1-7 - My CCSD
... The letter a has a value that can change, or vary. When a letter represents a number that can vary, it is called a variable. The year 1954 is a constant because the number cannot change. An algebraic expression consists of one or more variables. It usually contains constants and operations. For exam ...
... The letter a has a value that can change, or vary. When a letter represents a number that can vary, it is called a variable. The year 1954 is a constant because the number cannot change. An algebraic expression consists of one or more variables. It usually contains constants and operations. For exam ...
Exercise Sheet 4 - D-MATH
... b) More generally, any topological sheaf f : X Ñ Rn automatically acquires a smooth atlas consisting of its local homeomorphisms onto open subsets of Rn . c)* The sheaf of germs of holomorphic functions over C is Hausdorff and is a smooth manifold. The sheaf of germs of smooth real-valued functions ...
... b) More generally, any topological sheaf f : X Ñ Rn automatically acquires a smooth atlas consisting of its local homeomorphisms onto open subsets of Rn . c)* The sheaf of germs of holomorphic functions over C is Hausdorff and is a smooth manifold. The sheaf of germs of smooth real-valued functions ...
INTERSECTION THEORY IN ALGEBRAIC GEOMETRY: COUNTING
... We now deal with adding points at infinity. Definition 2. Define two nonzero points (x0 , x1 , x2 , . . . , xn ) and (y0 , y1 , y2 , . . . , yn ) in Cn+1 to be equivalent if there is a constant λ ∈ C× such that xi = λyi for 0 ≤ i ≤ n. Group the nonzero points of Cn+1 together into classes of equival ...
... We now deal with adding points at infinity. Definition 2. Define two nonzero points (x0 , x1 , x2 , . . . , xn ) and (y0 , y1 , y2 , . . . , yn ) in Cn+1 to be equivalent if there is a constant λ ∈ C× such that xi = λyi for 0 ≤ i ≤ n. Group the nonzero points of Cn+1 together into classes of equival ...
![Rings of constants of the form k[f]](http://s1.studyres.com/store/data/021729650_1-3a6201c0eec615e02140355abcc4b661-300x300.png)
Rings of constants of the form k[f]
... Lemma 2.3. If h ∈ k[X] \ k, then k[h] is a maximal element in the family M if and only if the algebra k[h] is integrally closed in k[X]. In particular, if f ∈ k[X] \ k, then the integral closure of k[f ] in k[X] is of the form k[g], for some g ∈ k[X] \ k. Note also the following obvious lemma. Lemma ...
... Lemma 2.3. If h ∈ k[X] \ k, then k[h] is a maximal element in the family M if and only if the algebra k[h] is integrally closed in k[X]. In particular, if f ∈ k[X] \ k, then the integral closure of k[f ] in k[X] is of the form k[g], for some g ∈ k[X] \ k. Note also the following obvious lemma. Lemma ...
CW Complexes and the Projective Space
... S ⊆ X a subset of X. The interior S̊ of S is the union of all open sets contained in S. The closure S of S is the intersection of all closed sets containing S. The boundary of S is δS = S \ S̊. A CW-complex X is given by the following data: {(X n , {ϕnα }α∈Λn )}n∈N ⊆N ...
... S ⊆ X a subset of X. The interior S̊ of S is the union of all open sets contained in S. The closure S of S is the intersection of all closed sets containing S. The boundary of S is δS = S \ S̊. A CW-complex X is given by the following data: {(X n , {ϕnα }α∈Λn )}n∈N ⊆N ...
Section 1.0.4.
... and the right-hand side is a piece of the zeroth homotopy group of KX localized with respect to the topological K-spectrum K under l-adic completion (LK KX)∧ , tensored with Q, ...
... and the right-hand side is a piece of the zeroth homotopy group of KX localized with respect to the topological K-spectrum K under l-adic completion (LK KX)∧ , tensored with Q, ...
x 2
... Finding the lowest common denominator (LCD) 1. Factor each denominator into its prime factors; that is, factor each denominator completely 2. Then the LCD is the product formed by using each of the different factors the greatest number of times that it occurs in any one of the given denominators ...
... Finding the lowest common denominator (LCD) 1. Factor each denominator into its prime factors; that is, factor each denominator completely 2. Then the LCD is the product formed by using each of the different factors the greatest number of times that it occurs in any one of the given denominators ...
Algebra Qualifying Exam January 2015
... non-zero prime ideal of R is maximal. (b) Give an example of a commutative ring R and a non-zero prime ideal I that is not maximal. (c) Let K a field which is NOT algebraically closed. Give an example of a maximal ideal of the ring R = K[X, Y ] which is NOT of the form (X − a, Y − b) with a, b ∈ K ( ...
