Morphisms of Algebraic Stacks

... best to treat them all separately. For morphisms of algebraic stacks which are representable by algebraic spaces we have already defined a large number of types of morphisms, see Properties of Stacks, Section 3. For each corresponding case in this chapter we have to make sure the definition in the g ...

... best to treat them all separately. For morphisms of algebraic stacks which are representable by algebraic spaces we have already defined a large number of types of morphisms, see Properties of Stacks, Section 3. For each corresponding case in this chapter we have to make sure the definition in the g ...

arXiv:math/0009100v1 [math.DG] 10 Sep 2000

... groupoid difference map, is continuous, where the pullbak G ×α G = {(a, b) ∈ G × G : α(a) = α(b)} has the subspace topology from G × G. Again if one of the maps α, β and the inversion are continuous, then the other map is continuous. Let X be a topological space. Then G = X × X is a topological grou ...

... groupoid difference map, is continuous, where the pullbak G ×α G = {(a, b) ∈ G × G : α(a) = α(b)} has the subspace topology from G × G. Again if one of the maps α, β and the inversion are continuous, then the other map is continuous. Let X be a topological space. Then G = X × X is a topological grou ...

Abstract Algebra - UCLA Department of Mathematics

... In abstract algebra, we attempt to provide lists of properties that common mathematical objects satisfy. Given such a list of properties, we impose them as “axioms”, and we study the properties of objects that satisfy these axioms. The objects that we deal with most in the first part of these notes ...

... In abstract algebra, we attempt to provide lists of properties that common mathematical objects satisfy. Given such a list of properties, we impose them as “axioms”, and we study the properties of objects that satisfy these axioms. The objects that we deal with most in the first part of these notes ...

THE GEOMETRIES OF 3

... central results in this area. This very naturally leads to the second section, on orbifolds, which is based on Thurston's notes [66]. This term is due to Thurston and is used in nice cases to describe the quotient of a manifold by a group which need not act freely. This is called a K-manifold by som ...

... central results in this area. This very naturally leads to the second section, on orbifolds, which is based on Thurston's notes [66]. This term is due to Thurston and is used in nice cases to describe the quotient of a manifold by a group which need not act freely. This is called a K-manifold by som ...

Contents 1. Introduction 2 2. The monoidal background 5 2.1

... of morphisms in cohomology. For example, the natural integration morphism : S ∗ (X) −→ Ω∗ (X) is not multiplicative, but the higher homotopies defined by acyclic models define a dash morphism which induces a multiplicative morphism between the singular cohomology and the De Rham cohomology of X, (se ...

... of morphisms in cohomology. For example, the natural integration morphism : S ∗ (X) −→ Ω∗ (X) is not multiplicative, but the higher homotopies defined by acyclic models define a dash morphism which induces a multiplicative morphism between the singular cohomology and the De Rham cohomology of X, (se ...

Commutative Algebra

... one-semester course for its study. The main motivation for this is its many applications in both algebraic geometry and (algebraic) number theory. Especially the connection between commutative algebra and algebraic geometry is very deep — in fact, to a certain extent one can say that these two field ...

... one-semester course for its study. The main motivation for this is its many applications in both algebraic geometry and (algebraic) number theory. Especially the connection between commutative algebra and algebraic geometry is very deep — in fact, to a certain extent one can say that these two field ...

COMPLEXES OF INJECTIVE kG-MODULES 1. Introduction Let k be

... We prove that the tensor product over k of complexes in K(Inj kG) corresponds 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 ...

... We prove that the tensor product over k of complexes in K(Inj kG) corresponds 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 ...

... one-semester course for its study. The main motivation for this is its many applications in both algebraic geometry and (algebraic) number theory. Especially the connection between commutative algebra and algebraic geometry is very deep — in fact, to a certain extent one can say that these two field ...

ON THE REPRESENTABILITY OF ACTIONS IN A SEMI

... contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove ﬁrst that the representability of Act(−, X) reduces to the preservation of binary coproducts. Next we give both a very ...

... contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove ﬁrst that the representability of Act(−, X) reduces to the preservation of binary coproducts. Next we give both a very ...

Geometry Concepts THEOREMS

... Theorem 5.1 Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side. Theorem 5.2 Perpendicular Bisector Theorem: If a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoint ...

... Theorem 5.1 Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side. Theorem 5.2 Perpendicular Bisector Theorem: If a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoint ...

Topolog´ıa Algebraica de Espacios Topológicos Finitos y Aplicaciones

... The main goal of this Thesis is to study and to delve deeper into the development of the theory of finite spaces and to investigate their applications to the homotopy theory and simple homotopy theory of polyhedra and general topological spaces. We use, in particular, some of the results that we obt ...

... The main goal of this Thesis is to study and to delve deeper into the development of the theory of finite spaces and to investigate their applications to the homotopy theory and simple homotopy theory of polyhedra and general topological spaces. We use, in particular, some of the results that we obt ...

barmakthesis.pdf

... conjecture. As a consequence of Perelman’s proof of Poincaré conjecture, one deduces that this conjecture is true for some complexes called standard spines, but the problem is still open for general polyhedra of dimension 2. With the results developed in this Thesis we substantially extend the clas ...

... conjecture. As a consequence of Perelman’s proof of Poincaré conjecture, one deduces that this conjecture is true for some complexes called standard spines, but the problem is still open for general polyhedra of dimension 2. With the results developed in this Thesis we substantially extend the clas ...

