Hodge Cycles on Abelian Varieties
... A. Let t1 ; : : : ; tN be absolute Hodge cycles on a smooth projective variety X and let G be the largest algebraic subgroup of GL.H .X; Q// GL.Q.1// fixing the ti ; then every cohomology class t on X fixed by G is an absolute Hodge cycle (see 3.8). B. If .Xs /s2S is an algebraic family of smoot ...
... A. Let t1 ; : : : ; tN be absolute Hodge cycles on a smooth projective variety X and let G be the largest algebraic subgroup of GL.H .X; Q// GL.Q.1// fixing the ti ; then every cohomology class t on X fixed by G is an absolute Hodge cycle (see 3.8). B. If .Xs /s2S is an algebraic family of smoot ...
A refinement of the Artin conductor and the base change conductor
... then de Shalit’s recipe [CY01, A1.9] shows that c(T ) = cGal (T̂ ), where c(T ) is the classical base change conductor of T , which is defined in terms of algebraic Néron models, and where cGal (T̂ ) is the base change conductor of T̂ , defined Galois-theoretically. It is easily seen that ˆ· induce ...
... then de Shalit’s recipe [CY01, A1.9] shows that c(T ) = cGal (T̂ ), where c(T ) is the classical base change conductor of T , which is defined in terms of algebraic Néron models, and where cGal (T̂ ) is the base change conductor of T̂ , defined Galois-theoretically. It is easily seen that ˆ· induce ...
12 Recognizing invertible elements and full ideals using finite
... Remarks. 1) The interest of of the last property is that, from the results of Section 11, we obtain the Proposition. Let M be a closed 3-dimensional manifold and ξ ∈ H 1 (M, R) \ {0}, such that every twisted Alexander polynomial ∆H M,u is unitary. Assume that Mn (Z[π1 (M )]ξ ) has finitely detectabl ...
... Remarks. 1) The interest of of the last property is that, from the results of Section 11, we obtain the Proposition. Let M be a closed 3-dimensional manifold and ξ ∈ H 1 (M, R) \ {0}, such that every twisted Alexander polynomial ∆H M,u is unitary. Assume that Mn (Z[π1 (M )]ξ ) has finitely detectabl ...
Moduli of elliptic curves
... One can have a bit of fun with the formulas, using them to find interesting curves with small conductor. For instance, in the N = 5 case, one can look for the choices of integer b that make ∆(b, b) = b5 (b2 − 11b − 1) as simple as possible, setting, say, b5 = 1 or b2 −11b−1 = −1. These choices give ...
... One can have a bit of fun with the formulas, using them to find interesting curves with small conductor. For instance, in the N = 5 case, one can look for the choices of integer b that make ∆(b, b) = b5 (b2 − 11b − 1) as simple as possible, setting, say, b5 = 1 or b2 −11b−1 = −1. These choices give ...
finitely generated powerful pro-p groups
... however, quite decisively distracted by one seemingly small result mentioned in passing in the first pages of the survey: “...a topological group G is compact p-adic analytic if and only if G is profinite, with an open subgroup which is pro-p of finite rank...” This led me down a somewhat skewed pat ...
... however, quite decisively distracted by one seemingly small result mentioned in passing in the first pages of the survey: “...a topological group G is compact p-adic analytic if and only if G is profinite, with an open subgroup which is pro-p of finite rank...” This led me down a somewhat skewed pat ...
Topological realizations of absolute Galois groups
... Descent along the cyclotomic extension. So far, all of our results were assuming that F contains all roots of unity. One may wonder whether the general case can be handled by a descent technique. This is, unfortunately, not automatic, as the construction for F involved the choice of roots of unity, ...
... Descent along the cyclotomic extension. So far, all of our results were assuming that F contains all roots of unity. One may wonder whether the general case can be handled by a descent technique. This is, unfortunately, not automatic, as the construction for F involved the choice of roots of unity, ...
Algebraic D-groups and differential Galois theory
... will see that such a generalized logarithmic derivative is essentially equivalent to an algebraic D-group structure on G (in the sense of Buium [3]). Our resulting exposition of the generalized differential Galois theory will be equivalent to that in [8] when the base field K is algebraically closed. ...
... will see that such a generalized logarithmic derivative is essentially equivalent to an algebraic D-group structure on G (in the sense of Buium [3]). Our resulting exposition of the generalized differential Galois theory will be equivalent to that in [8] when the base field K is algebraically closed. ...
An Introduction to Unitary Representations of Lie Groups
... if {0} and H are the only π(G)-invariant closed subspaces of H. The two fundamental problems in representation theory are: (FP1) To classify, resp., parameterize the irreducible representations of G, and (FP2) to explain how a general unitary representation can be decomposed into irreducible ones. T ...
... if {0} and H are the only π(G)-invariant closed subspaces of H. The two fundamental problems in representation theory are: (FP1) To classify, resp., parameterize the irreducible representations of G, and (FP2) to explain how a general unitary representation can be decomposed into irreducible ones. T ...