... non-zero prime ideal of R is maximal. (b) Give an example of a commutative ring R and a non-zero prime ideal I that is not maximal. (c) Let K a field which is NOT algebraically closed. Give an example of a maximal ideal of the ring R = K[X, Y ] which is NOT of the form (X − a, Y − b) with a, b ∈ K ( ...
An introduction to schemes - University of Chicago Math
... the base field. Let R = k[X1 , . . . Xm ]/(f1 , . . . fn ) be the coordinate ring of X and R0 = k0 [X1 , . . . Xm ]/(f1 , . . . fn ) be that of X0 . Then R = R0 ⊗k0 k, so by duality, X = X0 ×Spec(k0 ) Spec(k) In fact, the other way also holds. If we have affine varieties X over k and X0 over k0 sati ...
... the base field. Let R = k[X1 , . . . Xm ]/(f1 , . . . fn ) be the coordinate ring of X and R0 = k0 [X1 , . . . Xm ]/(f1 , . . . fn ) be that of X0 . Then R = R0 ⊗k0 k, so by duality, X = X0 ×Spec(k0 ) Spec(k) In fact, the other way also holds. If we have affine varieties X over k and X0 over k0 sati ...
geometric congruence
... therefore in neutral geometry). Also note that AAS is not valid in spherical geometry, but all of the other criteria are. On the other hand, in both hyperbolic geometry and spherical geometry, AAA is a criterion that indicates that two given triangles are congruent. Affine Congruence Two geometric f ...
... therefore in neutral geometry). Also note that AAS is not valid in spherical geometry, but all of the other criteria are. On the other hand, in both hyperbolic geometry and spherical geometry, AAA is a criterion that indicates that two given triangles are congruent. Affine Congruence Two geometric f ...
REPRESENTATION THEORY ASSIGNMENT 3 DUE FRIDAY
... flags in C2n and identify the image. (b) Recall that the compact form is K = SO2n (R). Give a linear algebra description of K/T and find a bijection between this set and the set of orthogonal flags described above. (Note that the maximal torus of K consists of 2 × 2 blocks of rotation matrices place ...
... flags in C2n and identify the image. (b) Recall that the compact form is K = SO2n (R). Give a linear algebra description of K/T and find a bijection between this set and the set of orthogonal flags described above. (Note that the maximal torus of K consists of 2 × 2 blocks of rotation matrices place ...
Section 2.1
... The remarks above show that the properties of R[x] are influenced by the properties of R. We will shortly assume that R is an integral domain, and later that R is a field. Definition 2.1.2 Let R be a ring. The degree of a polynomial f(x) in R[X] is defined to be the maximum i for which xi appears wi ...
... The remarks above show that the properties of R[x] are influenced by the properties of R. We will shortly assume that R is an integral domain, and later that R is a field. Definition 2.1.2 Let R be a ring. The degree of a polynomial f(x) in R[X] is defined to be the maximum i for which xi appears wi ...
Solutions to final review sheet
... under multiply (both of which are quite easy to see) it is not closed under addition. A better question would be “Is Ia an ideal of the ring R”! So let me answer that instead: NO! For example, consider the ring R = Z[x, y]/(x2 , y 2 ). This is the ring Z[x, y] of polynomials in two variables factore ...
... under multiply (both of which are quite easy to see) it is not closed under addition. A better question would be “Is Ia an ideal of the ring R”! So let me answer that instead: NO! For example, consider the ring R = Z[x, y]/(x2 , y 2 ). This is the ring Z[x, y] of polynomials in two variables factore ...
Algebraic Geometry I
... 4. Assume char k 6= 2, 3. Prove that a smooth plane cubic C has nine distinct inflection points. Equivalently prove that mp (C, Hess(C)) = 1 for every inflection point p of C. (Hint: you may assume that the inflection point p is [0, 0, 1] and the tangent line to C at p is y = 0. Then prove that if f ...
... 4. Assume char k 6= 2, 3. Prove that a smooth plane cubic C has nine distinct inflection points. Equivalently prove that mp (C, Hess(C)) = 1 for every inflection point p of C. (Hint: you may assume that the inflection point p is [0, 0, 1] and the tangent line to C at p is y = 0. Then prove that if f ...
PDF
... Theorem. The polynomial ring over a field is a Euclidean domain. Proof. Let K[X] be the polynomial ring over a field K in the indeterminate X. Since K is an integral domain and any polynomial ring over integral domain is an integral domain, the ring K[X] is an integral domain. The degree ν(f ), defi ...
