Morphisms of Algebraic Stacks

... (1) f is quasi-separated, (2) ∆f is quasi-compact, or (3) ∆f is of finite type. Proof. The statements “f is quasi-separated”, “∆f is quasi-compact”, and “∆f is of finite type” refer to the notions defined in Properties of Stacks, Section 3. Note that (2) and (3) are equivalent in view of the fact th ...

... (1) f is quasi-separated, (2) ∆f is quasi-compact, or (3) ∆f is of finite type. Proof. The statements “f is quasi-separated”, “∆f is quasi-compact”, and “∆f is of finite type” refer to the notions defined in Properties of Stacks, Section 3. Note that (2) and (3) are equivalent in view of the fact th ...

Algebra - University at Albany

... integers and rational numbers, Z[ζn ] and Q[ζn ], polynomials, group rings, and more. Free modules and chain conditions are studied, and the elementary theory of vector spaces and matrices is developed. The chapter closes with the study of rings and modules of fractions, which we shall apply to the ...

... integers and rational numbers, Z[ζn ] and Q[ζn ], polynomials, group rings, and more. Free modules and chain conditions are studied, and the elementary theory of vector spaces and matrices is developed. The chapter closes with the study of rings and modules of fractions, which we shall apply to the ...

Andr´e-Quillen (co)Homology, Abelianization and Stabilization

... If C and D are model categories and F : C −→ D takes fibrations to fibration and preserves weak equivalences between fibrant objects, define its total right derived functor by RF : HoC −→ HoD RF(X ) = F(f (X )) Maria Basterra ...

... If C and D are model categories and F : C −→ D takes fibrations to fibration and preserves weak equivalences between fibrant objects, define its total right derived functor by RF : HoC −→ HoD RF(X ) = F(f (X )) Maria Basterra ...

Hartshorne Ch. II, §3 First Properties of Schemes

... ϕ on the subset f (Zij ) = Xij , and so ϕ exists. ϕ is moreover unique since it is unique locally by the universal property in the affine case. Problem 3.9. The Topological Space of a Product. Recall that in the category of varieties, the Zariski topology on the product of two varieties is not equal ...

... ϕ on the subset f (Zij ) = Xij , and so ϕ exists. ϕ is moreover unique since it is unique locally by the universal property in the affine case. Problem 3.9. The Topological Space of a Product. Recall that in the category of varieties, the Zariski topology on the product of two varieties is not equal ...

Hovhannes Khudaverdian's notes

... The linear space of vectors with standard addition law and vector product × is ring where multiplication is anticommutative (a × b = −b × a), has not element e and product of two orthogonal vectors is equal to zero. The space C([0, 1]) of continuous functions on [0, 1] is ring but it is not integral ...

... The linear space of vectors with standard addition law and vector product × is ring where multiplication is anticommutative (a × b = −b × a), has not element e and product of two orthogonal vectors is equal to zero. The space C([0, 1]) of continuous functions on [0, 1] is ring but it is not integral ...

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 ...

Commutative ideal theory without finiteness

... R-submodule of Q is completely Q-irreducible and has the form pn ZpZ , where p is a prime integer and n is an integer. Thus for R = Z every nonzero proper Qirreducible R-submodule of Q is a fractional ideal of a valuation overring of R. Moreover, every nonzero fractional R-ideal has a unique represe ...

... R-submodule of Q is completely Q-irreducible and has the form pn ZpZ , where p is a prime integer and n is an integer. Thus for R = Z every nonzero proper Qirreducible R-submodule of Q is a fractional ideal of a valuation overring of R. Moreover, every nonzero fractional R-ideal has a unique represe ...

A First Course in Abstract Algebra: Rings, Groups, and Fields

... made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apo ...

... made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apo ...

Commutative Algebra

... of ring theory that you already know from earlier courses (e. g. ideals, quotient rings, the homomorphism theorem, and unique prime factorization in principal ideal domains such as the integers or polynomial rings in one variable over a field), and move on to more advanced topics, some of which will ...

... of ring theory that you already know from earlier courses (e. g. ideals, quotient rings, the homomorphism theorem, and unique prime factorization in principal ideal domains such as the integers or polynomial rings in one variable over a field), and move on to more advanced topics, some of which will ...

