Affine group schemes over symmetric monoidal categories
... presented in [Tate and Oort 1970]. Theorem 1.1 (Deligne’s lemma). Let G = Spec(A) be an affine commutative group scheme over a commutative, Noetherian ring k. Assume that A is a flat k-algebra of rank r ≥ 1. Then, for any k-algebra B, all elements in the group G(B) have an order dividing r . The pur ...
... presented in [Tate and Oort 1970]. Theorem 1.1 (Deligne’s lemma). Let G = Spec(A) be an affine commutative group scheme over a commutative, Noetherian ring k. Assume that A is a flat k-algebra of rank r ≥ 1. Then, for any k-algebra B, all elements in the group G(B) have an order dividing r . The pur ...
Incidence structures I. Constructions of some famous combinatorial
... An orthogonal array, OA(v, s, λ), is such (λv 2 ×s)dimensional matrix with v symbols, that each two columns each of v 2 possible pairs of symbols appears in exactly λ rows. This and to them equivalent structures (e.g. transversal designs, pairwise orthogonal Latin squares, nets,...) are part of desi ...
... An orthogonal array, OA(v, s, λ), is such (λv 2 ×s)dimensional matrix with v symbols, that each two columns each of v 2 possible pairs of symbols appears in exactly λ rows. This and to them equivalent structures (e.g. transversal designs, pairwise orthogonal Latin squares, nets,...) are part of desi ...
SCHEMES 01H8 Contents 1. Introduction 1 2. Locally ringed spaces
... is zero. If this is the case the morphism g : Y → Z such that f = i ◦ g is unique. Proof. Clearly if f factors as Y → Z → X then the map f ∗ I → OY is zero. Conversely suppose that f ∗ I → OY is zero. Pick any y ∈ Y , and consider the ring map fy] : OX,f (y) → OY,y . Since the composition If (y) → O ...
... is zero. If this is the case the morphism g : Y → Z such that f = i ◦ g is unique. Proof. Clearly if f factors as Y → Z → X then the map f ∗ I → OY is zero. Conversely suppose that f ∗ I → OY is zero. Pick any y ∈ Y , and consider the ring map fy] : OX,f (y) → OY,y . Since the composition If (y) → O ...
Symplectic structures -- a new approach to geometry.
... with geodesics is not far fetched. There is a very nice theory of these curves — one application is mentioned below — and they occur as an essential ingredient in many symplectic constructions, for example in Floer theory. In his 1998 Gibbs lecture, Witten discussed two “deformations” of classical ...
... with geodesics is not far fetched. There is a very nice theory of these curves — one application is mentioned below — and they occur as an essential ingredient in many symplectic constructions, for example in Floer theory. In his 1998 Gibbs lecture, Witten discussed two “deformations” of classical ...
Discovering Geometry An Investigative Approach
... C-86a Rectangular Prism Volume Conjecture If B is the area of the base of a right rectangular prism and H is the height of the solid, then the formula for the volume is V BH. (Lesson 10.2) C-86b Right Prism-Cylinder Volume Conjecture If B is the area of the base of a right prism (or cylinder) and ...
... C-86a Rectangular Prism Volume Conjecture If B is the area of the base of a right rectangular prism and H is the height of the solid, then the formula for the volume is V BH. (Lesson 10.2) C-86b Right Prism-Cylinder Volume Conjecture If B is the area of the base of a right prism (or cylinder) and ...
Conjecture - Miami Killian Senior High School
... The ordered pair rule (x, y)?(x, y) is a reflection over the y-axis. The ordered pair rule (x, y)?(x, y) is a reflection over the x-axis. The ordered pair rule (x, y)?(x, y) is a rotation about the origin. The ordered pair rule (x, y)?(y, x) is a reflection over y=x 93 Minimal Path Conjecture - If p ...
... The ordered pair rule (x, y)?(x, y) is a reflection over the y-axis. The ordered pair rule (x, y)?(x, y) is a reflection over the x-axis. The ordered pair rule (x, y)?(x, y) is a rotation about the origin. The ordered pair rule (x, y)?(y, x) is a reflection over y=x 93 Minimal Path Conjecture - If p ...
CONJECTURES - Discovering Geometry Chapter 2 C
... Reflection Line Conjecture - The line of reflection is the perpendicular bisector of every segment joining a point in the original figure with its image. Coordinate Transformations Conjecture The ordered pair rule (x, y)?(x, y) is a reflection over the y-axis. The ordered pair rule (x, y)?(x, y) is ...
