Chapter IV. Quotients by group schemes. When we work with group
... (i) A (left) action of G on X is a morphism ρ: G × X → X that induces, for every object T , a (left) action of the group G(T ) on the set X(T ). (ii) Let an action of G on X be given. A morphism q: X → Y in C is said to be G-invariant if q ◦ ρ = q ◦ prX : G × X → Y . By the Yoneda lemma this is equi ...
... (i) A (left) action of G on X is a morphism ρ: G × X → X that induces, for every object T , a (left) action of the group G(T ) on the set X(T ). (ii) Let an action of G on X be given. A morphism q: X → Y in C is said to be G-invariant if q ◦ ρ = q ◦ prX : G × X → Y . By the Yoneda lemma this is equi ...
PROJECTIVITY AND FLATNESS OVER THE
... k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra. Then we can consider the smash product Λ#H and the subring of invariants ΛH . Then we can give necessary and sufficient conditions for the projectivity and flatness over ΛH of a left Λ#H-module P . The results from [9] ...
... k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra. Then we can consider the smash product Λ#H and the subring of invariants ΛH . Then we can give necessary and sufficient conditions for the projectivity and flatness over ΛH of a left Λ#H-module P . The results from [9] ...
Abelian Varieties
... satisfies the separation axiom. If V is a variety over k and K k, then V .K/ is the set of points of V with coordinates in K and VK or V=K is the variety over K obtained from V by extension of scalars. An algebraic space is similar, except that Specm.R/ is an algebraic space for any finitely gener ...
... satisfies the separation axiom. If V is a variety over k and K k, then V .K/ is the set of points of V with coordinates in K and VK or V=K is the variety over K obtained from V by extension of scalars. An algebraic space is similar, except that Specm.R/ is an algebraic space for any finitely gener ...
last updated 2012-02-25 with Set 8
... We use the following (*) and (**) explained in Set 4. (*) For a finite field Fq of characteristic 6= 2, we have an analogue of quadratic reciprocity law for Fq [T ] explained in Preparation 1 for Problem 40. (You will prove that formula in Problem 50. ) (**) For a homomorphism χ : (Fq [T ]/(g))× → C ...
... We use the following (*) and (**) explained in Set 4. (*) For a finite field Fq of characteristic 6= 2, we have an analogue of quadratic reciprocity law for Fq [T ] explained in Preparation 1 for Problem 40. (You will prove that formula in Problem 50. ) (**) For a homomorphism χ : (Fq [T ]/(g))× → C ...
Modern geometry 2012.8.27 - 9. 5 Introduction to Geometry Ancient
... 4. All right angles are equal. 5. If a straight line meets two other lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles. ★ In a typical geom ...
... 4. All right angles are equal. 5. If a straight line meets two other lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles. ★ In a typical geom ...
Abelian Varieties - Harvard Math Department
... Assuming the theorem, we can embed the Jacobian J(X) of an algebraic curve X if we construct a symplectic pairing H1 (X, Z) × H1 (X, Z) → Z such that the corresponding form on H 0 (X, ΩX )∗ is positive definite. (This pairing is given by the intersection pairing). We will later see that the addition ...
... Assuming the theorem, we can embed the Jacobian J(X) of an algebraic curve X if we construct a symplectic pairing H1 (X, Z) × H1 (X, Z) → Z such that the corresponding form on H 0 (X, ΩX )∗ is positive definite. (This pairing is given by the intersection pairing). We will later see that the addition ...
Rings and modules
... Examples. Every abelian group is a Z -module, so the class of abelian groups coincide with the class of Z -modules. Every vector space over a field F is an F -module. 2.2. A map f : M → N is called a homomorphism of A -modules if f (x + y) = f (x) + f (y) for every x, y ∈ M and f (ax) = af (x) for e ...
... Examples. Every abelian group is a Z -module, so the class of abelian groups coincide with the class of Z -modules. Every vector space over a field F is an F -module. 2.2. A map f : M → N is called a homomorphism of A -modules if f (x + y) = f (x) + f (y) for every x, y ∈ M and f (ax) = af (x) for e ...
A.2 Polynomial Algebra over Fields
... The set F [x] equipped with the operations + and · is the polynomial ring in x over the field F . F is the field of coefficients of F [x]. Polynomial rings over fields have many of the properties enjoyed by fields. F [x] is closed and distributive nearly by definition. Commutativity and additive ass ...