A descending chain condition for groups definable in o
... equivalence classes does not get bigger when one passes to an elementary extension M 0 of M . By a definable group we mean a definable set G equipped with a group operation which is definable. By a type-definable group G we mean a typedefinable set equipped with a group operation whose graph is type ...
... equivalence classes does not get bigger when one passes to an elementary extension M 0 of M . By a definable group we mean a definable set G equipped with a group operation which is definable. By a type-definable group G we mean a typedefinable set equipped with a group operation whose graph is type ...
A course on finite flat group schemes and p
... 1.2. . . . and what they are good for. Special properties of p-divisible groups are used in: (1) Analysis of the local p-adic Galois action on p-torsion points of elliptic curves, see Serre’s theorem on open image for non-CM elliptic curves over number fields [Se72], and more recently in modularity ...
... 1.2. . . . and what they are good for. Special properties of p-divisible groups are used in: (1) Analysis of the local p-adic Galois action on p-torsion points of elliptic curves, see Serre’s theorem on open image for non-CM elliptic curves over number fields [Se72], and more recently in modularity ...
RELATIVE KAZHDAN PROPERTY
... We are interested in the question of determining, given a group G, subsets X such that (G, X) has relative Property (T). As a general result, we show, provided that G is compactly generated, that such subsets coincide with the bounded subsets for a well-defined, essentially left-invariant metric on ...
... We are interested in the question of determining, given a group G, subsets X such that (G, X) has relative Property (T). As a general result, we show, provided that G is compactly generated, that such subsets coincide with the bounded subsets for a well-defined, essentially left-invariant metric on ...
ETALE COHOMOLOGY AND THE WEIL CONJECTURES Sommaire 1.
... 2.2. Points. Let X be a separated scheme of finite type over Fq . We don’t know what the shape of X is, but we can count points. The basic idea of the Weil conjectures in this setting is that “shape= number of points”. But what are points? Recall that we have two notions of points for schemes. 2.3. ...
... 2.2. Points. Let X be a separated scheme of finite type over Fq . We don’t know what the shape of X is, but we can count points. The basic idea of the Weil conjectures in this setting is that “shape= number of points”. But what are points? Recall that we have two notions of points for schemes. 2.3. ...
Lecture Notes
... The scheme GR is also called the constant group scheme over R with fiber G. Lemma. Let def ...
... The scheme GR is also called the constant group scheme over R with fiber G. Lemma. Let def ...
13. Dedekind Domains
... examples for Dedekind domains. However, there is also a large class of examples in number theory, which explains why the concept of a Dedekind domain is equally important in number theory and geometry: it turns out that the ring of integral elements in a number field, i. e. in a finite field extensi ...
... examples for Dedekind domains. However, there is also a large class of examples in number theory, which explains why the concept of a Dedekind domain is equally important in number theory and geometry: it turns out that the ring of integral elements in a number field, i. e. in a finite field extensi ...
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 14
... We call a scheme of the form Proj S• (where S0 = A) a projective scheme over A, or a projective A-scheme. A quasiprojective A-scheme is an open subscheme of a projective A-scheme. The “A” is omitted if it is clear from the context; often A is some field. We now make a connection to classical termin ...
... We call a scheme of the form Proj S• (where S0 = A) a projective scheme over A, or a projective A-scheme. A quasiprojective A-scheme is an open subscheme of a projective A-scheme. The “A” is omitted if it is clear from the context; often A is some field. We now make a connection to classical termin ...
Quotient Modules in Depth
... certain subsets under multiplication and its opposite. About ten to twenty years ago, normality was extended to subrings succesfully by a “depth two” definition in [28] using only a tensor product of natural bimodules of the subring pair and module similarity in [2]. See for example the paper [6] fo ...
... certain subsets under multiplication and its opposite. About ten to twenty years ago, normality was extended to subrings succesfully by a “depth two” definition in [28] using only a tensor product of natural bimodules of the subring pair and module similarity in [2]. See for example the paper [6] fo ...
1 Binary Operations - Department of Mathematics | Illinois State
... did with addition, we usually denote this binary operation on Zn by the usual multiplication with the understanding that if this is to be a binary operation on Zn , it must be this modified one. An alternate way to view Zn that requires a little more background and is an example of what we will see ...
... did with addition, we usually denote this binary operation on Zn by the usual multiplication with the understanding that if this is to be a binary operation on Zn , it must be this modified one. An alternate way to view Zn that requires a little more background and is an example of what we will see ...
Commutative Algebra Notes Introduction to Commutative Algebra
... such that t ≤ x for every t ∈ T . Finally, a maximal element in S is an element x ∈ S so that for all y such that x ≤ y, we have x = y. Theorem 1.2 (Zorn’s Lemma). If every chain T of S has an upper bound in S then S has at least one maximal element. Zorn’s Lemma is equivalent to the axiom of choice ...
... such that t ≤ x for every t ∈ T . Finally, a maximal element in S is an element x ∈ S so that for all y such that x ≤ y, we have x = y. Theorem 1.2 (Zorn’s Lemma). If every chain T of S has an upper bound in S then S has at least one maximal element. Zorn’s Lemma is equivalent to the axiom of choice ...
