JACOBIANS AMONG ABELIAN THREEFOLDS
... (ii) We link Siegel modular forms, Teichmüller modular forms and invariants of plane curves. Once these two goals are achieved, Serre’s assertion can be rephrased as the following strategy — use (ii) to prove that a certain Siegel modular form f is a suitable n-th power with n > 1 on the Jacobian l ...
... (ii) We link Siegel modular forms, Teichmüller modular forms and invariants of plane curves. Once these two goals are achieved, Serre’s assertion can be rephrased as the following strategy — use (ii) to prove that a certain Siegel modular form f is a suitable n-th power with n > 1 on the Jacobian l ...
THE GEOMETRY OF COMPLEX CONJUGATE CONNECTIONS
... being a J-connection and as it is pointed out in [5, p. 105], CJ is the affine symmetry of the affine module C(M ) with respect to the affine submodule CJ (M ), made parallel with the linear submodule ker χJ . The present paper is devoted to a careful study of this connection CJ (∇), since all the a ...
... being a J-connection and as it is pointed out in [5, p. 105], CJ is the affine symmetry of the affine module C(M ) with respect to the affine submodule CJ (M ), made parallel with the linear submodule ker χJ . The present paper is devoted to a careful study of this connection CJ (∇), since all the a ...
2-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES
... a : 0 → 1 is sent to a map η : k → A, m : 2 → 1 is sent to a map µ : A ⊗ A → A, the identity i : 1 → 1 is sent to idA : A → A, d : 1 → 2 is sent to a map δ : A → A ⊗ A, e : 1 → 0 is sent to a map # : A → k, and finally s is sent to a map σ : A⊗A → A⊗A. These maps will be the structure maps making A ...
... a : 0 → 1 is sent to a map η : k → A, m : 2 → 1 is sent to a map µ : A ⊗ A → A, the identity i : 1 → 1 is sent to idA : A → A, d : 1 → 2 is sent to a map δ : A → A ⊗ A, e : 1 → 0 is sent to a map # : A → k, and finally s is sent to a map σ : A⊗A → A⊗A. These maps will be the structure maps making A ...
Harmonic Analysis on Finite Abelian Groups
... convergence in a certain norm, or in measure, or almost everywhere, or in a weak-∗ sense, or interpreted in terms of measures, distributions, or tempered distributions. Even seemingly simple formulae involving integrals are more complicated than first sight indicates. Harmonic analysis requires Lebe ...
... convergence in a certain norm, or in measure, or almost everywhere, or in a weak-∗ sense, or interpreted in terms of measures, distributions, or tempered distributions. Even seemingly simple formulae involving integrals are more complicated than first sight indicates. Harmonic analysis requires Lebe ...
INTRODUCTION TO RATIONAL CHEREDNIK ALGEBRAS
... 2.3. The center of Ht,c (W ) at t 6= 0. Let Z denote the center of H and Zt,c its specialization. Proposition 2.4. If t 6= 0, then the polynomial representation of Ht,c is faithful and Zt,c = C. Proof. Faithfulness follows in the same way as in Corollary 2.3, where we note that the polynomial repres ...
... 2.3. The center of Ht,c (W ) at t 6= 0. Let Z denote the center of H and Zt,c its specialization. Proposition 2.4. If t 6= 0, then the polynomial representation of Ht,c is faithful and Zt,c = C. Proof. Faithfulness follows in the same way as in Corollary 2.3, where we note that the polynomial repres ...
Notes - Mathematics and Statistics
... (10) Let S be a monoid (a semi-group with a two sided inverse). That is S has an associative composition law, (x, y ) 7→ xy , with a two-sided inverse. Define a category C , with a single object ∗ and with Mor(∗, ∗) = S, where composition is given by multiplication. (So morphisms need not be functio ...
... (10) Let S be a monoid (a semi-group with a two sided inverse). That is S has an associative composition law, (x, y ) 7→ xy , with a two-sided inverse. Define a category C , with a single object ∗ and with Mor(∗, ∗) = S, where composition is given by multiplication. (So morphisms need not be functio ...
