FILTERED MODULES WITH COEFFICIENTS 1. Introduction Let E
... 2.2. Fontaine’s rings. We wish to recall briefly the definition of some of Fontaine’s rings of periods (cf. [Fon94b], [Ber04]). For F a finite extension of Qp , we set GF = Gal(Q̄p /F ), and set Gp = GQp . Fontaine’s rings are topological Qp -algebras B equipped with an action of Gp such that BGF is ...
... 2.2. Fontaine’s rings. We wish to recall briefly the definition of some of Fontaine’s rings of periods (cf. [Fon94b], [Ber04]). For F a finite extension of Qp , we set GF = Gal(Q̄p /F ), and set Gp = GQp . Fontaine’s rings are topological Qp -algebras B equipped with an action of Gp such that BGF is ...
The Kazhdan-Lusztig polynomial of a matroid
... Having made these caveats, the observed phenomenon of positivity indicates that our KazhdanLusztig basis does hold interest. There are numerous other ways one could have used the KazhdanLusztig polynomials of a matroid as a change of basis matrix, but the corresponding bases do not have positive str ...
... Having made these caveats, the observed phenomenon of positivity indicates that our KazhdanLusztig basis does hold interest. There are numerous other ways one could have used the KazhdanLusztig polynomials of a matroid as a change of basis matrix, but the corresponding bases do not have positive str ...
On different notions of tameness in arithmetic geometry
... Let C̄ be a be a proper, connected and regular curve (i.e. dim C = 1) of finite type over Spec(Z) and let C ⊂ C̄ be a nonempty open subscheme. Every point x ∈ C̄ r C defines a discrete rank one valuation v x on the function field k(C ). One says that an étale covering C 0 → C is tamely ramified alon ...
... Let C̄ be a be a proper, connected and regular curve (i.e. dim C = 1) of finite type over Spec(Z) and let C ⊂ C̄ be a nonempty open subscheme. Every point x ∈ C̄ r C defines a discrete rank one valuation v x on the function field k(C ). One says that an étale covering C 0 → C is tamely ramified alon ...
1 Definability in classes of finite structures
... Most of the families of finite simple groups are uniformly parameter bi-interpretable (even bi-definable), in a natural sense, with finite fields (see Chapter 4 of [52]). Using results of Elwes and Ryten, it follows that the property of being an asymptotic class transfers from the fields to the grou ...
... Most of the families of finite simple groups are uniformly parameter bi-interpretable (even bi-definable), in a natural sense, with finite fields (see Chapter 4 of [52]). Using results of Elwes and Ryten, it follows that the property of being an asymptotic class transfers from the fields to the grou ...
An Introduction to K-theory
... Grothendieck group of finitely generated projective R-modules for a (commutative) ring R if Spec R = X, of topological vector vector bundles over X if X is a finite dimensional C.W. complex, and of coherent, locally free OX -modules if X is a scheme. Without a doubt, a primary goal (if not the prima ...
... Grothendieck group of finitely generated projective R-modules for a (commutative) ring R if Spec R = X, of topological vector vector bundles over X if X is a finite dimensional C.W. complex, and of coherent, locally free OX -modules if X is a scheme. Without a doubt, a primary goal (if not the prima ...
On finite primary rings and their groups of units
... PROOF OF (*). We can assume i k since we already know that Ni is cyclic for i > k. We show that every element of order p in Ni is in Ni+1; this will establish that Ni has a unique subgroup of order p - since by assumption Ni+1 is cyclic. Indeed, let x E Nz and assume that px 0. Then (1+x)p 1+xp ...
... PROOF OF (*). We can assume i k since we already know that Ni is cyclic for i > k. We show that every element of order p in Ni is in Ni+1; this will establish that Ni has a unique subgroup of order p - since by assumption Ni+1 is cyclic. Indeed, let x E Nz and assume that px 0. Then (1+x)p 1+xp ...
Algebraic Groups
... Definition 1.1. A closed subgroup G ⊆ GLn is called an algebraic group or a linear algebraic group. The identity matrix in GLn is denoted by En or E, and the identity element of an arbitrary group G mostly by e or eG . Examples 1.2. We start with some well-known examples of matrix groups. (1) The sp ...
... Definition 1.1. A closed subgroup G ⊆ GLn is called an algebraic group or a linear algebraic group. The identity matrix in GLn is denoted by En or E, and the identity element of an arbitrary group G mostly by e or eG . Examples 1.2. We start with some well-known examples of matrix groups. (1) The sp ...
Solvable Groups
... Solvable Groups Mathematics 581, Fall 2012 In many ways, abstract algebra began with the work of Abel and Galois on the solvability of polynomial equations by radicals. The key idea Galois had was to transform questions about fields and polynomials into questions about finite groups. For the proof t ...
... Solvable Groups Mathematics 581, Fall 2012 In many ways, abstract algebra began with the work of Abel and Galois on the solvability of polynomial equations by radicals. The key idea Galois had was to transform questions about fields and polynomials into questions about finite groups. For the proof t ...
