Morphisms of Algebraic Stacks

... Let X = [U/R] be a presentation of an algebraic stack. Then the properties of the diagonal of X over S, are the properties of the morphism j : R → U ×S U . For example, if X = [S/G] for some smooth group G in algebraic spaces over S then j is the structure morphism G → S. Hence the diagonal is not a ...

... Let X = [U/R] be a presentation of an algebraic stack. Then the properties of the diagonal of X over S, are the properties of the morphism j : R → U ×S U . For example, if X = [S/G] for some smooth group G in algebraic spaces over S then j is the structure morphism G → S. Hence the diagonal is not a ...

An Introduction to Topological Groups

... Since G → G, x 7→ yx is a homeomorphism, there exists a neighborhood W of e such that yW ⊆ V . Observe, that yW is a neighborhood of y ! By the continuity of the multiplication at (e, e), there is an open neighborhood U of e such that U · U ⊆ W . W.l.o.g. we may assume that U is symmetric (i.e. U = ...

... Since G → G, x 7→ yx is a homeomorphism, there exists a neighborhood W of e such that yW ⊆ V . Observe, that yW is a neighborhood of y ! By the continuity of the multiplication at (e, e), there is an open neighborhood U of e such that U · U ⊆ W . W.l.o.g. we may assume that U is symmetric (i.e. U = ...

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

... In some elementary presentations, the representability hypothesis for K∗ is replaced by asking it to be free on models, see for instance [D]. In [EM1], Eilenberg-MacLane apply their theorem to compare singular chains S∗ (X) and normalized cubical chains C∗ (X): for p ≥ 0, the abelian group Sp (X) is ...

... In some elementary presentations, the representability hypothesis for K∗ is replaced by asking it to be free on models, see for instance [D]. In [EM1], Eilenberg-MacLane apply their theorem to compare singular chains S∗ (X) and normalized cubical chains C∗ (X): for p ≥ 0, the abelian group Sp (X) is ...

Class Field Theory

... H ENSEL (1861–1941). He defined the field of p-adic numbers (as the set of infinite sums P ...

... H ENSEL (1861–1941). He defined the field of p-adic numbers (as the set of infinite sums P ...

Abelian Varieties

... the case of an elliptic curve, E _ D E. In general, A and A_ are isogenous, but they are not equal (and usually not even isomorphic). Appropriately interpreted, most of the statements in Silverman’s books on elliptic curves hold for abelian varieties, but because we don’t have equations, the proofs ...

... the case of an elliptic curve, E _ D E. In general, A and A_ are isogenous, but they are not equal (and usually not even isomorphic). Appropriately interpreted, most of the statements in Silverman’s books on elliptic curves hold for abelian varieties, but because we don’t have equations, the proofs ...

A Concise Course in Algebraic Topology J. P. May

... standard approach. However, to the best of my knowledge, there exists no rigorous exposition of this approach in the literature, at any level. More centrally, there now exist axiomatic treatments of large swaths of homotopy theory based on Quillen’s theory of closed model categories. While I do not ...

... standard approach. However, to the best of my knowledge, there exists no rigorous exposition of this approach in the literature, at any level. More centrally, there now exist axiomatic treatments of large swaths of homotopy theory based on Quillen’s theory of closed model categories. While I do not ...

A Concise Course in Algebraic Topology JP May

... standard approach. However, to the best of my knowledge, there exists no rigorous exposition of this approach in the literature, at any level. More centrally, there now exist axiomatic treatments of large swaths of homotopy theory based on Quillen’s theory of closed model categories. While I do not ...

... standard approach. However, to the best of my knowledge, there exists no rigorous exposition of this approach in the literature, at any level. More centrally, there now exist axiomatic treatments of large swaths of homotopy theory based on Quillen’s theory of closed model categories. While I do not ...

Algebraic D-groups and differential Galois theory

... In particular, the above map deﬁnes a K-rational isomorphism between the vector groups τ (G)e and T (G)e = L(G). Note that we have again an exact sequence 0 → τ (G)e → τ (G) → G → e of algebraic groups over K, which by virtue of the (canonical) isomorphism between τ (G)e and L(G) given by Lemma 2.3 ...

... In particular, the above map deﬁnes a K-rational isomorphism between the vector groups τ (G)e and T (G)e = L(G). Note that we have again an exact sequence 0 → τ (G)e → τ (G) → G → e of algebraic groups over K, which by virtue of the (canonical) isomorphism between τ (G)e and L(G) given by Lemma 2.3 ...