... of ring theory that you already know from earlier courses (e. g. ideals, quotient rings, the homomorphism theorem, and unique prime factorization in principal ideal domains such as the integers or polynomial rings in one variable over a field), and move on to more advanced topics, some of which will ...

Abelian Varieties

... other than C, but are written in a language which makes them difficult to read. Mumford’s book (1970) is the only modern account of the subject, but as an introduction it is rather difficult. It treats only abelian varieties over algebraically closed fields; in particular, it does not cover the arit ...

... other than C, but are written in a language which makes them difficult to read. Mumford’s book (1970) is the only modern account of the subject, but as an introduction it is rather difficult. It treats only abelian varieties over algebraically closed fields; in particular, it does not cover the arit ...

Dilation Theory, Commutant Lifting and Semicrossed Products

... and Solel [40]. The language used there is a module theoretic approach, while we will mostly talk about representations instead. But the general constructs can, of course, be formulated in either language. Douglas and Paulsen focus on Shilov modules as a primary building block. Muhly and Solel adopt ...

... and Solel [40]. The language used there is a module theoretic approach, while we will mostly talk about representations instead. But the general constructs can, of course, be formulated in either language. Douglas and Paulsen focus on Shilov modules as a primary building block. Muhly and Solel adopt ...

HEIGHTS OF VARIETIES IN MULTIPROJECTIVE SPACES AND

... As for many central results in commutative algebra and in algebraic geometry, it is a non-effective statement. By the end of the 1980s, the estimation of the degree and the height of polynomials satisfying such an identity became a widely considered question in connection with problems in computer a ...

... As for many central results in commutative algebra and in algebraic geometry, it is a non-effective statement. By the end of the 1980s, the estimation of the degree and the height of polynomials satisfying such an identity became a widely considered question in connection with problems in computer a ...

Class Field Theory

... He studied the Gaussian integers ZŒi in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields. D IRICHLET (1805–1859). He introduced L-series, and used them t ...

... He studied the Gaussian integers ZŒi in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields. D IRICHLET (1805–1859). He introduced L-series, and used them t ...

A Book of Abstract Algebra

... During the seven years that have elapsed since publication of the first edition of A Book of Abstract Algebra, I have received letters from many readers with comments and suggestions. Moreover, a number of reviewers have gone over the text with the aim of finding ways to increase it ...

... During the seven years that have elapsed since publication of the first edition of A Book of Abstract Algebra, I have received letters from many readers with comments and suggestions. Moreover, a number of reviewers have gone over the text with the aim of finding ways to increase it ...

Sans titre

... (1) Let M be a coherent DX -module which is R-specializable along H. Let us first show the coherence of M(⇤H). This is a local problem; moreover, by induction on the cardinal of a generators system of M, we can assume that M is generated by one section m 2 M. After Exercise 9.2.4, there exists a non ...

... (1) Let M be a coherent DX -module which is R-specializable along H. Let us first show the coherence of M(⇤H). This is a local problem; moreover, by induction on the cardinal of a generators system of M, we can assume that M is generated by one section m 2 M. After Exercise 9.2.4, there exists a non ...

SEMIDEFINITE DESCRIPTIONS OF THE CONVEX HULL OF

... this paper is the polar of SO(n), the set of linear functionals that take value at most one on SO(n), i.e., SO(n)◦ = {Y ∈ Rn×n : Y, X ≤ 1 for all X ∈ SO(n)}, where we have identiﬁed Rn×n with its dual space via the trace inner product Y, X = tr(Y T X). These two convex bodies are closely related ...

... this paper is the polar of SO(n), the set of linear functionals that take value at most one on SO(n), i.e., SO(n)◦ = {Y ∈ Rn×n : Y, X ≤ 1 for all X ∈ SO(n)}, where we have identiﬁed Rn×n with its dual space via the trace inner product Y, X = tr(Y T X). These two convex bodies are closely related ...

Notes on Galois Theory

... Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by Aut E. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of Aut ...

... Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by Aut E. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of Aut ...

higher algebra

... 9.5 Principal Ideal Domains are Unique Factorization Domains . . 306 9.6 If D is a UFD, then so is D[x]. . . . . . . . . . . . . . . . . . 306 9.7 Appendix to Chapter 8: The Arithmetic of Quadratic Domains.313 9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 313 ...