... Reflection Line Conjecture - The line of reflection is the perpendicular bisector of every segment joining a point in the original figure with its image. Coordinate Transformations Conjecture The ordered pair rule (x, y)?(x, y) is a reflection over the y-axis. The ordered pair rule (x, y)?(x, y) is ...
geometry - Swampscott High School
... A diagram of part of a baseball field and some of its dimensions are shown below. Point F represents First Base, point S represents Second Base, point T represents Third Base, point H represents Home Plate, and point P represents another location on the baseball field. ...
... A diagram of part of a baseball field and some of its dimensions are shown below. Point F represents First Base, point S represents Second Base, point T represents Third Base, point H represents Home Plate, and point P represents another location on the baseball field. ...
Derived Algebraic Geometry XI: Descent
... will henceforth denote simply by ModR ). This was proven in [40] as a consequence of the following more general result: if A is a connective E∞ -ring and C is an A-linear ∞-category which admits an excellent t-structure, then the construction B 7→ LModB (C) satisfies descent for the flat topology (T ...
... will henceforth denote simply by ModR ). This was proven in [40] as a consequence of the following more general result: if A is a connective E∞ -ring and C is an A-linear ∞-category which admits an excellent t-structure, then the construction B 7→ LModB (C) satisfies descent for the flat topology (T ...
360
... If two chords in a circle are congruent, then they determine two central angles that are congruent. Chord Arcs Conjecture If two chords in a circle are congruent, then their intercepted arcs are congruent. Perpendicular to a Chord Conjecture The perpendicular from the center of a circle to a chord i ...
... If two chords in a circle are congruent, then they determine two central angles that are congruent. Chord Arcs Conjecture If two chords in a circle are congruent, then their intercepted arcs are congruent. Perpendicular to a Chord Conjecture The perpendicular from the center of a circle to a chord i ...
Conjectures for Geometry for Math 70 By I. L. Tse Chapter 2
... 13. Circumference Conjecture: If C is the circumference and d is the diameter of a circle, then there is a number such as C= d. If d = 2r where r = is the radius, then C = 2 r. 14. Arc Length Conjecture: The arc length equals the arc measure divided by 360° times the circumference arc measure ar ...
... 13. Circumference Conjecture: If C is the circumference and d is the diameter of a circle, then there is a number such as C= d. If d = 2r where r = is the radius, then C = 2 r. 14. Arc Length Conjecture: The arc length equals the arc measure divided by 360° times the circumference arc measure ar ...
CLUSTER ALGEBRAS AND CLUSTER CATEGORIES
... grouped into overlapping subsets (the clusters) of constant cardinality (the rank) which are constructed recursively via mutation from an initial cluster. The set of cluster variables can be finite or infinite. Theorem 3.1. [38]. The cluster algebras having only a finite number of cluster variables ...
... grouped into overlapping subsets (the clusters) of constant cardinality (the rank) which are constructed recursively via mutation from an initial cluster. The set of cluster variables can be finite or infinite. Theorem 3.1. [38]. The cluster algebras having only a finite number of cluster variables ...
on the ubiquity of simplicial objects
... In the second chapter we specialise to simplicial sets. We will develop some simplicial homotopy theory and then establish an equivalence between simplicial homotopy theory and the classical homotopy theory of topological spaces. Most of our results in this section come from May’s Simplicial Objects ...
... In the second chapter we specialise to simplicial sets. We will develop some simplicial homotopy theory and then establish an equivalence between simplicial homotopy theory and the classical homotopy theory of topological spaces. Most of our results in this section come from May’s Simplicial Objects ...
Slide 1
... the electric eels used to make the wallet holds an electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? The conclusion is based on logical reasoning from scientific research. I ...
... the electric eels used to make the wallet holds an electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? The conclusion is based on logical reasoning from scientific research. I ...
Motivic Homotopy Theory
... is determined by the cohomology of its pieces X, X 0 and X0 . If we start with some category of schemes, there are a few problems. First, what will be the notion of a weak equivalence of schemes and a homotopy between morphisms of schemes ? The notion of a weak equivalence between the underlying top ...
... is determined by the cohomology of its pieces X, X 0 and X0 . If we start with some category of schemes, there are a few problems. First, what will be the notion of a weak equivalence of schemes and a homotopy between morphisms of schemes ? The notion of a weak equivalence between the underlying top ...
AUTOMORPHISM GROUPS AND PICARD GROUPS OF ADDITIVE
... modules N . Then Pic(C) is naturally isomorphic to AutA (C). Such construction agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as develop ...
... modules N . Then Pic(C) is naturally isomorphic to AutA (C). Such construction agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as develop ...