FINITE SIMPLICIAL MULTICOMPLEXES
... Definition 1.1. A finite subset Γ ⊂ Nn is called a finite simplicial multicomplex if for all a ∈ Γ and all b ∈ Nn with b ≤ a, it follows that b ∈ Γ. The elements of Γ are called faces. An element m ∈ Γ is called a maximal facet if it does not exist a ∈ Γ with a > m; in other words, if m is maximal w ...
... Definition 1.1. A finite subset Γ ⊂ Nn is called a finite simplicial multicomplex if for all a ∈ Γ and all b ∈ Nn with b ≤ a, it follows that b ∈ Γ. The elements of Γ are called faces. An element m ∈ Γ is called a maximal facet if it does not exist a ∈ Γ with a > m; in other words, if m is maximal w ...
Algebra I (Math 200)
... Note that gXg −1 = X ⇐⇒ cg (X) = X ⇐⇒ gX = Xg. One always has NG (X) 6 G. Moreover, the centralizer of X is defined as CG (X) := {g ∈ G | gxg −1 = x for all x ∈ X} . Note that g ∈ CG (X) ⇐⇒ cg is the identity on X ⇐⇒ gx = xg for all x ∈ X. It is easy to check that CG (X) 6 NG (X) is again a subgrou ...
... Note that gXg −1 = X ⇐⇒ cg (X) = X ⇐⇒ gX = Xg. One always has NG (X) 6 G. Moreover, the centralizer of X is defined as CG (X) := {g ∈ G | gxg −1 = x for all x ∈ X} . Note that g ∈ CG (X) ⇐⇒ cg is the identity on X ⇐⇒ gx = xg for all x ∈ X. It is easy to check that CG (X) 6 NG (X) is again a subgrou ...
´Etale cohomology of schemes and analytic spaces
... two arrows, namely, the two projections, from Xi ×X Xi to X. Therefore when i = j the condition si|Xi ×X Xj = sj|Xi ×X Xj should be understood as follows: the images of s in F (Xi ×X Xi ) under the arrows induced by the two projections coincide; this is not automatic in general, as we will see furth ...
... two arrows, namely, the two projections, from Xi ×X Xi to X. Therefore when i = j the condition si|Xi ×X Xj = sj|Xi ×X Xj should be understood as follows: the images of s in F (Xi ×X Xi ) under the arrows induced by the two projections coincide; this is not automatic in general, as we will see furth ...
The Spectrum of a Ring as a Partially Ordered Set.
... in contrast to Hochster1s development, a direct, constructive proof for finite sets is given which yields some information about R itself. It is known that if R is a Prh'fer domain, then Spec R is a tree with unique minimal element and which, of course, satisfies properties (Kl) and (K2). ...
... in contrast to Hochster1s development, a direct, constructive proof for finite sets is given which yields some information about R itself. It is known that if R is a Prh'fer domain, then Spec R is a tree with unique minimal element and which, of course, satisfies properties (Kl) and (K2). ...
the usual castelnuovo s argument and special subhomaloidal
... octic surface in P6 (see [19], [12]). These special homaloidal systems have also the property that the map they de�ne is an isomorphism on Pr \ Sec(X ). Several interesting examples of special subhomaloidal systems are discussed in [18], [8] and [12] and most of the examples of homaloidal systems ar ...
... octic surface in P6 (see [19], [12]). These special homaloidal systems have also the property that the map they de�ne is an isomorphism on Pr \ Sec(X ). Several interesting examples of special subhomaloidal systems are discussed in [18], [8] and [12] and most of the examples of homaloidal systems ar ...
Sylow`s Subgroup Theorem
... by definition of the centre, h ∈ Z ( G ). Let Ci be the i-th conjugacy class. The group G acts on Ci by conjugation g ∈ G : h ∈ Ci 7→ ghg−1 . If h ∈ Ci , then the stabiliser Stab(h) is subgroup in G satisfying | G | = |Orb(h)||Stab(h)| = ni |Stab(h)|. The stabiliser Stab(h) is called the centraliser ...
... by definition of the centre, h ∈ Z ( G ). Let Ci be the i-th conjugacy class. The group G acts on Ci by conjugation g ∈ G : h ∈ Ci 7→ ghg−1 . If h ∈ Ci , then the stabiliser Stab(h) is subgroup in G satisfying | G | = |Orb(h)||Stab(h)| = ni |Stab(h)|. The stabiliser Stab(h) is called the centraliser ...
Some topics in the theory of finite groups
... only if the numbers m1 , . . . , mr and k are the same for the two groups. Alternatively, all finite abelian groups are direct products of cyclic groups of prime power order. This follows from the fact that if m and n are relatively prime then Cm ×Cn ∼ ...
... only if the numbers m1 , . . . , mr and k are the same for the two groups. Alternatively, all finite abelian groups are direct products of cyclic groups of prime power order. This follows from the fact that if m and n are relatively prime then Cm ×Cn ∼ ...