On function field Mordell-Lang: the semiabelian case and the
... and impenetrable for non model-theorists and (ii) in the positive characteristic case, it is “type-definable” Zariski geometries which are used and for which there is no really comprehensive exposition, although the proofs in [11] are correct. In [2] we concentrated on the abelian variety case, and ...
... and impenetrable for non model-theorists and (ii) in the positive characteristic case, it is “type-definable” Zariski geometries which are used and for which there is no really comprehensive exposition, although the proofs in [11] are correct. In [2] we concentrated on the abelian variety case, and ...
Chapter I, Section 6
... Definition (6.1.1). — X is locally Noetherian if it has a covering by open affines Spec(R) with R Noetherian. X is Noetherian if it has a finite such covering [Liu, 2.3.45]. If X is locally Noetherian, then OX is coherent, a quasi-coherent sheaf of OX modules is coherent iff it is locally finitely g ...
... Definition (6.1.1). — X is locally Noetherian if it has a covering by open affines Spec(R) with R Noetherian. X is Noetherian if it has a finite such covering [Liu, 2.3.45]. If X is locally Noetherian, then OX is coherent, a quasi-coherent sheaf of OX modules is coherent iff it is locally finitely g ...
On different notions of tameness in arithmetic geometry
... Let X be a normal, noetherian scheme and let X 0 ⊂ X be a dense open subscheme. Assume we are given an an étale covering Y 0 → X 0 . Definition. Let x ∈ X r X 0 be a point. We say that Y 0 → X 0 is unramified along x if it extends to an étale covering of some open subscheme U ⊂ X which contains X 0 ...
... Let X be a normal, noetherian scheme and let X 0 ⊂ X be a dense open subscheme. Assume we are given an an étale covering Y 0 → X 0 . Definition. Let x ∈ X r X 0 be a point. We say that Y 0 → X 0 is unramified along x if it extends to an étale covering of some open subscheme U ⊂ X which contains X 0 ...
THREE APPROACHES TO CHOW`S THEOREM 1. Statement and
... I don’t know when this result was first conjectured; Chow offers only that it is “classical.” ...
... I don’t know when this result was first conjectured; Chow offers only that it is “classical.” ...
Topological types of Algebraic stacks - IBS-CGP
... schemes or simplicial schemes to algebraic stacks. We achieve this by studying relationship among various topoi. Especially, localized topoi play a pivotal role. The main strength of our approach is that one can systematically deal with étale homotopy types of algebraic stacks thanks to the use of m ...
... schemes or simplicial schemes to algebraic stacks. We achieve this by studying relationship among various topoi. Especially, localized topoi play a pivotal role. The main strength of our approach is that one can systematically deal with étale homotopy types of algebraic stacks thanks to the use of m ...
Chapter 7 Duality
... of singularities for k-varieties, then the category DM(k) has a duality involution, making DM(k) a rigid triangulated tensor category. We then give some applications of the duality involution: in (7.4.4)-(7.4.6) we show that, in case the base scheme S is Spec of a field of characteristic zero, the m ...
... of singularities for k-varieties, then the category DM(k) has a duality involution, making DM(k) a rigid triangulated tensor category. We then give some applications of the duality involution: in (7.4.4)-(7.4.6) we show that, in case the base scheme S is Spec of a field of characteristic zero, the m ...
Sheaf Theory (London Mathematical Society Lecture Note Series)
... Chapter 3 defines the categorical viewpoint, shows that the categories of sheaves and presheaves of abelian groups on a fixed topological space are abelian, and investigates the relations between them. It also covers the processes of change of base space of a sheaf, both for the inclusion of a subsp ...
... Chapter 3 defines the categorical viewpoint, shows that the categories of sheaves and presheaves of abelian groups on a fixed topological space are abelian, and investigates the relations between them. It also covers the processes of change of base space of a sheaf, both for the inclusion of a subsp ...
