1. Divisors Let X be a complete non-singular curve. Definition 1.1. A
... induces a homomorphism of abelian groups Z(X) → Z. For example, if P, Q, R are points on X, then D = −P + 3Q − 4R is a divisor on X, of degree −2. Let f ∈ k(X)∗ be a non-zero rational function on X. For a point P ∈ X, let vP : Frac OX,P → Z ∪ {∞} be the discrete valuation of OX,P . Lemma 1.2. We hav ...
... induces a homomorphism of abelian groups Z(X) → Z. For example, if P, Q, R are points on X, then D = −P + 3Q − 4R is a divisor on X, of degree −2. Let f ∈ k(X)∗ be a non-zero rational function on X. For a point P ∈ X, let vP : Frac OX,P → Z ∪ {∞} be the discrete valuation of OX,P . Lemma 1.2. We hav ...
On the second dominant eigenvalue affecting the power method for
... Because each term πi1 i2 in (3.16) is nonnegative, we see that x(t+1) converges if and only if Πt+1,t converges as t goes to infinity. If suffices to consider the limiting behavior of the iteration (3.15). For each fixed i2 , we may rewrite this updating mechanism in the matrix-to-vector multiplicat ...
... Because each term πi1 i2 in (3.16) is nonnegative, we see that x(t+1) converges if and only if Πt+1,t converges as t goes to infinity. If suffices to consider the limiting behavior of the iteration (3.15). For each fixed i2 , we may rewrite this updating mechanism in the matrix-to-vector multiplicat ...
Lecture 3
... for all objects X, Y , Z, W and all morphisms f : X −→ Y , g : Y −→ Z and h : Z −→ W . (2) For every object X, there is a special morphism i = iX ∈ Hom(X, X) which acts as an identity under composition. That is for all f ∈ Hom(X, Y ), f ◦ iX = f = iY ◦ f. We say that a category C is called locally s ...
... for all objects X, Y , Z, W and all morphisms f : X −→ Y , g : Y −→ Z and h : Z −→ W . (2) For every object X, there is a special morphism i = iX ∈ Hom(X, X) which acts as an identity under composition. That is for all f ∈ Hom(X, Y ), f ◦ iX = f = iY ◦ f. We say that a category C is called locally s ...
Notes - Math Berkeley
... reducing to an isomorphism modulo m. Since the dimension of every fiber of M is d2 , the map ⇡ induces an isomorphism modulo every prime ideal of R. This implies in particular that at each generic point of R the map ⇡ is an isomorphism. Since R is reduced, this implies that ⇡ is also injective, when ...
... reducing to an isomorphism modulo m. Since the dimension of every fiber of M is d2 , the map ⇡ induces an isomorphism modulo every prime ideal of R. This implies in particular that at each generic point of R the map ⇡ is an isomorphism. Since R is reduced, this implies that ⇡ is also injective, when ...
Riemannian manifolds with a semi-symmetric metric connection
... manifold be conformally flat. For some properties of Riemannian manifolds with a semi-symmetric metric connection see also [1,4,10,16,17]. In [14,15], Szabó studied semisymmetric Riemannian manifolds, that is Riemannian manifolds satisfying the condition R · R = 0. It is well known that locally sym ...
... manifold be conformally flat. For some properties of Riemannian manifolds with a semi-symmetric metric connection see also [1,4,10,16,17]. In [14,15], Szabó studied semisymmetric Riemannian manifolds, that is Riemannian manifolds satisfying the condition R · R = 0. It is well known that locally sym ...
Algebra I: Section 3. Group Theory 3.1 Groups.
... 3.1.14 Example (Ωn = nth Roots of Unity). Recall that every complex number z 6= 0 can be written in polar form z = reiθ = (r cos θ)+ i(r sin θ) as shown in Figure 3.1. Here r = |z| and θ is the angle variable (angle from positive x-axis to the ray from the origin to z), in radians. An nth root of un ...
... 3.1.14 Example (Ωn = nth Roots of Unity). Recall that every complex number z 6= 0 can be written in polar form z = reiθ = (r cos θ)+ i(r sin θ) as shown in Figure 3.1. Here r = |z| and θ is the angle variable (angle from positive x-axis to the ray from the origin to z), in radians. An nth root of un ...
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1
... An example of a module S for which the conditions of the theorem do not hold is given in [9] 4.4. Condition (1) of the theorem is independent of the ring R, so if, for given S, k and G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying ...
... An example of a module S for which the conditions of the theorem do not hold is given in [9] 4.4. Condition (1) of the theorem is independent of the ring R, so if, for given S, k and G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying ...
Collated Notes on TQFT.pdf
... to that of a homotopy equivalence. We will formalize this intuition. Definition 6.1. Let X and Y be topological spaces, and let I = [0, 1] be the unit interval. Let f, g : X −→ Y be continuous maps. A homotopy from f to g is a continuous map h : X × I −→ Y such that h(x, 0) = f (x) and h(x, 1) = g(x ...