... 9.5 Principal Ideal Domains are Unique Factorization Domains . . 306 9.6 If D is a UFD, then so is D[x]. . . . . . . . . . . . . . . . . . 306 9.7 Appendix to Chapter 8: The Arithmetic of Quadratic Domains.313 9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 313 ...

Lecture Notes

... automorphism of G; with inverse Cx 1 : The map C W x ! Cx is a group homomorphism from G into Aut.G/: Its kernel is the subgroup of G consisting of the elements x 2 G with the property that xyx 1 D y for all y 2 G; or, equivalently, that xy D yx for all y 2 G: Thus, the kernel of C equals the center ...

... automorphism of G; with inverse Cx 1 : The map C W x ! Cx is a group homomorphism from G into Aut.G/: Its kernel is the subgroup of G consisting of the elements x 2 G with the property that xyx 1 D y for all y 2 G; or, equivalently, that xy D yx for all y 2 G: Thus, the kernel of C equals the center ...

Lie groups, lecture notes

... automorphism of G; with inverse Cx 1 : The map C W x ! Cx is a group homomorphism from G into Aut.G/: Its kernel is the subgroup of G consisting of the elements x 2 G with the property that xyx 1 D y for all y 2 G; or, equivalently, that xy D yx for all y 2 G: Thus, the kernel of C equals the center ...

... automorphism of G; with inverse Cx 1 : The map C W x ! Cx is a group homomorphism from G into Aut.G/: Its kernel is the subgroup of G consisting of the elements x 2 G with the property that xyx 1 D y for all y 2 G; or, equivalently, that xy D yx for all y 2 G: Thus, the kernel of C equals the center ...

An introduction to matrix groups and their applications

... In Chapter 1 the general linear groups GLn (k) where k = R, the real numbers, or k = C, the complex numbers, are introduced and studied as both groups and topological spaces. Matrix groups are defined and a number of standard examples are discussed, including the unimodular groups SLn (k), orthogona ...

... In Chapter 1 the general linear groups GLn (k) where k = R, the real numbers, or k = C, the complex numbers, are introduced and studied as both groups and topological spaces. Matrix groups are defined and a number of standard examples are discussed, including the unimodular groups SLn (k), orthogona ...

Lectures on Etale Cohomology

... a ringed space .X; OX / admitting a finite open covering X D [Ui such that .Ui ; OX jUi / is an affine variety for each i and which satisfies the separation axiom. We often use X to denote .X; OX / as well as the underlying topological space. A regular map of varieties will sometimes be called a mor ...

... a ringed space .X; OX / admitting a finite open covering X D [Ui such that .Ui ; OX jUi / is an affine variety for each i and which satisfies the separation axiom. We often use X to denote .X; OX / as well as the underlying topological space. A regular map of varieties will sometimes be called a mor ...

Rational points on Shimura curves and Galois representations Carlos de Vera Piquero

... models of the affine quartic curve 2y 2 = 1 − 17x4 , due to Lind and Reichardt, and of Selmer’s cubic 3x3 +4y 3 = 5. These curves have rational points locally everywhere, but fail to have global Qrational points, thus they are counterexamples to the Hasse principle (see [Lin40, Rei42, Sel51]). Genus ...

... models of the affine quartic curve 2y 2 = 1 − 17x4 , due to Lind and Reichardt, and of Selmer’s cubic 3x3 +4y 3 = 5. These curves have rational points locally everywhere, but fail to have global Qrational points, thus they are counterexamples to the Hasse principle (see [Lin40, Rei42, Sel51]). Genus ...

Geometric structures on 3–manifolds - bcf.usc.edu

... to CP1 : We saw that in this case S must be homeomorphic to the sphere S2 or to the real projective plane RP2 = S2 / {±Id}. On the other hand, the case where Se is biholomorphically isomorphic to H2 covers all the other surfaces. In this ‘generic’ case, we saw that S admits a complete hyperbolic met ...

... to CP1 : We saw that in this case S must be homeomorphic to the sphere S2 or to the real projective plane RP2 = S2 / {±Id}. On the other hand, the case where Se is biholomorphically isomorphic to H2 covers all the other surfaces. In this ‘generic’ case, we saw that S admits a complete hyperbolic met ...