How to quantize infinitesimally-braided symmetric monoidal categories
... other coherency, you get knot invariants. The constructions above have duals and coherency, because you can’t break this structure by deformation-quantizing [33], and g-rep has them. So given an element V ∈ g-rep, the structure (g-rep[[~]], ⊗, α, s) turns the knot into a number in K[[~]]. To get K-v ...
... other coherency, you get knot invariants. The constructions above have duals and coherency, because you can’t break this structure by deformation-quantizing [33], and g-rep has them. So given an element V ∈ g-rep, the structure (g-rep[[~]], ⊗, α, s) turns the knot into a number in K[[~]]. To get K-v ...
Ringoids (Pre%Talk Notes) By Edward Burkard Question: Consider
... Notation 27 Instead of writing ei j R in the decomposition of a semi-simple a a ringoid, I will instead write eiij j R. This is because we will be dealing with matrices in the following theorems. ...
... Notation 27 Instead of writing ei j R in the decomposition of a semi-simple a a ringoid, I will instead write eiij j R. This is because we will be dealing with matrices in the following theorems. ...
Lecture Notes - Mathematics
... is a well-defined regular function at all points of its domain. So regular functions need not be represented globally by one rational function; the rational expressions witnessing regularity can be different at different points. Remark 2.16. Let V be an affine algebraic variety in Cn . An interestin ...
... is a well-defined regular function at all points of its domain. So regular functions need not be represented globally by one rational function; the rational expressions witnessing regularity can be different at different points. Remark 2.16. Let V be an affine algebraic variety in Cn . An interestin ...
Some definable Galois theory and examples
... Several papers and/or preprints have appeared recently, giving a version of the Galois correspondence when the “constants” of the base field are not necessarily “closed” in the appropriate senses. The point of our current paper is to explain how these results are implicit or explicit in the model-th ...
... Several papers and/or preprints have appeared recently, giving a version of the Galois correspondence when the “constants” of the base field are not necessarily “closed” in the appropriate senses. The point of our current paper is to explain how these results are implicit or explicit in the model-th ...
SINGULARITIES ON COMPLETE ALGEBRAIC VARIETIES 1
... the constraints imposed on the singularities that can be afforded on a given class of algebraic varieties. A general result in this direction appeared in [CG]. There it was shown that for any algebraic family of algebraic varieties there are isolated singular points which can not be afforded on a an ...
... the constraints imposed on the singularities that can be afforded on a given class of algebraic varieties. A general result in this direction appeared in [CG]. There it was shown that for any algebraic family of algebraic varieties there are isolated singular points which can not be afforded on a an ...
Algebraic Numbers and Algebraic Integers
... by multiplicativity of the norm. By the above corollary, both NK/Q (α) and its inverse belong to Z, meaning that the only possible values are ±1. Conversely, let us assume that α ∈ OK has norm ±1, which means that the constant term of its minimal polynomial f (X) is ±1: f (X) = X n + an−1 X n−1 + · ...
... by multiplicativity of the norm. By the above corollary, both NK/Q (α) and its inverse belong to Z, meaning that the only possible values are ±1. Conversely, let us assume that α ∈ OK has norm ±1, which means that the constant term of its minimal polynomial f (X) is ±1: f (X) = X n + an−1 X n−1 + · ...
1736 - RIMS, Kyoto University
... Throughout this article R.V. on an algebraic surface X means the vanishing of H 1 (X, L−1 ) for all nef and big line bundles on X. Conversely to the above counterexample, using Theorem 1 and [LM], we prove the following. Theorem 3. In the case where X is of dimension two, we have the following: (a) ...
... Throughout this article R.V. on an algebraic surface X means the vanishing of H 1 (X, L−1 ) for all nef and big line bundles on X. Conversely to the above counterexample, using Theorem 1 and [LM], we prove the following. Theorem 3. In the case where X is of dimension two, we have the following: (a) ...
Slides
... µT = τ ◦ E ∈ Prob(C), supp(µT ) = σ(T ). Fuglede-Kadison determinant(1952) M finite vN algebra with trace τ ,T ∈ M invertible ∆(T ) = exp(τ (log |T |)) Extend to any T ∈ M : Z ∆(T ) = exp ...
... µT = τ ◦ E ∈ Prob(C), supp(µT ) = σ(T ). Fuglede-Kadison determinant(1952) M finite vN algebra with trace τ ,T ∈ M invertible ∆(T ) = exp(τ (log |T |)) Extend to any T ∈ M : Z ∆(T ) = exp ...
Chapter 3
... (always with the understanding that neither A nor B lies on the line itself). (Greenberg does not point out (∗) explicitly, but mentions “excluded middle” every time it is used.) Axiom B-4 (plane separation property) and corollary (pp. 110–111). [(iii) follows from (i) and (∗) and the fact that “sa ...
... (always with the understanding that neither A nor B lies on the line itself). (Greenberg does not point out (∗) explicitly, but mentions “excluded middle” every time it is used.) Axiom B-4 (plane separation property) and corollary (pp. 110–111). [(iii) follows from (i) and (∗) and the fact that “sa ...
nnpc – fstp- maths_eng 1
... 1.If n is not a +ve integer, the series will be infinite in ascending power of x as none of the nos (n-1), (n-2), (n-3), ………… in the coeffs. will never be zero. 2. The coeffs can no longer be written in the form nCr. 3. The expansion is only valid if the numerical value of x is less than 1; ie -1 ˂ ...
... 1.If n is not a +ve integer, the series will be infinite in ascending power of x as none of the nos (n-1), (n-2), (n-3), ………… in the coeffs. will never be zero. 2. The coeffs can no longer be written in the form nCr. 3. The expansion is only valid if the numerical value of x is less than 1; ie -1 ˂ ...