... Theorem. The polynomial ring over a field is a Euclidean domain. Proof. Let K[X] be the polynomial ring over a field K in the indeterminate X. Since K is an integral domain and any polynomial ring over integral domain is an integral domain, the ring K[X] is an integral domain. The degree ν(f ), defi ...
SOLUTIONS TO EXERCISES 1.3, 1.12, 1.14, 1.16 Exercise 1.3: Let
... b) ((i) =⇒ (ii):) Suppose every nonzero element of k is a root of unity. Then every nonzero element satisfies the polynomial xm − 1 for some positive integer m. Let n = (1 + 1 + · · · + 1), the n-fold sum of 1. Then either 2 = 0 (hence k has characteristic 2), or there exists m ∈ N such that 2m = 1. ...
... b) ((i) =⇒ (ii):) Suppose every nonzero element of k is a root of unity. Then every nonzero element satisfies the polynomial xm − 1 for some positive integer m. Let n = (1 + 1 + · · · + 1), the n-fold sum of 1. Then either 2 = 0 (hence k has characteristic 2), or there exists m ∈ N such that 2m = 1. ...
Curves and Manifolds
... curves), and algebraic plane curves. A smooth plane curve is a curve in a real Euclidian plane R2 is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x,y) = 0, where f is a smooth function of two variables, and the partial derivatives fx and ...
... curves), and algebraic plane curves. A smooth plane curve is a curve in a real Euclidian plane R2 is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x,y) = 0, where f is a smooth function of two variables, and the partial derivatives fx and ...
Document
... Investigate number patterns leading to those where there is a constant difference between consecutive terms, and the general term is therefore linear. WITHOUT USING A FORMULA Linear equations. Quadratic equations (by factorisation). Literal equations (change the subject of the formula). Simultaneous ...
... Investigate number patterns leading to those where there is a constant difference between consecutive terms, and the general term is therefore linear. WITHOUT USING A FORMULA Linear equations. Quadratic equations (by factorisation). Literal equations (change the subject of the formula). Simultaneous ...
1 Polynomial Rings
... – Two different polynomials can define the same function on R, but we still treat them as different elements of R[x]. For example p(x) = x · (x − 1) · · · (x − p + 1) defines the zero function on Zp , but is not the zero polynomial (why?). – For polynomials of degree at most n, their sum can be comp ...
... – Two different polynomials can define the same function on R, but we still treat them as different elements of R[x]. For example p(x) = x · (x − 1) · · · (x − p + 1) defines the zero function on Zp , but is not the zero polynomial (why?). – For polynomials of degree at most n, their sum can be comp ...
![A NOTE ON A THEOREM OF AX 1. Introduction In [1]](http://s1.studyres.com/store/data/002606018_1-a040f94f7e203e575591799a9dc26ca7-300x300.png)
A NOTE ON A THEOREM OF AX 1. Introduction In [1]
... generated by V (K) − c is of the form H(K), where H is a connected algebraic subgroup of A defined over C. By the |C|+ -saturation of K, Ox is Zariski dense in V . Therefore, Ox − c generates H(K) as well. By 2.2, ωK vanishes on Ox . By 2.4, ωK vanishes on H(K). Clearly, x ∈ H(K) + A(C). We need the ...
... generated by V (K) − c is of the form H(K), where H is a connected algebraic subgroup of A defined over C. By the |C|+ -saturation of K, Ox is Zariski dense in V . Therefore, Ox − c generates H(K) as well. By 2.2, ωK vanishes on Ox . By 2.4, ωK vanishes on H(K). Clearly, x ∈ H(K) + A(C). We need the ...
On functions with zero mean over a finite group
... 7/.,-invariant, f has at least 2n roots counted according to their multiplicities. At the same time ] belongs to a Chebyshev system of order 2n - 1, and therefore f = 0. 3. We give three analogs of Theorem 3 for functions of several variables. (i) Let P ( z x , . . . , zt) be a complex polynomial of ...
... 7/.,-invariant, f has at least 2n roots counted according to their multiplicities. At the same time ] belongs to a Chebyshev system of order 2n - 1, and therefore f = 0. 3. We give three analogs of Theorem 3 for functions of several variables. (i) Let P ( z x , . . . , zt) be a complex polynomial of ...
field 53: elementary mathematics
... properties of tessellations; transformations in the coordinate plane; and proofs of theorem and solutions of problems through coordinate and transformational geometry. ...
... properties of tessellations; transformations in the coordinate plane; and proofs of theorem and solutions of problems through coordinate and transformational geometry. ...
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.