Wedge products and determinants
... The goal of these lecture notes are the following: • To give a basis-free definition of the determinant det(T ) of a linear map T : V → V , using wedge products. • Define the characteristic polynomial of a linear operator T in a way that can be calculated using any basis of V , not just one for whic ...
... The goal of these lecture notes are the following: • To give a basis-free definition of the determinant det(T ) of a linear map T : V → V , using wedge products. • Define the characteristic polynomial of a linear operator T in a way that can be calculated using any basis of V , not just one for whic ...
MA3412 Section 3
... J under the quotient homomorphism ν: M → M/L, where ν(x) = L + x for all x ∈ M . Then ν(J) is a submodule of the Noetherian module M/L and is therefore finitely-generated. It follows that there exist elements x1 , x2 , . . . , xk of J such that ν(J) is generated by L + x1 , L + x2 , . . . , L + xk . ...
... J under the quotient homomorphism ν: M → M/L, where ν(x) = L + x for all x ∈ M . Then ν(J) is a submodule of the Noetherian module M/L and is therefore finitely-generated. It follows that there exist elements x1 , x2 , . . . , xk of J such that ν(J) is generated by L + x1 , L + x2 , . . . , L + xk . ...
9 Solutions for Section 2
... by an I mean the reduction of a modulo n. (Thus θ(0) = (0, 0), θ(1) = (1, 1), θ(2) = (0, 2), θ(3) = (1, 0), θ(4) = (0, 1), θ(5) = (1, 2) is a complete definition of θ.) Each of the reduction modulo n maps from Z6 to Zn (n = 2, 3) is a homomorphism (by the notes) so it follows directly (I won’t write ...
... by an I mean the reduction of a modulo n. (Thus θ(0) = (0, 0), θ(1) = (1, 1), θ(2) = (0, 2), θ(3) = (1, 0), θ(4) = (0, 1), θ(5) = (1, 2) is a complete definition of θ.) Each of the reduction modulo n maps from Z6 to Zn (n = 2, 3) is a homomorphism (by the notes) so it follows directly (I won’t write ...
The support of local cohomology modules
... the support of F -finite F -modules. That algorithm requires the calculation for roots of these modules, and this relies on the calculation of Grobner bases; these are often too complex to be computed in practice. Crucially, our algorithm does not involve Gröbner bases, and consists essentially of ...
... the support of F -finite F -modules. That algorithm requires the calculation for roots of these modules, and this relies on the calculation of Grobner bases; these are often too complex to be computed in practice. Crucially, our algorithm does not involve Gröbner bases, and consists essentially of ...
Effective descent morphisms for Banach modules
... unital Banach algebras, it follows from Theorem 3.1 that the canonical inclusion c0 → ℓ∞ of unital Banach algebras is an effective descent morphism. We conclude the note by giving a result which shows how to construct an effective descent morphism for Banach modules from any commutative unital Banac ...
... unital Banach algebras, it follows from Theorem 3.1 that the canonical inclusion c0 → ℓ∞ of unital Banach algebras is an effective descent morphism. We conclude the note by giving a result which shows how to construct an effective descent morphism for Banach modules from any commutative unital Banac ...
Isomorphisms of the Unitriangular Groups and Associated Lie Rings
... investigate isomorphisms between the adjoint groups G(R), G(R ) and associated Lie rings (R) and (R ) at = and || = 3 or 4. 1. Certain Isomorphisms and the Case n = 3 First we need to define certain automorphisms and isomorphisms. Let K and S be associative rings with identities R = NT(n, ...
... investigate isomorphisms between the adjoint groups G(R), G(R ) and associated Lie rings (R) and (R ) at = and || = 3 or 4. 1. Certain Isomorphisms and the Case n = 3 First we need to define certain automorphisms and isomorphisms. Let K and S be associative rings with identities R = NT(n, ...
The plus construction, Bousfield localization, and derived completion Tyler Lawson June 28, 2009
... In Section 2 we recall the definition of the (derived) Quillen homology object. In Section 3 we indicate how this can be computed as a simplicial object, and obtain the relevant obstruction theory in Section 4 to define a plusconstruction. The constructions of this paper are carried out in a “based” ...
... In Section 2 we recall the definition of the (derived) Quillen homology object. In Section 3 we indicate how this can be computed as a simplicial object, and obtain the relevant obstruction theory in Section 4 to define a plusconstruction. The constructions of this paper are carried out in a “based” ...