Professor Farb's course notes
... However, a direct computation of the simplicial homology groups of this ∆-complex becomes a complicated affair for large n, since in order to compute we need an actual ∆-complex structure, and for this we need to throw in all faces of σ and τ , and the faces of these faces, etc. It is possible to do ...
... However, a direct computation of the simplicial homology groups of this ∆-complex becomes a complicated affair for large n, since in order to compute we need an actual ∆-complex structure, and for this we need to throw in all faces of σ and τ , and the faces of these faces, etc. It is possible to do ...
Some applications of the ultrafilter topology on spaces of valuation
... a basis for the open sets, the subsets BF := {V ∈ Z | V ⊇ F }, for F varying in the family of all finite subsets of K. When no confusion can arise, we will simply denote by Bx the basic open set B{x} of Z. This topology is now called the Zariski topology on Z and the set Z, equipped with this topolo ...
... a basis for the open sets, the subsets BF := {V ∈ Z | V ⊇ F }, for F varying in the family of all finite subsets of K. When no confusion can arise, we will simply denote by Bx the basic open set B{x} of Z. This topology is now called the Zariski topology on Z and the set Z, equipped with this topolo ...
INDEPENDENCE, MEASURE AND PSEUDOFINITE FIELDS 1
... , on one hand to stress the connection between the conjugated character and the contragredient representation, on the other to distinguish it from the notation k̄ for the algebraic closure of a field k. The residue field at a point s of a scheme is denoted k(s). For a scheme S, a variety over S is a ...
... , on one hand to stress the connection between the conjugated character and the contragredient representation, on the other to distinguish it from the notation k̄ for the algebraic closure of a field k. The residue field at a point s of a scheme is denoted k(s). For a scheme S, a variety over S is a ...
VISIBLE EVIDENCE FOR THE BIRCH AND SWINNERTON
... M2 (Γ0 (N )) ⊗ Q using Manin symbols, which are a finite set of generators for M2 (Γ0 (N )). In general, the easiest way we have found to compute M2 (Γ0 (N )) is to compute M2 (Γ0 (N ))⊗ Q and then to compute the Z-submodule of M2 (Γ0 (N ))⊗ Q generated by the Manin symbols. 3.2. Enumerating newform ...
... M2 (Γ0 (N )) ⊗ Q using Manin symbols, which are a finite set of generators for M2 (Γ0 (N )). In general, the easiest way we have found to compute M2 (Γ0 (N )) is to compute M2 (Γ0 (N ))⊗ Q and then to compute the Z-submodule of M2 (Γ0 (N ))⊗ Q generated by the Manin symbols. 3.2. Enumerating newform ...
Homology With Local Coefficients
... such as axz. Its inverseis denoted by ax-yor avx. The elementsof Fx are abbreviated ax , Ox, etc. The class acx determinesan isomorphismFx-F(denotedby ax,) definedby ax,(Ox) = ayx~xa,. In keepingwiththis notation, the productaCY3xmeans the elementof Fx obtained by traversingfirsta curve of the class ...
... such as axz. Its inverseis denoted by ax-yor avx. The elementsof Fx are abbreviated ax , Ox, etc. The class acx determinesan isomorphismFx-F(denotedby ax,) definedby ax,(Ox) = ayx~xa,. In keepingwiththis notation, the productaCY3xmeans the elementof Fx obtained by traversingfirsta curve of the class ...
Derived Representation Theory and the Algebraic K
... Kk ∧ → KF ∧ is a weak equivalence in the sense that it induces an isomorphism of homotopy pro-groups. We also will free to use the standard results concerning higher algebraic Ktheory, such as localization sequences, devissage, reduction by resolution, etc., as presented in [26]. We also recall the ...
