Dimension theory
... PL (t) − tks PK (t) 1 − tks Since xs annihilates K and L, these are graded modules over the graded ring A = A/(xs ) whose positive grade ideal has s − 1 generators x1 , x2 , · · · , xs−1 , the images of i ∈ A. So, PL (t) and PK (t) are rational �xs−1 functions with denominator i=1 (1 − tki ). The th ...
... PL (t) − tks PK (t) 1 − tks Since xs annihilates K and L, these are graded modules over the graded ring A = A/(xs ) whose positive grade ideal has s − 1 generators x1 , x2 , · · · , xs−1 , the images of i ∈ A. So, PL (t) and PK (t) are rational �xs−1 functions with denominator i=1 (1 − tki ). The th ...
Selected Exercises 1. Let M and N be R
... or x = 0. Show that a torsion-free divisible R-module is injective. Conclude that K is an injective R-module, for any field K containing R. 20. Let R be a Noetherian commutative ring and Q an injective R-module. Fix an ideal I ⊆ R and set ΓI (Q) := {x ∈ Q | I n x = 0, for some n ≥ 0}. Show that ΓI ( ...
... or x = 0. Show that a torsion-free divisible R-module is injective. Conclude that K is an injective R-module, for any field K containing R. 20. Let R be a Noetherian commutative ring and Q an injective R-module. Fix an ideal I ⊆ R and set ΓI (Q) := {x ∈ Q | I n x = 0, for some n ≥ 0}. Show that ΓI ( ...
PM 464
... Remark. Equality does not always hold in point (3) of the last proposition. 1. Consider I = 〈x 2 + 1〉 ⊆ R[x]. Then 1 ∈ / I, so I 6= R[x]. But V (I) = ∅, so I(V (I)) = R[x] % I. 2. Consider X = [0, 1] ⊆ R. Then I(X ) = 0 and V (I(X )) = R % X . 1.14 Definition. Let X ⊆ An (|) and I ⊆ |[x 1 , . . . , ...
... Remark. Equality does not always hold in point (3) of the last proposition. 1. Consider I = 〈x 2 + 1〉 ⊆ R[x]. Then 1 ∈ / I, so I 6= R[x]. But V (I) = ∅, so I(V (I)) = R[x] % I. 2. Consider X = [0, 1] ⊆ R. Then I(X ) = 0 and V (I(X )) = R % X . 1.14 Definition. Let X ⊆ An (|) and I ⊆ |[x 1 , . . . , ...
SINGULARITIES ON COMPLETE ALGEBRAIC VARIETIES 1
... hence it’s image F0 (R0 ) = D1 is defined by a monic polynomial f1 (zn ) of degree d1 ≤ (d − 1) in zn . Let F1 : Cn → Cn be the branch cover of degree d1 defined by F1 (z1 , ..., zn ) = (z1 , ..., zn−1 , f1 (zn )) and g2 = F1 ◦ g1 . The ramification divisor is the union of two sections of p, F1 ({zn ...
... hence it’s image F0 (R0 ) = D1 is defined by a monic polynomial f1 (zn ) of degree d1 ≤ (d − 1) in zn . Let F1 : Cn → Cn be the branch cover of degree d1 defined by F1 (z1 , ..., zn ) = (z1 , ..., zn−1 , f1 (zn )) and g2 = F1 ◦ g1 . The ramification divisor is the union of two sections of p, F1 ({zn ...
THE BRAUER GROUP: A SURVEY Introduction Notation
... space of dimension n − 1). Of course, there are those projective bundles which come from trivial bundles, namely projectivizations of vector bundles. We wish to consider such bundles as trivial. The cohomology set H 1 (X, GLn ) is in bijective correspondence with vector bundles. There is a map H 1 ( ...
... space of dimension n − 1). Of course, there are those projective bundles which come from trivial bundles, namely projectivizations of vector bundles. We wish to consider such bundles as trivial. The cohomology set H 1 (X, GLn ) is in bijective correspondence with vector bundles. There is a map H 1 ( ...