Some structure theorems for algebraic groups

... The above theorems have a long history. Theorem 1 was first obtained by Rosenlicht in 1956 for smooth connected algebraic groups, see [48, Sec. 5]. The version presented here is due to Demazure and Gabriel, see [22, III.3.8]. In the setting of smooth connected algebraic groups again, Theorem 2 was a ...

... The above theorems have a long history. Theorem 1 was first obtained by Rosenlicht in 1956 for smooth connected algebraic groups, see [48, Sec. 5]. The version presented here is due to Demazure and Gabriel, see [22, III.3.8]. In the setting of smooth connected algebraic groups again, Theorem 2 was a ...

Chapter IV. Quotients by group schemes. When we work with group

... there does not exist a geometric quotient. (Indeed, the orbits in M2 (k) would be the preimages of the k-valued points of the geometric quotient. Cf. the second example in (4.6).) This suggests that the points in a subvariety of the form N (2λ, λ2 ) ⊂ M2,k are the “bad” points for the given action o ...

... there does not exist a geometric quotient. (Indeed, the orbits in M2 (k) would be the preimages of the k-valued points of the geometric quotient. Cf. the second example in (4.6).) This suggests that the points in a subvariety of the form N (2λ, λ2 ) ⊂ M2,k are the “bad” points for the given action o ...

A SHEAF THEORETIC APPROACH TO MEASURE THEORY Matthew Jackson by

... The categories S, G, and T share the same taxonomic classification from Definition 2; they are all large, locally small, categories. They are also all examples of concrete categories (categories whose objects are “sets with structure” and whose arrows are functions from these underlying sets). ...

... The categories S, G, and T share the same taxonomic classification from Definition 2; they are all large, locally small, categories. They are also all examples of concrete categories (categories whose objects are “sets with structure” and whose arrows are functions from these underlying sets). ...

Hodge Cycles on Abelian Varieties

... consequences for the algebraicity of the periods of abelian integrals: briefly, it allows one to prove all arithmetic properties of abelian periods that would follow from knowing the Hodge conjecture for abelian varieties.3 —————————————————In more detail, the main theorem proved in these notes is t ...

... consequences for the algebraicity of the periods of abelian integrals: briefly, it allows one to prove all arithmetic properties of abelian periods that would follow from knowing the Hodge conjecture for abelian varieties.3 —————————————————In more detail, the main theorem proved in these notes is t ...

A Coherence Criterion for Fréchet Modules

... In the literature, one finds essentially two general criteria to get the finiteness of the cohomology groups of complexes of locally convex topological vector spaces. They are (a) If u· : G· −→ F · is a compact morphism of complexes of Fréchet spaces then dim H k (F · ) < +∞ for any k ∈ ZZ such tha ...

... In the literature, one finds essentially two general criteria to get the finiteness of the cohomology groups of complexes of locally convex topological vector spaces. They are (a) If u· : G· −→ F · is a compact morphism of complexes of Fréchet spaces then dim H k (F · ) < +∞ for any k ∈ ZZ such tha ...

Math 248A. Homework 10 1. (optional) The purpose of this (optional

... (iii) Assume K/k is Galois, and let Σ denote the set of subgroups of Gal(K/k) that arise in the form Gal(K/K 0 ) for intermediate extensions K 0 . By (ii), K 0 7→ Gal(K/K 0 ) is a bijection from the set of intermediate extensions to the set Σ, with K-Galois subextensions corresponding to normal subg ...

... (iii) Assume K/k is Galois, and let Σ denote the set of subgroups of Gal(K/k) that arise in the form Gal(K/K 0 ) for intermediate extensions K 0 . By (ii), K 0 7→ Gal(K/K 0 ) is a bijection from the set of intermediate extensions to the set Σ, with K-Galois subextensions corresponding to normal subg ...

Abelian group

... Тhe simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary orders. Even though the de ...

... Тhe simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary orders. Even though the de ...

Basic Modern Algebraic Geometry

... a tuple of objects (A1 , A2 , . . . , An ) from categories Ci , i = 1, 2, . . . , n assigns an object T (A1 , A2 , . . . , An ) if a category D, and which is covariant in some of the variables, contravariant in othes, and such that the obvious generalization of the naturallity condition holds. In pa ...

... a tuple of objects (A1 , A2 , . . . , An ) from categories Ci , i = 1, 2, . . . , n assigns an object T (A1 , A2 , . . . , An ) if a category D, and which is covariant in some of the variables, contravariant in othes, and such that the obvious generalization of the naturallity condition holds. In pa ...

Examples - Stacks Project

... (using multi-index notation) such that for each d ≥ 0 there are only finitely many nonzero aI for |I| = d. The maximal ideal m0 ⊂ R∧ is the collection of f with zero constant term. In particular, the element f = x1 + x22 + x33 + . . . is in m0 but not in mR∧ which shows that (1) is false in this exa ...

