A. Case Structures
... To select a case, type the values in the case selector identifier or use the Labeling tool to edit the values. Specify a single value or lists and ranges of values to select the case. For lists, use commas to separate values. Specify a range as 10..20, meaning all numbers from 10 to 20 inclusively. ...
... To select a case, type the values in the case selector identifier or use the Labeling tool to edit the values. Specify a single value or lists and ranges of values to select the case. For lists, use commas to separate values. Specify a range as 10..20, meaning all numbers from 10 to 20 inclusively. ...
2 Lecture 2: Spaces of valuations
... Letting pv be the support of v, naturally v induces a well-defined valuation ve on the residue field κ(pv ) at pv . Let Γv := ve(κ(pv )× ). Write Rv for the valuation ring of ve in κ(pv ), so that Γv = κ(pv )× /Rv× . It is not at all true that, in general, v(A) ⊂ Γ≤1 ∪ {0}, as shown by the following ...
... Letting pv be the support of v, naturally v induces a well-defined valuation ve on the residue field κ(pv ) at pv . Let Γv := ve(κ(pv )× ). Write Rv for the valuation ring of ve in κ(pv ), so that Γv = κ(pv )× /Rv× . It is not at all true that, in general, v(A) ⊂ Γ≤1 ∪ {0}, as shown by the following ...
CHAP03 Examples of Groups
... the groups are isomorphic. So ℤ12# is isomorphic to the mattress group but neither of these is isomorphic to ℤ10#. (We’ll define isomorphism a little more formally later.) So there are at least two, essentially different, groups of order 4. In fact, as we’ll see later, these are the only two. There ...
... the groups are isomorphic. So ℤ12# is isomorphic to the mattress group but neither of these is isomorphic to ℤ10#. (We’ll define isomorphism a little more formally later.) So there are at least two, essentially different, groups of order 4. In fact, as we’ll see later, these are the only two. There ...
On derivatives of polynomials over finite fields through integration
... k ranges from 1 to r, taking k = 2v−1 implies that r/2v−1 is also even. It means that 2v | r and therefore r is of the form 2v . ...
... k ranges from 1 to r, taking k = 2v−1 implies that r/2v−1 is also even. It means that 2v | r and therefore r is of the form 2v . ...
Infinite Galois Theory
... topology, which is the most natural nontrivial topology for a Galois group, on the infinite Galois groups. After doing that, we will get a result, similar to the one above, that is for the infinite Galois extension. Given an Galois extension E/F , we are going to use T to denote the collection of al ...
... topology, which is the most natural nontrivial topology for a Galois group, on the infinite Galois groups. After doing that, we will get a result, similar to the one above, that is for the infinite Galois extension. Given an Galois extension E/F , we are going to use T to denote the collection of al ...
Structured ring spectra and displays
... groups by a large Hopf algebroid. In section 4 we study the deformation theory of nilpotent displays in matrix form over a ring R and use this to give a criterion for these to satisfy the universal deformation property. Specifically, a display in matrix form determines a map to projective space Ph−1 ...
... groups by a large Hopf algebroid. In section 4 we study the deformation theory of nilpotent displays in matrix form over a ring R and use this to give a criterion for these to satisfy the universal deformation property. Specifically, a display in matrix form determines a map to projective space Ph−1 ...
Full Groups and Orbit Equivalence in Cantor Dynamics
... As a corollary, we get that every transitive system (X, G) with infinite orbits has many involutions. Furthermore, the clopen supports of the involutions from [G] generate CO(X). Theorem 2.5. Let (X1 , G1 ) and (X2 , G2 ) be Cantor dynamical systems such that each Gi -orbit contains at least three p ...
... As a corollary, we get that every transitive system (X, G) with infinite orbits has many involutions. Furthermore, the clopen supports of the involutions from [G] generate CO(X). Theorem 2.5. Let (X1 , G1 ) and (X2 , G2 ) be Cantor dynamical systems such that each Gi -orbit contains at least three p ...
Morita equivalence for regular algebras
... Corollary 2.2 The biequivalence of proposition 2.1 restricts to a biequivalence between regular equivalences and invertible regular bimodules. In other words, two regular algebras are Morita equivalent (= there exists a regular equivalence B-modreg -> A-modreg) iff they are equivalent in the b ...
... Corollary 2.2 The biequivalence of proposition 2.1 restricts to a biequivalence between regular equivalences and invertible regular bimodules. In other words, two regular algebras are Morita equivalent (= there exists a regular equivalence B-modreg -> A-modreg) iff they are equivalent in the b ...
pdf
... for all r > 0. (Here Γ denotes the group of units in Zp congruent to one modulo p, Γr denotes the kernel of the reduction of Γ modulo pr , and Λ denotes the completed group ring Zp [[Γ]] = lim←r− Zp [Γ/Γr ].) Hida proves this through a series of group cohomological calculations combined with his the ...
... for all r > 0. (Here Γ denotes the group of units in Zp congruent to one modulo p, Γr denotes the kernel of the reduction of Γ modulo pr , and Λ denotes the completed group ring Zp [[Γ]] = lim←r− Zp [Γ/Γr ].) Hida proves this through a series of group cohomological calculations combined with his the ...
Higher regulators and values of L
... Q -+@ff~-J(X,Q(i)). The corresponding constructions are recalled in Sec. 2. Let l-l~-J(X, Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists ...
... Q -+@ff~-J(X,Q(i)). The corresponding constructions are recalled in Sec. 2. Let l-l~-J(X, Q (~))cKj(X)~Q be the eigenspace of weight i relative to the Adams operator [2]; then ch~ defines a r e g u l a t o r - a morphism r~:HJs~(X,Q(i))-+I-I~(X,Q(i)). [It is thought that for any schemes there exists ...
The congruence subgroup problem
... A k-group T is a torus if it is connected and can be conjugated into diagonal matrices in GL(n, C). It is again a basic result that if G is a reductive k-group, G contains a central k-torus T such that G/T has no connected abelian normal subgroups. Information on T and G/T separately can be pieced t ...
... A k-group T is a torus if it is connected and can be conjugated into diagonal matrices in GL(n, C). It is again a basic result that if G is a reductive k-group, G contains a central k-torus T such that G/T has no connected abelian normal subgroups. Information on T and G/T separately can be pieced t ...
− CA Π and Order Types of Countable Ordered Groups 1
... Π11 − CA0 consists of the axioms of ACA0 plus the scheme of Π11 comprehension. An important property of Π11 − CA0 is that it is strong enough to define sets by the transfinite recursion of an arithmetic formula over any well ordered set. In the next section we will give formal definitions for a well ...
... Π11 − CA0 consists of the axioms of ACA0 plus the scheme of Π11 comprehension. An important property of Π11 − CA0 is that it is strong enough to define sets by the transfinite recursion of an arithmetic formula over any well ordered set. In the next section we will give formal definitions for a well ...
LOCALLY COMPACT FIELDS Contents 5. Locally compact fields 1
... (i) =⇒ (v): Let (K, | |) be a discretely valued locally compact field. First suppose that K has characteristic 0. Thus Q ,→ K and the norm on K restricts to a nonArchimedean norm on Q. But we have classified all such and know that they are (up to equivalence, which is harmless here) all of the form ...
... (i) =⇒ (v): Let (K, | |) be a discretely valued locally compact field. First suppose that K has characteristic 0. Thus Q ,→ K and the norm on K restricts to a nonArchimedean norm on Q. But we have classified all such and know that they are (up to equivalence, which is harmless here) all of the form ...