LA ACCION DEL ALGEBRA DE STEENROD SOBRE LAS
... These results are applied to find obstructions to the existence of sections of multiples of the Hopf bundle over real projective spaces. The content of this article is part of the thesis [1] written under the direction of Samuel Gitler, whom I express my gratitude. I also want to thank Pedro Armenda ...
... These results are applied to find obstructions to the existence of sections of multiples of the Hopf bundle over real projective spaces. The content of this article is part of the thesis [1] written under the direction of Samuel Gitler, whom I express my gratitude. I also want to thank Pedro Armenda ...
SUFFICIENTLY GENERIC ORTHOGONAL GRASSMANNIANS 1
... Given projective homogeneous varieties X1 and X2 (under possibly different algebraic groups) over finite separable field extensions K1 and K2 of F , the upper motives of the F -varieties X1 and X2 satisfy the following isomorphism criterion: Lemma 2.1 ([11, Corollary 2.15]). The upper motives U (X1 ) a ...
... Given projective homogeneous varieties X1 and X2 (under possibly different algebraic groups) over finite separable field extensions K1 and K2 of F , the upper motives of the F -varieties X1 and X2 satisfy the following isomorphism criterion: Lemma 2.1 ([11, Corollary 2.15]). The upper motives U (X1 ) a ...
Projective p-adic representations of the K-rational geometric fundamental group (with G. Frey).
... A first condition for the existence of such a system {ρ̃Vi } is given by the criterion of Néron-Ogg-Shafarevich: A has to have good reduction everywhere, i.e. at all points of C. A second condition is that the fixed fields of (finite quotients of) the kernels of the ρ̃Vi ’s are regular extensions ...
... A first condition for the existence of such a system {ρ̃Vi } is given by the criterion of Néron-Ogg-Shafarevich: A has to have good reduction everywhere, i.e. at all points of C. A second condition is that the fixed fields of (finite quotients of) the kernels of the ρ̃Vi ’s are regular extensions ...
Rank conjecture revisited
... kn ; since (3.6) has one solution in integers numbers, it has infinitely many such solutions. In other words, the projective closure VE of an affine variety defined by the Euler equations (3.4) is a fiber bundle over the projective line CP 1 . Let us show that the fiber p−1 (x) of a flat morphism p ...
... kn ; since (3.6) has one solution in integers numbers, it has infinitely many such solutions. In other words, the projective closure VE of an affine variety defined by the Euler equations (3.4) is a fiber bundle over the projective line CP 1 . Let us show that the fiber p−1 (x) of a flat morphism p ...
Axiomatising the modal logic of affine planes
... A1. any two distinct points lie on a unique line ∀x, y ∈ P (x 6= y → ∃ ! l ∈ L(x E l ∧ y E l)) A2. there is a unique line through any point parallel to any given line ∀x ∈ P ∀l ∈ L ∃ ! m ∈ L(x E m ∧ m || l) A3. there exist three non-collinear points ∃x0 x1 x2 ∈ P ¬∃l ∈ L(x0 E l ∧ x1 E l ∧ x2 E l) (e ...
... A1. any two distinct points lie on a unique line ∀x, y ∈ P (x 6= y → ∃ ! l ∈ L(x E l ∧ y E l)) A2. there is a unique line through any point parallel to any given line ∀x ∈ P ∀l ∈ L ∃ ! m ∈ L(x E m ∧ m || l) A3. there exist three non-collinear points ∃x0 x1 x2 ∈ P ¬∃l ∈ L(x0 E l ∧ x1 E l ∧ x2 E l) (e ...
Lecture Notes - Mathematics
... called the analytic index of singularities or the log canonical threshold of f , and is an important invariant in algebraic geometry and complex analysis of several variables. Resolution of singularities is a huge topic with a classical history that has been re-interpreted many times over into the p ...