... (using multi-index notation) such that for each d ≥ 0 there are only finitely many nonzero aI for |I| = d. The maximal ideal m0 ⊂ R∧ is the collection of f with zero constant term. In particular, the element f = x1 + x22 + x33 + . . . is in m0 but not in mR∧ which shows that (1) is false in this exa ...

Towards a p-adic theory of harmonic weak Maass forms

... After Ramanujan’s premature death the notion of mock modular forms continued to baffle mathematicians for several decades. The resemblance with the theory of modular forms suggested a relation between the two, but there was no obvious way to relate the transformation properties of one family to anot ...

... After Ramanujan’s premature death the notion of mock modular forms continued to baffle mathematicians for several decades. The resemblance with the theory of modular forms suggested a relation between the two, but there was no obvious way to relate the transformation properties of one family to anot ...

Galois actions on homotopy groups of algebraic varieties

... The étale and `–adic homotopy types carry natural Galois actions, and the main aim of this paper is to study their structure. In many respects, the analogous question for XC has already been addressed, with Katzarkov, Pantev and Toën [22] and the author [34] describing mixed Hodge structures on the ...

... The étale and `–adic homotopy types carry natural Galois actions, and the main aim of this paper is to study their structure. In many respects, the analogous question for XC has already been addressed, with Katzarkov, Pantev and Toën [22] and the author [34] describing mixed Hodge structures on the ...

THE INVERSE PROBLEM OF GALOIS THEORY 1. Introduction Let

... the creation of a database for number fields. It encompasses roughly 100 000 polynomials generating distinct number fields over the rationals of degrees up to 15. The database contains polynomials for all transitive permutation groups up to that degree, and is accessed via the computer algebra syste ...

... the creation of a database for number fields. It encompasses roughly 100 000 polynomials generating distinct number fields over the rationals of degrees up to 15. The database contains polynomials for all transitive permutation groups up to that degree, and is accessed via the computer algebra syste ...

PDF

... sets of Spec(R) in the Zariski topology. This isomorphism between these two lattices of open sets can be used to equate the sheaf Spec(R) with the structure sheaf of the variety V , showing that the two objects are identical in every respect except for the minor detail of Spec(R) having more points ...

... sets of Spec(R) in the Zariski topology. This isomorphism between these two lattices of open sets can be used to equate the sheaf Spec(R) with the structure sheaf of the variety V , showing that the two objects are identical in every respect except for the minor detail of Spec(R) having more points ...

Several approaches to non-archimedean geometry

... L : V → V 0 between k-Banach spaces has bounded inverse. (Hint: copy the classical proof over R.) This theorem is fundamental in non-archimedean analysis and geometry, and it fails if the absolute value on k is trivial. If we choose two different presentations Tn /I ' A and Tm /J ' A of a k-affinoid ...

... L : V → V 0 between k-Banach spaces has bounded inverse. (Hint: copy the classical proof over R.) This theorem is fundamental in non-archimedean analysis and geometry, and it fails if the absolute value on k is trivial. If we choose two different presentations Tn /I ' A and Tm /J ' A of a k-affinoid ...

Higher regulators and values of L

... mixed Hodge structure on H ~ ( X ) . For example, if X is compact, then ch (Kj(X)) = 0 for j > 0. It turns out that the Hodge conditions can be used, and, untangling them, it is possible to obtain finer analytic invariants of the elements of K.(X) than the usual cohomology classes. For the case of C ...

... mixed Hodge structure on H ~ ( X ) . For example, if X is compact, then ch (Kj(X)) = 0 for j > 0. It turns out that the Hodge conditions can be used, and, untangling them, it is possible to obtain finer analytic invariants of the elements of K.(X) than the usual cohomology classes. For the case of C ...

Vector bundles and torsion free sheaves on degenerations of elliptic

... In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projectiv ...

... In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projectiv ...

My notes - Harvard Mathematics Department

... takes values in Z. Using this theorem, let’s see (loosely) why the Jacobian J(X) admits the structure of an algebraic variety. The fourth statement of the quoted theorem is the easiest, so let’s try to find a hermitian form on H 0 (X, ΩX )∨ satisfying the above properties. So, we’ll need a lemma fro ...

... takes values in Z. Using this theorem, let’s see (loosely) why the Jacobian J(X) admits the structure of an algebraic variety. The fourth statement of the quoted theorem is the easiest, so let’s try to find a hermitian form on H 0 (X, ΩX )∨ satisfying the above properties. So, we’ll need a lemma fro ...