... to that of a homotopy equivalence. We will formalize this intuition. Definition 6.1. Let X and Y be topological spaces, and let I = [0, 1] be the unit interval. Let f, g : X −→ Y be continuous maps. A homotopy from f to g is a continuous map h : X × I −→ Y such that h(x, 0) = f (x) and h(x, 1) = g(x ...
Symmeric self-adjoint Hopf categories and a categorical Heisenberg double June 17, 2014
... of certain squares - that are part of the SSH structure - satisfying the BeckChevalley condition, which in Catex is the same as being comma. In fact, we notice that these squares are images of comma squares in the category Heis(H), defined in §5.3, via a canonical projection Heis(H) → Catex . From t ...
... of certain squares - that are part of the SSH structure - satisfying the BeckChevalley condition, which in Catex is the same as being comma. In fact, we notice that these squares are images of comma squares in the category Heis(H), defined in §5.3, via a canonical projection Heis(H) → Catex . From t ...
Autonomous categories and linear logic
... the category of finite dimensional vector spaces over a field. The (unique) dualizing object is the field itself. The details are familiar. Notable is the fact that this duality does not extend to infinite dimensional spaces. In fact V ∼ = V ⊥⊥ is equivalent to V being finite dimensional. The finite ...
... the category of finite dimensional vector spaces over a field. The (unique) dualizing object is the field itself. The details are familiar. Notable is the fact that this duality does not extend to infinite dimensional spaces. In fact V ∼ = V ⊥⊥ is equivalent to V being finite dimensional. The finite ...
On zero product determined algebras
... Proof Recall that the tensor algebra T (V ) over a vector space V may be thought of as the free unital associative K -algebra on dim V generators. Likewise, the symmetric algebra S(V ) over V may be thought of as the free commutative unital associative K -algebra on dim V generators. We will denote ...
... Proof Recall that the tensor algebra T (V ) over a vector space V may be thought of as the free unital associative K -algebra on dim V generators. Likewise, the symmetric algebra S(V ) over V may be thought of as the free commutative unital associative K -algebra on dim V generators. We will denote ...
HOPF ALGEBRAS AND QUADRATIC FORMS 1. Introduction Let Y
... immediate. We now use the fact that the map ϕ : A ⊗R AD → HomR (A, A), with ϕ(h ⊗ f )(a) =< f, a > h, is an isomorphism. Therefore there exist elements {h1 , ..., hn } of A and {f1 , ..., fn } of AD such that X Id = ϕ(hi ⊗ fi ) . 1≤i≤n ...
... immediate. We now use the fact that the map ϕ : A ⊗R AD → HomR (A, A), with ϕ(h ⊗ f )(a) =< f, a > h, is an isomorphism. Therefore there exist elements {h1 , ..., hn } of A and {f1 , ..., fn } of AD such that X Id = ϕ(hi ⊗ fi ) . 1≤i≤n ...
ON BOUNDED MODULE MAPS BETWEEN HILBERT MODULES OVER LOCALLY C -ALGEBRAS
... A locally C ∗ -algebra is a complete Hausdorff complex topological ∗-algebra A whose topology is determined by its continuous C ∗ -seminorms in the sense that the net {ai }i converges to 0 if and only if the net {p(ai )}i converges to 0 for every continuous C ∗ -seminorm p on A. In fact a locally C ...
... A locally C ∗ -algebra is a complete Hausdorff complex topological ∗-algebra A whose topology is determined by its continuous C ∗ -seminorms in the sense that the net {ai }i converges to 0 if and only if the net {p(ai )}i converges to 0 for every continuous C ∗ -seminorm p on A. In fact a locally C ...
Topological modules over strictly minimal topological
... into F is a homeomorphism from (R, τR ) into (F, τ ); (c) for every free R-module F with a basis of 1 element, there is only one separated (R, τR )-module topology on F ; (d) for every topological (R, τR )-module (E, τ ) and for every separated topological (R, τR )-module (F, τ ) which is a free ...
... into F is a homeomorphism from (R, τR ) into (F, τ ); (c) for every free R-module F with a basis of 1 element, there is only one separated (R, τR )-module topology on F ; (d) for every topological (R, τR )-module (E, τ ) and for every separated topological (R, τR )-module (F, τ ) which is a free ...
MATH 8253 ALGEBRAIC GEOMETRY HOMEWORK 1 1.2.10. Let A
... We know that in general for a multiplicative subset S of A the prime ideals of S −1 A are in one-to-one correspondence with the prime ideals of A that do not intersect S. When S = A − p this means that the prime ideals of Ap are in one-to-one correspondence with the prime ideals of A contained in p. ...
... We know that in general for a multiplicative subset S of A the prime ideals of S −1 A are in one-to-one correspondence with the prime ideals of A that do not intersect S. When S = A − p this means that the prime ideals of Ap are in one-to-one correspondence with the prime ideals of A contained in p. ...
Remarks on dual vector spaces and scalar products
... f , i.e. B = A−1 . Hence, the isomorphism g ∗ is fully determined by f −1 . To also demonstrate this more explicitly, we mention that every linear map g : V −→ W between real vector spaces V and W uniquely determines a correspondingly linear map g ∗ : W ∗ −→ V ∗ , via g ∗ (α)(~v ) := α(g(~v )), for ...