PERIODS OF GENERIC TORSORS OF GROUPS OF
... where P is connected, reductive and special, is called a resolution of Q. Since we can embed Q into GLn for some positive integer n, resolutions of Q exist. The generic fiber of the morphism φ : P → S is a Q-torsor over the function field F (S) of S, moreover it is a generic Q-torsor (see [1, sectio ...
... where P is connected, reductive and special, is called a resolution of Q. Since we can embed Q into GLn for some positive integer n, resolutions of Q exist. The generic fiber of the morphism φ : P → S is a Q-torsor over the function field F (S) of S, moreover it is a generic Q-torsor (see [1, sectio ...
Chapter 3
... not accidentally “holes” in the space. For√example, the rational plane Q2 lacks points with irrational coordinates, so “ y = 1 − x2 ” may be problematical. We do not commit to any one continuity axiom. A recent paper by Greenberg, Amer. Math. Monthly 117 (2010) 198–219, reports some more recent and/ ...
... not accidentally “holes” in the space. For√example, the rational plane Q2 lacks points with irrational coordinates, so “ y = 1 − x2 ” may be problematical. We do not commit to any one continuity axiom. A recent paper by Greenberg, Amer. Math. Monthly 117 (2010) 198–219, reports some more recent and/ ...
13 Lecture 13: Uniformity and sheaf properties
... quasi-compact open set,1 we see that the category of adic spaces fully faithfully contains the category of locally noetherian schemes. Example 13.3.2 The adic space Spa(Z, Z) associated to the noetherian scheme Spec(Z) (where A+ = Z is given the discrete topology) is the final object in the categor ...
... quasi-compact open set,1 we see that the category of adic spaces fully faithfully contains the category of locally noetherian schemes. Example 13.3.2 The adic space Spa(Z, Z) associated to the noetherian scheme Spec(Z) (where A+ = Z is given the discrete topology) is the final object in the categor ...
Fibre products
... Theorem 4.2.1. Fibre products exist in the category of schemes. Before proving this, let us understand some consequences. First of all, it tells us that products exist. Since Spec Z is the terminal object in the category of schemes. The product is X ⇥ Y = X ⇥Spec Z Y . Secondly, given a point s 2 S ...
... Theorem 4.2.1. Fibre products exist in the category of schemes. Before proving this, let us understand some consequences. First of all, it tells us that products exist. Since Spec Z is the terminal object in the category of schemes. The product is X ⇥ Y = X ⇥Spec Z Y . Secondly, given a point s 2 S ...
Simplicial Objects and Singular Homology
... e i (X) and Hi (X) are isomorphic. more readily used and for i > 0 the groups H e p (S n ) is as claimed for all To prove the results above we induct on n ≥ 0 to show that H p ≥ 0. For n = 0, we have the 0-sphere S 0 , we should note that it consists of two points {1, −1} and is therefore a discrete ...
... e i (X) and Hi (X) are isomorphic. more readily used and for i > 0 the groups H e p (S n ) is as claimed for all To prove the results above we induct on n ≥ 0 to show that H p ≥ 0. For n = 0, we have the 0-sphere S 0 , we should note that it consists of two points {1, −1} and is therefore a discrete ...
File
... -State the definition for the term (auxiliary line) -Write and use the conjectures (Triangle Sum Conjecture, Third Angle Conjecture) -Use the conjectures to solve for angle measurements in triangles -Do pages 201-203 problems #2-9, 14-21 4.2 Properties of Special Triangles I Can -State the definitio ...
... -State the definition for the term (auxiliary line) -Write and use the conjectures (Triangle Sum Conjecture, Third Angle Conjecture) -Use the conjectures to solve for angle measurements in triangles -Do pages 201-203 problems #2-9, 14-21 4.2 Properties of Special Triangles I Can -State the definitio ...
Notes on Tate's article on p-divisible groups
... fractions K is of characteristic 0. Let G and H be p-divisible groups over R. Then, the map HomR (G, H) → HomK (G ⊗R K, H ⊗R K) is bijective. Before we begin the proof, note that for any v, the natural map HomR (Gv , Hv ) → HomK (Gv ⊗R K, Hv ⊗R K) is injective because the coordinate rings of Gv and ...
... fractions K is of characteristic 0. Let G and H be p-divisible groups over R. Then, the map HomR (G, H) → HomK (G ⊗R K, H ⊗R K) is bijective. Before we begin the proof, note that for any v, the natural map HomR (Gv , Hv ) → HomK (Gv ⊗R K, Hv ⊗R K) is injective because the coordinate rings of Gv and ...
EQUIVALENT NOTIONS OF ∞-TOPOI Seminar on Higher Category
... a morphism f : C → D in C, then C is also in C(0) . (2) Let C be an object of C. A sieve on C is a sieve on C/C. (3) Let (C/C)(0) be a sieve on the object C and f : D → C be a morphism in C. We let f ∗ (C/C)(0) be the full subcategory of C/D spanned by those g : D0 → D in C such that f g is equivale ...
... a morphism f : C → D in C, then C is also in C(0) . (2) Let C be an object of C. A sieve on C is a sieve on C/C. (3) Let (C/C)(0) be a sieve on the object C and f : D → C be a morphism in C. We let f ∗ (C/C)(0) be the full subcategory of C/D spanned by those g : D0 → D in C such that f g is equivale ...
The separated extensional Chu category.
... MICHAEL BARR Transmitted by ABSTRACT. This paper shows that, given a factorization system, E/M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also ∗-autonomous under weaker conditions than had been given previously ([Barr, 1991)] ...
... MICHAEL BARR Transmitted by ABSTRACT. This paper shows that, given a factorization system, E/M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also ∗-autonomous under weaker conditions than had been given previously ([Barr, 1991)] ...
