31,583 research outputs found
GL-equivariant modules over polynomial rings in infinitely many variables
Consider the polynomial ring in countably infinitely many variables over a
field of characteristic zero, together with its natural action of the infinite
general linear group G. We study the algebraic and homological properties of
finitely generated modules over this ring that are equipped with a compatible
G-action. We define and prove finiteness properties for analogues of Hilbert
series, systems of parameters, depth, local cohomology, Koszul duality, and
regularity. We also show that this category is built out of a simpler, more
combinatorial, quiver category which we describe explicitly.
Our work is motivated by recent papers in the literature which study
finiteness properties of infinite polynomial rings equipped with group actions.
(For example, the paper by Church, Ellenberg and Farb on the category of
FI-modules, which is equivalent to our category.) Along the way, we see several
connections with the character polynomials from the representation theory of
the symmetric groups. Several examples are given to illustrate that the
invariants we introduce are explicit and computable.Comment: 59 pages, uses ytableau.sty; v2: expanded details in many proofs
especially in Sections 2 and 4, Section 6 substantially expanded, added
references; v3: corrected typos and Remark 4.3.3 from published versio
From divisibility and factorization to Iwasawa theory of Zp-extensions
This thesis covers the factorization properties of number fields, and presents the structures necessary for understanding a proof on Iwasawa's theorem. The first three chapters aim to construct a ring of integers for arbitrary number fields, and prove that such a ring exists.
We prove that our ring of integers is a Dedekind ring, giving us unique factorization on the set of prime ideals. We prove that there exists an isomorphism between principal and factorial divisors and ideals, define an equivalence relation on the set of all divisors, and show that the equivalence classes form the ideal class group. The class number of a field is defined as the order of the ideal class group. We define ramification of primes, and the invariants related to a prime P called the ramification index, inertia degree and decomposition number.
We expand on the Galois theory of finite extensions, by introducing a topology on an infinite algebraic Galois extension, and a Galois correspondence between closed subgroups and intermediate fields. We show how to define the decomposition- and inertia group in the infinite case. The maximal unramified field extension, the Hilbert class field, whose Galois group is isomorphic to the ideal class group, is introduced.
We introduce a p-adic metric on the ring of integers with the help of valuations, and construct the p-adic integers as a completion with regards to the metric. We prove some structure results for this ring. The lambda-modules are constructed as a limit of modules over group rings, where the group rings are generated by the p-adic integers, and a suitable multiplicative cyclic group.
The final result is a proof of Iwasawa’s theorem as found in Washington, Introduction to Cyclotomic fields. We view the Galois group of the p-adic extension as a lambda-module, and from the structure theorems of lambda-modules, we prove results that carry on to the galois groups of the intermediate fields, culminating in a formula for the exact power of p, that divides the class number of the n-th intermediate field
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