N AS AN AEC Very Preliminary We show the concept of an Abstract
... (2) Can the question of whether a class(e.g. Whitehead groups) is a P CΓclass (defined as the reducts of models of say a countable theory that omitting a family of types be independent of ZFC? (Note that under V = L, ‘Whitehead=free’ and the class is easily P CΓ. Lemma 1.16. For a cotorsion Abelian ...
... (2) Can the question of whether a class(e.g. Whitehead groups) is a P CΓclass (defined as the reducts of models of say a countable theory that omitting a family of types be independent of ZFC? (Note that under V = L, ‘Whitehead=free’ and the class is easily P CΓ. Lemma 1.16. For a cotorsion Abelian ...
Chapter 22 Tensor Algebras, Symmetric Algebras and Exterior
... this, we investigate special kinds of tensors, namely, symmetric tensors and skew-symmetric tensors. Tensor products of modules over a commutative ring with identity will be discussed very briefly. They show up naturally when we consider the space of sections of a tensor product of vector bundles. G ...
... this, we investigate special kinds of tensors, namely, symmetric tensors and skew-symmetric tensors. Tensor products of modules over a commutative ring with identity will be discussed very briefly. They show up naturally when we consider the space of sections of a tensor product of vector bundles. G ...
PRIMITIVE ELEMENTS FOR p-DIVISIBLE GROUPS 1. Introduction
... (for example if G = E[pr ] for an elliptic curve E), a satisfactory theory of full level structures has been built out of the ideas of Drinfeld [Dri74]. However, Drinfeld’s definition crucially uses the fact that G is a Cartier divisor in C. Katz and Mazur developed a notion of “full set of sections ...
... (for example if G = E[pr ] for an elliptic curve E), a satisfactory theory of full level structures has been built out of the ideas of Drinfeld [Dri74]. However, Drinfeld’s definition crucially uses the fact that G is a Cartier divisor in C. Katz and Mazur developed a notion of “full set of sections ...
LECTURE 12: HOPF ALGEBRA (sl ) Introduction
... group algebra CG (resp., over the universal enveloping algebra U (g)). Both CG, U (g) are associative algebras. Note, however, that if A is an associative algebra, then we do not have natural A-module structures on V1 ⊗ V2 , V1∗ , C (where V1 , V2 are A-modules). Indeed, V1 ⊗ V2 carries a natural st ...
... group algebra CG (resp., over the universal enveloping algebra U (g)). Both CG, U (g) are associative algebras. Note, however, that if A is an associative algebra, then we do not have natural A-module structures on V1 ⊗ V2 , V1∗ , C (where V1 , V2 are A-modules). Indeed, V1 ⊗ V2 carries a natural st ...
Formal power series rings, inverse limits, and I
... R[[x1 , . . . , xn ]] → T [[x1 , . . . , xn ]]: apply h to every coefficient. Let T = R[y1 , . . . , ym ] and h be the inclusion R ⊆ T to get an injection R[[x1 , . . . , xn ]] → (R[y1 , . . . , ym ])[[x1 , . . . , xn ]]. Now extend this homomorphism of R[[x1 , . . . , xn ]]-algebras to the polynomi ...
... R[[x1 , . . . , xn ]] → T [[x1 , . . . , xn ]]: apply h to every coefficient. Let T = R[y1 , . . . , ym ] and h be the inclusion R ⊆ T to get an injection R[[x1 , . . . , xn ]] → (R[y1 , . . . , ym ])[[x1 , . . . , xn ]]. Now extend this homomorphism of R[[x1 , . . . , xn ]]-algebras to the polynomi ...
MATH 436 Notes: Finitely generated Abelian groups.
... It is a simple check using that A is Abelian, that fˆ defined in this way is indeed a homomorphism which proves existance. The rest of the Theorem is a rephrasing of the diagram. This yields the following important corollary! Corollary 2.6 (Abelian groups are quotients of free Abelian groups). Let A ...
... It is a simple check using that A is Abelian, that fˆ defined in this way is indeed a homomorphism which proves existance. The rest of the Theorem is a rephrasing of the diagram. This yields the following important corollary! Corollary 2.6 (Abelian groups are quotients of free Abelian groups). Let A ...