... called the analytic index of singularities or the log canonical threshold of f , and is an important invariant in algebraic geometry and complex analysis of several variables. Resolution of singularities is a huge topic with a classical history that has been re-interpreted many times over into the p ...
SYNTHETIC PROJECTIVE GEOMETRY
... Y1 , · · · Yk respectively. Prove that the points Y 1 , · · · Yk are distinct if and only if the lines M1 , · · · Mk are distinct. 5. Let (S, Π, d) be a regular incidence space of dimension ≥ 3, and assume that every plane in S is projective (so it follows that (P-1) holds). Prove that S is projecti ...
... Y1 , · · · Yk respectively. Prove that the points Y 1 , · · · Yk are distinct if and only if the lines M1 , · · · Mk are distinct. 5. Let (S, Π, d) be a regular incidence space of dimension ≥ 3, and assume that every plane in S is projective (so it follows that (P-1) holds). Prove that S is projecti ...
Motivic interpretation of Milnor K
... 2.1 Let k be a field, and X a smooth quasi-projective varieties over k. We use the notation CH0 (X) for the group of zero-cycles on X modulo rational equivalence. If G is a group scheme defined over k and A is k-algebra, we use the notation G(A) for the group of A-rational points, i.e., the set of m ...
... 2.1 Let k be a field, and X a smooth quasi-projective varieties over k. We use the notation CH0 (X) for the group of zero-cycles on X modulo rational equivalence. If G is a group scheme defined over k and A is k-algebra, we use the notation G(A) for the group of A-rational points, i.e., the set of m ...
fpp revised
... We start with x1 , it has a non-zero coefficient in at least one y and one z, assume these are y1 and z1 . If it has different coefficients in these two expressions, we impose the condition: y1 = z1 , otherwise, we impose: y1 = −z1 . Here is an implicit example: z1 = a11 x1 + a21 x2 + . . . + aN 1 x ...
... We start with x1 , it has a non-zero coefficient in at least one y and one z, assume these are y1 and z1 . If it has different coefficients in these two expressions, we impose the condition: y1 = z1 , otherwise, we impose: y1 = −z1 . Here is an implicit example: z1 = a11 x1 + a21 x2 + . . . + aN 1 x ...
(), Marina HARALAMPIDOU Department of Mathematics, University of Athens
... Marina HARALAMPIDOU ([email protected]), Department of Mathematics, University of Athens Panepistimioupolis, GR-157 84, Athens, Greece, The Krull nature of locally C ∗ -algebras. ABSTRACT. Any complete locally m-convex algebra, whose normed factors in its Arens-Michael decomposition are Krull algeb ...
... Marina HARALAMPIDOU ([email protected]), Department of Mathematics, University of Athens Panepistimioupolis, GR-157 84, Athens, Greece, The Krull nature of locally C ∗ -algebras. ABSTRACT. Any complete locally m-convex algebra, whose normed factors in its Arens-Michael decomposition are Krull algeb ...
PDF on arxiv.org - at www.arxiv.org.
... will identify Gal(f ) with its image and consider it as a permutation group of Rf . Clearly, Gal(f ) is transitive if and only if f is irreducible in K[x]. We refer the reader to [6, 20, 3, 8] for the definition and properties of the heart R (Fp f )00 ...
... will identify Gal(f ) with its image and consider it as a permutation group of Rf . Clearly, Gal(f ) is transitive if and only if f is irreducible in K[x]. We refer the reader to [6, 20, 3, 8] for the definition and properties of the heart R (Fp f )00 ...
NOTES ON FINITE LINEAR PROJECTIVE PLANES 1. Projective
... Lemma 17. Let Q be a finite double loop in which addition is associative and the right distributive law holds. Then (1) (−a)b = −ab for all a, b ∈ Q, (2) hQ, +, 0i is an elementary abelian p-group, i.e., there is a prime p such that |Q| = pk and every element of Q has additive order p, and (3) prope ...