... f , i.e. B = A−1 . Hence, the isomorphism g ∗ is fully determined by f −1 . To also demonstrate this more explicitly, we mention that every linear map g : V −→ W between real vector spaces V and W uniquely determines a correspondingly linear map g ∗ : W ∗ −→ V ∗ , via g ∗ (α)(~v ) := α(g(~v )), for ...
Gates accept concurrent behavior
... Besides the general contribution made by the overall gates-as-acceptors framework, we also make the following specific technical contributions. 0 Schedules and automata constitute respectively declarative and imperative programming styles. We pair these up by representing both as binary relations, w ...
... Besides the general contribution made by the overall gates-as-acceptors framework, we also make the following specific technical contributions. 0 Schedules and automata constitute respectively declarative and imperative programming styles. We pair these up by representing both as binary relations, w ...
Algebraic Topology Lecture Notes Jarah Evslin and
... Global information is important. For example, global data can tell you that it is impossible to define a spinor on a certain space, and so a given spacetime cannot be inhabited by fermions. In general there are both local and global obstructions, with the local obstructions determined by differentia ...
... Global information is important. For example, global data can tell you that it is impossible to define a spinor on a certain space, and so a given spacetime cannot be inhabited by fermions. In general there are both local and global obstructions, with the local obstructions determined by differentia ...
GROUPS AND THEIR REPRESENTATIONS 1. introduction
... A group G is abelian if the operation ? is commutative, that is if g ? h = h ? g for all g and h in G. The order of a group (G, ?) is the cardinality of the set G. The order of G is denoted |G|. The quintessential example of a group is the set of symmetries of the square under composition, already m ...
... A group G is abelian if the operation ? is commutative, that is if g ? h = h ? g for all g and h in G. The order of a group (G, ?) is the cardinality of the set G. The order of G is denoted |G|. The quintessential example of a group is the set of symmetries of the square under composition, already m ...
Contents - Harvard Mathematics Department
... and abelian groups) sending a space X to its nth homology group Hn (X). We know that given a map of spaces f : X → Y , we get a map of abelian groups f∗ : Hn (X) → Hn (Y ). See [Hat02], for instance. We shall often need to compose functors. For instance, we will want to see, for instance, that the t ...
... and abelian groups) sending a space X to its nth homology group Hn (X). We know that given a map of spaces f : X → Y , we get a map of abelian groups f∗ : Hn (X) → Hn (Y ). See [Hat02], for instance. We shall often need to compose functors. For instance, we will want to see, for instance, that the t ...
Lecture 8
... Lecture VIII - Categories and Functors Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qua ...
... Lecture VIII - Categories and Functors Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qua ...
4. Categories and Functors We recall the definition of a category
... It is easy to check that F is indeed a functor; for obvious reasons it is called a forgetful functor and there are many such functors. Note that we may compose functors in the obvious way and that there is an identity functor. Slightly more interestingly there is an obvious contravariant functor fro ...
... It is easy to check that F is indeed a functor; for obvious reasons it is called a forgetful functor and there are many such functors. Note that we may compose functors in the obvious way and that there is an identity functor. Slightly more interestingly there is an obvious contravariant functor fro ...
Derived Representation Theory and the Algebraic K
... monoidal structure is coherently commutative, the spectrum will be a commutative S-algebra. See [11] or [17] for these results. Since the tensor product of finitely generated projective modules over a commutative ring R is coherently commutative, we have Proposition 2.2 For any commutative ring A, t ...
... monoidal structure is coherently commutative, the spectrum will be a commutative S-algebra. See [11] or [17] for these results. Since the tensor product of finitely generated projective modules over a commutative ring R is coherently commutative, we have Proposition 2.2 For any commutative ring A, t ...
![arXiv:math/0005256v2 [math.QA] 21 Jun 2000](http://s1.studyres.com/store/data/017029559_1-d28b6a03fef36ff14eaa2fb575ab89be-300x300.png)
arXiv:math/0005256v2 [math.QA] 21 Jun 2000
... These notes contain almost no proof, many results are classical or easy. There are two notable exceptions, namely Theorem 2 and Theorem 3 the proof of which are absolutely non trivial although their meaning is transparent. Let us say some words on our conventions. For sake of completeness we have gi ...
... These notes contain almost no proof, many results are classical or easy. There are two notable exceptions, namely Theorem 2 and Theorem 3 the proof of which are absolutely non trivial although their meaning is transparent. Let us say some words on our conventions. For sake of completeness we have gi ...
Lecture 38: Unitary operators
... A linear map T : V −→ W preserves inner products if for all v, v0 ∈ V we have (T v|T v0 ) = (v|v0 ). In this case T is injective. If T is also bijective, then we say that T is an isomorphism of inner product spaces. In this case T −1 also preserves inner products. A co-ordinate system C : Fn −→ V is ...
... A linear map T : V −→ W preserves inner products if for all v, v0 ∈ V we have (T v|T v0 ) = (v|v0 ). In this case T is injective. If T is also bijective, then we say that T is an isomorphism of inner product spaces. In this case T −1 also preserves inner products. A co-ordinate system C : Fn −→ V is ...
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.