TRIANGLE CONGRUENCE

... 7. Key question: Can a given region tessellate the plane? 8. One museum where many tessellations can be found is the Alhambra. 9. Yes. Since the sum of the angle measures in ABCD is 360, it is possible to have a different angle from each of the four congruent quadrilaterals meeting at a single point ...

... 7. Key question: Can a given region tessellate the plane? 8. One museum where many tessellations can be found is the Alhambra. 9. Yes. Since the sum of the angle measures in ABCD is 360, it is possible to have a different angle from each of the four congruent quadrilaterals meeting at a single point ...

An Introduction to Topological Groups

... Let G be a group and τ a topology on G which is Čech–complete (e.g. complete and metrizable). If all left and right translations are continuous, then (G, τ) is a topological group. ...

... Let G be a group and τ a topology on G which is Čech–complete (e.g. complete and metrizable). If all left and right translations are continuous, then (G, τ) is a topological group. ...

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 ...

EQUIVARIANT SYMMETRIC MONOIDAL STRUCTURES 1

... that H/K ⊂ F 0 and any X ∈ F 0 can be expressed as a disjoint union of copies of various H/K. These parameterize the norms that an F 0 -commutative monoid has. The remaining axioms for an indexing coefficient system guarantee that (1) M is a commutative monoid (in a traditional sense) and (2) norm m ...

... that H/K ⊂ F 0 and any X ∈ F 0 can be expressed as a disjoint union of copies of various H/K. These parameterize the norms that an F 0 -commutative monoid has. The remaining axioms for an indexing coefficient system guarantee that (1) M is a commutative monoid (in a traditional sense) and (2) norm m ...

Class Field Theory

... B RAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. W EIL (1906–1998). Defined the Weil group, which enabled him to give a common generalization of Artin L-series and Hecke L-series. C HEVALLEY (1909–84). The main statem ...

... B RAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. W EIL (1906–1998). Defined the Weil group, which enabled him to give a common generalization of Artin L-series and Hecke L-series. C HEVALLEY (1909–84). The main statem ...

Hyperbolic Geometry and 3-Manifold Topology

... In particular we will work with simplicial hyperbolic surfaces instead of the more general Cat(−1) ones and replace Shrinkwrapping in the smooth category by PL Shrinkwrapping. We will also give an elementary proof of the Tameness Criterion that eliminates the use of results about the Thurston and Gr ...

... In particular we will work with simplicial hyperbolic surfaces instead of the more general Cat(−1) ones and replace Shrinkwrapping in the smooth category by PL Shrinkwrapping. We will also give an elementary proof of the Tameness Criterion that eliminates the use of results about the Thurston and Gr ...

Vector Bundles And F Theory

... of flat G-valued connections. The moduli space of such flat connections has an elementary, explicit description: a flat connection on the two-torus is given by a pair of commuting elements in the gauge group G. Two such connections are equivalent if and only if they are isomorphic, which is the same ...

... of flat G-valued connections. The moduli space of such flat connections has an elementary, explicit description: a flat connection on the two-torus is given by a pair of commuting elements in the gauge group G. Two such connections are equivalent if and only if they are isomorphic, which is the same ...

Constructible Sheaves, Stalks, and Cohomology

... over i. The morphisms are the obvious thing, namely local R-homomorphisms respecting the embeddings of the residue fields into ks . Since local schemes are connected, by the elementary rigidity for pointed étale maps (see Lemma 3.1, which does not rely on the present considerations) and denominator- ...

... over i. The morphisms are the obvious thing, namely local R-homomorphisms respecting the embeddings of the residue fields into ks . Since local schemes are connected, by the elementary rigidity for pointed étale maps (see Lemma 3.1, which does not rely on the present considerations) and denominator- ...

Real Algebraic Sets

... Semialgebraic Sets In this chapter we present some basic topological facts concerning semialgebraic sets, which are subsets of Rn defined by combinations of polynomial equations and inequalities. One of the main properties is the fact that a compact semialgebraic set can be triangulated. We introduc ...

... Semialgebraic Sets In this chapter we present some basic topological facts concerning semialgebraic sets, which are subsets of Rn defined by combinations of polynomial equations and inequalities. One of the main properties is the fact that a compact semialgebraic set can be triangulated. We introduc ...

Abelian Varieties

... .A.k/ W Zei /.A_ .k/ W Zfj / is independent of the choice of the ei and fj . [This is an exercise in linear algebra.] The first chapter of these notes covers the basic (geometric) theory of abelian varieties over arbitrary fields, the second chapter discusses some of the arithmetic of abelian variet ...

... .A.k/ W Zei /.A_ .k/ W Zfj / is independent of the choice of the ei and fj . [This is an exercise in linear algebra.] The first chapter of these notes covers the basic (geometric) theory of abelian varieties over arbitrary fields, the second chapter discusses some of the arithmetic of abelian variet ...

The local structure of compactified Jacobians

... have nice properties: often it has inﬁnitely many connected components (that is, is not of ﬁnite type), and these components fail to be proper. To construct a well-behaved compactiﬁed Jacobian, one must modify the moduli problem. There are a number of diﬀerent ways to do this, and the literature on ...

... have nice properties: often it has inﬁnitely many connected components (that is, is not of ﬁnite type), and these components fail to be proper. To construct a well-behaved compactiﬁed Jacobian, one must modify the moduli problem. There are a number of diﬀerent ways to do this, and the literature on ...

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 ...