Essential Normality and Boundary Representations
... This paper is concerned with the essential normality of a class of quotient modules including Beurling-type quotient modules of H 2 (Dn ), n ≥ 3. An added benefit of this consideration is the study of boundary representations, in the sense of Arveson [1], [2], of algebras generated by {IQ , Cz1 , . . ...
... This paper is concerned with the essential normality of a class of quotient modules including Beurling-type quotient modules of H 2 (Dn ), n ≥ 3. An added benefit of this consideration is the study of boundary representations, in the sense of Arveson [1], [2], of algebras generated by {IQ , Cz1 , . . ...
1. Lecture 1 1.1. Differential operators. Let k be an algebraically
... the OX -linear map ∇2 : M → M ⊗OX Ω2X given on local sections by ∇2(m)(v, w) := ([∇v , ∇w ]−∇[v,w])m. This map is called the curvature of ∇. We say that ∇ is flat if its curvature vanishes: ∇2 = 0. Proposition 1.9. A left DX -module is the same thing as an OX -module with a flat connection. Proof. G ...
... the OX -linear map ∇2 : M → M ⊗OX Ω2X given on local sections by ∇2(m)(v, w) := ([∇v , ∇w ]−∇[v,w])m. This map is called the curvature of ∇. We say that ∇ is flat if its curvature vanishes: ∇2 = 0. Proposition 1.9. A left DX -module is the same thing as an OX -module with a flat connection. Proof. G ...
Rings and modules
... Proof. To prove (i) we first define an A -homomorphism f : L → M, lm,a 7→ am where L is a free A -module with a basis lm,a , m ∈ M, a ∈ A. Then K (which is the submodule of L defined as in 3.3) is in the kernel of f . So f induces an A -homomorphism g: M ⊗A A = L/K → M , m ⊗ a 7→ am. Define h: M → M ...
... Proof. To prove (i) we first define an A -homomorphism f : L → M, lm,a 7→ am where L is a free A -module with a basis lm,a , m ∈ M, a ∈ A. Then K (which is the submodule of L defined as in 3.3) is in the kernel of f . So f induces an A -homomorphism g: M ⊗A A = L/K → M , m ⊗ a 7→ am. Define h: M → M ...
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... Each module Γ1 (N pr ) ⊗ Zp is free of finite rank over Zp , and so is compact in its p-adic topology. Thus if we give the limiting module W the topology which ab is the projective limit of the p-adic topology on each module Γ1 (N pr ) ⊗ Zp it becomes a compact Λ-module. Furthermore, Λ acts continuo ...
... Each module Γ1 (N pr ) ⊗ Zp is free of finite rank over Zp , and so is compact in its p-adic topology. Thus if we give the limiting module W the topology which ab is the projective limit of the p-adic topology on each module Γ1 (N pr ) ⊗ Zp it becomes a compact Λ-module. Furthermore, Λ acts continuo ...
Calculating with Vectors in Plane Geometry Introduction Vector
... long time: almost all results are obtained using century- or millenia-old techniques. This venerable tradition serves remarkably well for exposing a vast amount of the subject of geometry, but not all of it. A number of geometrical problems consist of finding or comparing point locations, distances, ...
... long time: almost all results are obtained using century- or millenia-old techniques. This venerable tradition serves remarkably well for exposing a vast amount of the subject of geometry, but not all of it. A number of geometrical problems consist of finding or comparing point locations, distances, ...
TAG Lecture 2: Schemes
... 1. Let A be an E∞ -ring spectrum and M an A-module. Assume we can define the symmetric A-algebra SymA (M) and that it has the appropriate universal property. (What would that be?) Let A = S be the sphere spectrum and let M = ∨n S (∨ = coproduct or wedge). What is Spec(SymS (M))? That is, what functo ...
... 1. Let A be an E∞ -ring spectrum and M an A-module. Assume we can define the symmetric A-algebra SymA (M) and that it has the appropriate universal property. (What would that be?) Let A = S be the sphere spectrum and let M = ∨n S (∨ = coproduct or wedge). What is Spec(SymS (M))? That is, what functo ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.