... The set F [x] equipped with the operations + and · is the polynomial ring in x over the field F . F is the field of coefficients of F [x]. Polynomial rings over fields have many of the properties enjoyed by fields. F [x] is closed and distributive nearly by definition. Commutativity and additive ass ...
EPH-classifications in Geometry, Algebra, Analysis and Arithmetic
... eigenvector with eigenvalue 1, i.e. such that Av = v. All the planes {v T Jx = c} are invariant under A , since v T JAx = v T (AT )−1 Jx = (A−1 v)T Jx = v T Jx. This is a family of parallel planes all invariant under A. Then we say that A is elliptic, parabolic or hyperbolic if and only if the corre ...
... eigenvector with eigenvalue 1, i.e. such that Av = v. All the planes {v T Jx = c} are invariant under A , since v T JAx = v T (AT )−1 Jx = (A−1 v)T Jx = v T Jx. This is a family of parallel planes all invariant under A. Then we say that A is elliptic, parabolic or hyperbolic if and only if the corre ...
Checking Polynomial Identities over any Field: Towards a
... We show that the error probability of this test can be reduced, in polynomial time, to any inverse polynomial quantity by using approximations modulo larger powers of x. 1.4 Layout of the Paper Section 3 describes some standard algebraic tools which our algorithm uses. Section 4 gives a more detaile ...
... We show that the error probability of this test can be reduced, in polynomial time, to any inverse polynomial quantity by using approximations modulo larger powers of x. 1.4 Layout of the Paper Section 3 describes some standard algebraic tools which our algorithm uses. Section 4 gives a more detaile ...
lecture notes as PDF
... • A ring is called a ring with identity if the ring has a multiplicative identity (usually denoted e or 1). • A ring is called commutative if ∗ is commutative. • A ring is called an integral domain if it is a commutative ring with identity e 6= 0 in which ab = 0 implies a = 0 or b = 0 (i.e. no zero ...
... • A ring is called a ring with identity if the ring has a multiplicative identity (usually denoted e or 1). • A ring is called commutative if ∗ is commutative. • A ring is called an integral domain if it is a commutative ring with identity e 6= 0 in which ab = 0 implies a = 0 or b = 0 (i.e. no zero ...
Finite Fields
... • A ring is called a ring with identity if the ring has a multiplicative identity (usually denoted e or 1). • A ring is called commutative if ∗ is commutative. • A ring is called an integral domain if it is a commutative ring with identity e 6= 0 in which ab = 0 implies a = 0 or b = 0 (i.e. no zero ...
... • A ring is called a ring with identity if the ring has a multiplicative identity (usually denoted e or 1). • A ring is called commutative if ∗ is commutative. • A ring is called an integral domain if it is a commutative ring with identity e 6= 0 in which ab = 0 implies a = 0 or b = 0 (i.e. no zero ...
Classical Period Domains - Stony Brook Mathematics
... 2.1), and then give a representation theoretic description of variations of Hodge structure on locally symmetric domains (Section 2.2) following [Mil13]. Using the description, we discuss the classification of variations of Hodge structure of abelian variety type and Calabi-Yau type following [Del79 ...
... 2.1), and then give a representation theoretic description of variations of Hodge structure on locally symmetric domains (Section 2.2) following [Mil13]. Using the description, we discuss the classification of variations of Hodge structure of abelian variety type and Calabi-Yau type following [Del79 ...
Group Actions and Representations
... representation of G on Tx X called the tangent representation in the fixed point x. We will see in section 2.3 (Corollary 2.6) that this is a rational representation of G. Example 2.5. The representations ρ : C+ → GLn of the additive group C+ are in one-to-one correspondence with the nilpotent matri ...
... representation of G on Tx X called the tangent representation in the fixed point x. We will see in section 2.3 (Corollary 2.6) that this is a rational representation of G. Example 2.5. The representations ρ : C+ → GLn of the additive group C+ are in one-to-one correspondence with the nilpotent matri ...
Principal bundles on the projective line
... of P1k . Then c goes to zero under the map H 1 (k, T ) → H 1 (k, G). We now twist ET by the class c−1 to obtain a new T bundle ET0 which is trivial at the origin. The twisted T bundle ET0 is contained in the twist of E by c−1 . However since the image of c in H 1 (k, G) is trivial, we see that the t ...