... Kk ∧ → KF ∧ is a weak equivalence in the sense that it induces an isomorphism of homotopy pro-groups. We also will free to use the standard results concerning higher algebraic Ktheory, such as localization sequences, devissage, reduction by resolution, etc., as presented in [26]. We also recall the ...
dmodules ja
... F is called -torsion if, for all f ∈ F, there exists > 0 such that f = 0. Let -Tors denote the full subcategory of -torsion modules. Theorem (Cox). (1) The category -Mod is equivalent to the quotient category S-GrMod/-Tors. (2) The variety X is a geometric quotient of SpecS\Var by a s ...
... F is called -torsion if, for all f ∈ F, there exists > 0 such that f = 0. Let -Tors denote the full subcategory of -torsion modules. Theorem (Cox). (1) The category -Mod is equivalent to the quotient category S-GrMod/-Tors. (2) The variety X is a geometric quotient of SpecS\Var by a s ...
On Noether`s Normalization Lemma for projective schemes
... rst half of last century. The intuitive idea, which is going to be founding for the sheme theory, is that some geometric objects may carry some algebraic structures intrisecally connected to their geometric nature. It is the case of, as an example, rings of functions dened over open subsets of a u ...
... rst half of last century. The intuitive idea, which is going to be founding for the sheme theory, is that some geometric objects may carry some algebraic structures intrisecally connected to their geometric nature. It is the case of, as an example, rings of functions dened over open subsets of a u ...
Universal unramified cohomology of cubic fourfolds containing a plane
... Assuming that the discriminant divisor D is smooth, the discriminant double cover S → P2 branched along D is then a K3 surface of degree 2 and the even Clifford e → P2 gives rise to a Brauer class β ∈ Br(S), called algebra of the quadric fibration X the Clifford invariant of X. This invariant does n ...
... Assuming that the discriminant divisor D is smooth, the discriminant double cover S → P2 branched along D is then a K3 surface of degree 2 and the even Clifford e → P2 gives rise to a Brauer class β ∈ Br(S), called algebra of the quadric fibration X the Clifford invariant of X. This invariant does n ...
7.1. Sheaves and sheafification. The first thing we have to do to
... A quasi-coherent sheaf on X is called locally free of rank r if it is locally isomorphic to OX⊕r . Locally free sheaves are the most well-behaved sheaves; they correspond to vector bundles in topology. Any construction and theorem valid for vector spaces can be carried over to the category of locall ...
... A quasi-coherent sheaf on X is called locally free of rank r if it is locally isomorphic to OX⊕r . Locally free sheaves are the most well-behaved sheaves; they correspond to vector bundles in topology. Any construction and theorem valid for vector spaces can be carried over to the category of locall ...
String topology and the based loop space.
... that this is also isomorphic to HH ∗ (kG ) and shows that, when k is a field of characteristic 0, HH ∗ (kG ) admits a BV structure isomorphic to that of string topology. Our main result is a generalization of this family of results, replacing the group ring kG with the chain algebra C∗ ΩM. When M = ...
... that this is also isomorphic to HH ∗ (kG ) and shows that, when k is a field of characteristic 0, HH ∗ (kG ) admits a BV structure isomorphic to that of string topology. Our main result is a generalization of this family of results, replacing the group ring kG with the chain algebra C∗ ΩM. When M = ...
Notes on étale cohomology
... The adic formalism is developed in §1.4, and it is used to define étale cohomology with `-adic coefficients; we discuss the Künneth isomorphism and Poincaré duality with Q` -coefficients, and extend the comparison isomorphism with topological cohomology to the `-adic case. We conclude in §1.5 by ...
... The adic formalism is developed in §1.4, and it is used to define étale cohomology with `-adic coefficients; we discuss the Künneth isomorphism and Poincaré duality with Q` -coefficients, and extend the comparison isomorphism with topological cohomology to the `-adic case. We conclude in §1.5 by ...
Topological Models for Arithmetic William G. Dwyer and Eric M
... support this interpretation [4, 5.1]. Using the fact that both algebraic K-theory and etale K-theory are the homotopy groups of infinite loop spaces and that the map between these spaces is an infinite loop map [4, 4.4], one can extend the above definition to include K0 and K1 . Moreover, one can ex ...
