometric introduction to K-theory

... (x1 − p1 , . . . , xn − pn ) where P = (p1 , . . . , pn ). The localization is necessary because Z ∩W might have points other than P in it, and our definition needs to only depend on what is happening near P . The best way to get a feeling for the above definition is via some easy examples: Example ...

... (x1 − p1 , . . . , xn − pn ) where P = (p1 , . . . , pn ). The localization is necessary because Z ∩W might have points other than P in it, and our definition needs to only depend on what is happening near P . The best way to get a feeling for the above definition is via some easy examples: Example ...

Hodge Cycles on Abelian Varieties

... violate the sign conventions for complexes.) R A number !, 2 Hn .X; Q/, is called a period of !. The map in (1.2) identifies H n .X; Q/ with the space of classes of closed forms whose periods are all rational. Theorem 1.2 can be restated as follows: a closed differential form is exact if all its ...

... violate the sign conventions for complexes.) R A number !, 2 Hn .X; Q/, is called a period of !. The map in (1.2) identifies H n .X; Q/ with the space of classes of closed forms whose periods are all rational. Theorem 1.2 can be restated as follows: a closed differential form is exact if all its ...

Galois actions on homotopy groups of algebraic varieties

... the Mittag–Leffler condition (for instance if the groups are finite). When working with the étale homotopy type XeK t , we will usually apply this construction to Z` –local systems fMi D M=`i gi . In that case, the exact sequence above becomes the comparison between étale cohomology and Jannsen’s co ...

... the Mittag–Leffler condition (for instance if the groups are finite). When working with the étale homotopy type XeK t , we will usually apply this construction to Z` –local systems fMi D M=`i gi . In that case, the exact sequence above becomes the comparison between étale cohomology and Jannsen’s co ...