... Lemma 17. Let Q be a finite double loop in which addition is associative and the right distributive law holds. Then (1) (−a)b = −ab for all a, b ∈ Q, (2) hQ, +, 0i is an elementary abelian p-group, i.e., there is a prime p such that |Q| = pk and every element of Q has additive order p, and (3) prope ...
ON PROJECTIVE THREEFOLDS OF GENERAL TYPE 1
... When dim Y = 1, it was classically known that |mKY | gives an embedding of Y into a projective space if m ≥ 3. When dim Y = 2, Bombieri’s theorem [1] says that ϕm gives a birational map onto the image for m ≥ 5. When dim Y ≥ 3, a recent remarkable result of Hacon and Mc Kernan [7], Takayama [16] and ...
... When dim Y = 1, it was classically known that |mKY | gives an embedding of Y into a projective space if m ≥ 3. When dim Y = 2, Bombieri’s theorem [1] says that ϕm gives a birational map onto the image for m ≥ 5. When dim Y ≥ 3, a recent remarkable result of Hacon and Mc Kernan [7], Takayama [16] and ...
2 - arXiv
... we can replace V by its normalisation. There is a smooth open subvariety U of V such that V \U has codimension 2, for which gU : XU = g −1 (U) → U is a morphism of varieties. As V is reduced, gU is a well defined family of algebraic cycles. It is therefore an element of Chow big (U). As U is normal ...
... we can replace V by its normalisation. There is a smooth open subvariety U of V such that V \U has codimension 2, for which gU : XU = g −1 (U) → U is a morphism of varieties. As V is reduced, gU is a well defined family of algebraic cycles. It is therefore an element of Chow big (U). As U is normal ...
Perspective - Faculty Web Pages
... The standard upon which school mathematics was based since the middle of the 19th century ...
... The standard upon which school mathematics was based since the middle of the 19th century ...
1.5.4 Every abelian variety is a quotient of a Jacobian
... Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety. The modular abelian varieties that we will encounter later are, by definition, exactly the quotients of the Jacobian J1 (N ) of X1 (N ) for some N . In this section we see that merely being a quotient o ...
... Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety. The modular abelian varieties that we will encounter later are, by definition, exactly the quotients of the Jacobian J1 (N ) of X1 (N ) for some N . In this section we see that merely being a quotient o ...
Lecture 20 1 Point Set Topology
... Thus we are allowed to localize B at s ∈ B in order to simplify the problem. If s ∈ A, s 6= 0, consider a map φs : As → Bs , then the corresponding variety Xs is an open set because Xs = X − (set of zeros ofs in X) and the set of zeros of s in X is a closed set. So if the image is constructible in X ...
... Thus we are allowed to localize B at s ∈ B in order to simplify the problem. If s ∈ A, s 6= 0, consider a map φs : As → Bs , then the corresponding variety Xs is an open set because Xs = X − (set of zeros ofs in X) and the set of zeros of s in X is a closed set. So if the image is constructible in X ...
Chakravarti, I.MAssociation Schemes, Orthogonal Arrays and Codes from Non-denerate Quadrics and Hermitian Varieties in Finite Projective Geometries"
... by Mesner (1967) is to classify points (according to some geometrical criterion) in ...
... by Mesner (1967) is to classify points (according to some geometrical criterion) in ...
The Picard group
... ? X(Q) X(Fp ) o X(Qp ) o Figure 1: Some of the sets of points associated to a Diophantine equation • On X(C), we have all the tools available to study complex analytic varieties. For example, X(C) has cohomology groups which give much information about its geometry, and these come with Hodge decompo ...
... ? X(Q) X(Fp ) o X(Qp ) o Figure 1: Some of the sets of points associated to a Diophantine equation • On X(C), we have all the tools available to study complex analytic varieties. For example, X(C) has cohomology groups which give much information about its geometry, and these come with Hodge decompo ...