... support this interpretation [4, 5.1]. Using the fact that both algebraic K-theory and etale K-theory are the homotopy groups of infinite loop spaces and that the map between these spaces is an infinite loop map [4, 4.4], one can extend the above definition to include K0 and K1 . Moreover, one can ex ...
Group actions on manifolds - Department of Mathematics, University
... complex structure is a Lie group, provided M is compact. (By contrast, the group of symplectomorphisms of a symplectic manifold Diff(M, ω) is of course infinite-dimensional!) The general setting for this type of problem is explained in detail in Kobayashi’s book on transformation groups. Let M be a ...
... complex structure is a Lie group, provided M is compact. (By contrast, the group of symplectomorphisms of a symplectic manifold Diff(M, ω) is of course infinite-dimensional!) The general setting for this type of problem is explained in detail in Kobayashi’s book on transformation groups. Let M be a ...
A Grothendieck site is a small category C equipped with a
... 1) f ∗ is left adjoint to f∗, and 2) f ∗ preserves finite limits. The left adjoint f ∗ is called the inverse image functor, while f∗ is called the direct image. The inverse image f ∗ is left and right exact in the sense that it preserves all finite limits and colimits. The direct image f∗ is usuall ...
... 1) f ∗ is left adjoint to f∗, and 2) f ∗ preserves finite limits. The left adjoint f ∗ is called the inverse image functor, while f∗ is called the direct image. The inverse image f ∗ is left and right exact in the sense that it preserves all finite limits and colimits. The direct image f∗ is usuall ...
Sheaves of Groups and Rings
... Px be the usual stalk and define ΛP = x∈X Px (this is the usual disjoint union). Given an open set U ⊆ X we say a function t : U −→ ΛP is regular if satisfies the following two conditions (i) t(x) ∈ Px for every x ∈ U . (ii) For each x ∈ U there is an open neighborhood x ∈ V ⊆ U and s ∈ P (V ) such ...
... Px be the usual stalk and define ΛP = x∈X Px (this is the usual disjoint union). Given an open set U ⊆ X we say a function t : U −→ ΛP is regular if satisfies the following two conditions (i) t(x) ∈ Px for every x ∈ U . (ii) For each x ∈ U there is an open neighborhood x ∈ V ⊆ U and s ∈ P (V ) such ...
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR
... Z ⊗ A → Q ⊗ A → Q/Z ⊗ A → 0. But for non-trivial torsion groups the first map is certainly not injective since it is isomorphic to the map A → 0. (3) Let F be a free abelian group and let 0 → A0 → A → A00 → 0 be exact. Then also 0 → A0 ⊗ F → A ⊗ F → A00 ⊗ F → 0 is a short exact sequence. This is imm ...
... Z ⊗ A → Q ⊗ A → Q/Z ⊗ A → 0. But for non-trivial torsion groups the first map is certainly not injective since it is isomorphic to the map A → 0. (3) Let F be a free abelian group and let 0 → A0 → A → A00 → 0 be exact. Then also 0 → A0 ⊗ F → A ⊗ F → A00 ⊗ F → 0 is a short exact sequence. This is imm ...
Math 8211 Homework 1 PJW
... We say that a functor F : C → D is an isomorphism of categories if there is a functor G : D → C so that GF = 1C and F G = 1D . We say that a functor F : C → D is an equivalence of categories if there is a functor G : D → C so that GF is naturally isomorphic to the identity functor 1C and F G is natu ...
... We say that a functor F : C → D is an isomorphism of categories if there is a functor G : D → C so that GF = 1C and F G = 1D . We say that a functor F : C → D is an equivalence of categories if there is a functor G : D → C so that GF is naturally isomorphic to the identity functor 1C and F G is natu ...