... of P1k . Then c goes to zero under the map H 1 (k, T ) → H 1 (k, G). We now twist ET by the class c−1 to obtain a new T bundle ET0 which is trivial at the origin. The twisted T bundle ET0 is contained in the twist of E by c−1 . However since the image of c in H 1 (k, G) is trivial, we see that the t ...
Basic Modern Algebraic Geometry
... 1.2.1 Definition of covariant and contravariant functors . . . 11 1.2.2 Forgetful functors . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 The category of functors Fun(C, D) . . . . . . . . . . . 12 1.2.4 Functors of several variables . . . . . . . . . . . . . . . 13 1.3 Isomorphic and equivalent ...
... 1.2.1 Definition of covariant and contravariant functors . . . 11 1.2.2 Forgetful functors . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 The category of functors Fun(C, D) . . . . . . . . . . . 12 1.2.4 Functors of several variables . . . . . . . . . . . . . . . 13 1.3 Isomorphic and equivalent ...
Grade 8 Mathematics Item Descriptions
... Solves a word problem by using patterns in a two-column table to determine the number in the second column that would correspond to a number midway between two entries in the first column ...
... Solves a word problem by using patterns in a two-column table to determine the number in the second column that would correspond to a number midway between two entries in the first column ...
Semirings Modeling Confidence and Uncertainty in
... 1. The feeling or belief that one can have faith in or rely on someone or something. 2. The telling of private matters or secrets with mutual trust. As usual in speech recognition, in this paper I restrict attention to part of the first meaning, namely to “the belief that one can rely on something”. ...
... 1. The feeling or belief that one can have faith in or rely on someone or something. 2. The telling of private matters or secrets with mutual trust. As usual in speech recognition, in this paper I restrict attention to part of the first meaning, namely to “the belief that one can rely on something”. ...
The Relationship Between Two Commutators
... Conditions (1) and (2) imply that for any A ∈ V ∗ we have p(x, y, z) = x − y + z with respect to some abelian group structure on A. We call a structure of the form hA; pi where p satisfies (1) and (2) an affine abelian group. Condition (3) says that every basic τ –operation is multilinear with resp ...
... Conditions (1) and (2) imply that for any A ∈ V ∗ we have p(x, y, z) = x − y + z with respect to some abelian group structure on A. We call a structure of the form hA; pi where p satisfies (1) and (2) an affine abelian group. Condition (3) says that every basic τ –operation is multilinear with resp ...
Can there be efficient and natural FHE schemes?
... of the work we do not assume a public encryption key, only that the adversary can perform evaluation of ciphertexts. Furthermore, we extend the characterisation of Armknecht et al. [4] to handle all public key FHE schemes over any algebraic structure. This yields a simple transformation from any sch ...
... of the work we do not assume a public encryption key, only that the adversary can perform evaluation of ciphertexts. Furthermore, we extend the characterisation of Armknecht et al. [4] to handle all public key FHE schemes over any algebraic structure. This yields a simple transformation from any sch ...
Use the FOIL Method
... (2x + -3)(4x + 5) F : Multiply the First term in each binomial. 2x • 4x = 8x2 O : Multiply the Outer terms in the binomials. 2x • 5 = 10x I : Multiply the Inner terms in the binomials. -3 • 4x = -12x L : Multiply the Last term in each binomial. -3 • 5 = -15 (2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 ...
... (2x + -3)(4x + 5) F : Multiply the First term in each binomial. 2x • 4x = 8x2 O : Multiply the Outer terms in the binomials. 2x • 5 = 10x I : Multiply the Inner terms in the binomials. -3 • 4x = -12x L : Multiply the Last term in each binomial. -3 • 5 = -15 (2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 ...
Derived splinters in positive characteristic
... Definition 1.3. A scheme S is called a derived splinter, or simply a D-splinter, if for any proper surjective map f : X → S, the pullback map OS → R f∗ OX is split in derived category D(Coh(S)) of coherent sheaves on S. D-splinters are, in fact, well-known to complex geometers, but under a different ...
... Definition 1.3. A scheme S is called a derived splinter, or simply a D-splinter, if for any proper surjective map f : X → S, the pullback map OS → R f∗ OX is split in derived category D(Coh(S)) of coherent sheaves on S. D-splinters are, in fact, well-known to complex geometers, but under a different ...
Algebraic variety
In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry. Classically, an algebraic variety was defined to be the set of solutions of a system of polynomial equations, over the real or complex numbers. Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition.Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an ""algebraic variety"" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraic varieties are called algebraic sets.The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry among the other subareas of geometry.