Linearly Compact Research Articles

Linear Compactness and Combinatorial Bialgebras

We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose equivalence classes generate linearly compact bialgebras under shifted shuffling and deconcatenation. We also extend some of the theory of combinatorial Hopf algebras to bialgebras that are not connected or of finite graded dimension. Finally, we discuss several examples of quasi-symmetric functions, not necessarily of bounded degree, that may be constructed via terminal properties of combinatorial bialgebras.

Linearly compact modules of continuous mappings

A criterion for the linear compactness of submodules of certain linearly topologized modules of continuous mappings is established.

AB-5* and Linear Compactness

It is well known that there is a close relation between linear compactness and the Grothendieck condition AB-5*. In fact, for any cocomplete abelian category these conditions are equivalent to the exactness of inverse limits. However, the module categories are not selfdual and therefore Ž . these conditions are, in general, not equivalent for a given fixed module. In this paper, we emphasize the difference between these two important properties and show that in general, a ‘‘proper’’ condition AB-5* on a module M, i.e., M has AB-5*, but it is not linearly compact, is a very strong requirement on M.

Quasi-Duality, Linear Compactness and Morita Duality for Power Series Rings

AbstractAS a generalization of Morita duality, Kraemer introduced the notion of quasi-duality and showed that each left linearly compact ring has a quasi-duality. Let R be an associative ring with identity and R[[x]] the power series ring. We prove that (1) R[[x]] has a quasi-duality if and only if R has a quasi-duality; (2) R[[x]] is left linearly compact if and only if R is left linearly compact and left noetherian; and (3) R[[x]] has a Morita duality if and only if R is left noetherian and has a Morita duality induced by a bimodule RUS such that S is right noetherian.

The Topological Completion and the Linearly Compact Hull of an Abelian l -Group

The Topological Completion and the Linearly Compact Hull of an Abelian <i>l</i> -Group

Determination of quadratic extensions of linearly compact fields by norm groups

It is shown that quadratic extensions of a field not of characteristic two, which is linearly compact at a valuation, are determined by their groups of norms, provided the residue field has a unique quadratic extension and is perfect if of characteristic two. It is indicated that Henselian can replace linearly compact in some cases. Necessity of the condition on the residue field is shown. 1. In this brief paper we shall apply the techniques and results of the paper Quadratic extensions of linearly compact fields by Ron Brown and myself (referred to below as [BW]) to prove the following result: THEOREM 1. Let F be a field of characteristic char(F)$2. Let v be a (nonarchimedean) valuation on F with arbitrary value group rF and residue field kF; assume only that kF is perfect if char(kF)=2. Suppose that F is linearly compact at v and that kF has a unique quadratic extension. Then for K1 and K2 quadratic extensions of F, K1 K2 if and only if N1K1 = N2K2. (Here Ni denotes the norm map Ki-*F.) For definition and properties of linear compactness, see [BW] or [Bour]. All the hypotheses of Theorem 1 are satisfied by any classical local field of characteristic not two. Indeed Theorem 1 is a generalization of a special case of the local class field theorem which says that an abelian extension of a local field is determined by its group of norms. The conclusion of Theorem 1 is equivalent to the assertion that a binary quadratic form over F is determined up to equivalence by the elements of F which it represents. A straightforward application of the Global Squares Theorem extends this result to the global case, obtaining the well-known result that over any local or global field of characteristic not two, binary quadratic forms are equivalent if and only if they represent the same elements. Received by the editors July 9, 1971. AMS 1970 subject classifications. Primary 12B10, 12J10, 12B25; Secondary 10C05, 12A25, 12J20.

Duality theory for Grothendieck categories and linearly compact rings

l’he main new definition is that of linear compactness (more general than in the literature): An R-module X is called algebraically linearly compact if for each decreasing family (Xi ; i E I), I directed, of finitely generated submodules Xi of X the intersection nj Si is again finitely generated and the canonical map X lim X/Xi is surjective. A topological R-left module X is called topologically linearly compact if it has a basis of neighborhoods of 0 consisting of submodules X’ such that S/X’ is algebraically linearly compact. A comp!ete topologically coherent R-module is called strict (Jan-Erik Roos suggests the term “proper”) if each topologically coherent closed submodule Iof -\I, such that (X/Y)dis (X/Y with the discrete topology) is coherent, is open. Strictness is a notion of a technical nature which is necessary to insure that the relevant categories of topological modules are Abelian. As a consequence of Theorem A one sees that a category \!I is a locally nocthcrian Grothendieck category i f f it is dual to the category of complete topologically coherent R-left modules over some complete topologically left

Note on Linearly Compact Abelian Groups

By a group A is meant throughout an additively written abelian group. A is said to have linear topology if there is a system of subgroups Ui (i∈I) of A such that, for a ∈ A, the cosets a+Ui(i∈I) form a fundamental system of neighborhoods of a. The group operations are continuous in any linear topology; the topologies are always assumed to be Hausdorff, that is, ∩iUi = 0. A linearly compact group is a group A with a linear topology such that if aj+Aj (j∈I) is a system of cosets modulo closed subgroups Aj with the finite intersection property (i.e. any finite number of aj+Aj have a non-void intersection), then the intersection ∩j(aj+Aj) of all of them is not empty.

Some remarks on 𝜈-transitive rings and linear compactness

Johnson in [3] has introduced the concept of a v-transitive ring which generalizes the notion of a dense ring of linear transformations. We give necessary and sufficient conditions that an abstract ring be isomorphic to a v-transitive ring which contains finite-valued linear transformations. The condition (2) used here is a modification of one used by Baer [I ] in his characterization of the endomorphism ring of a primary Abelian operator group. This condition is also related to the linear compactness of a ring considered as a right module over itself. This enables one to conclude that a primitive ring with minimal ideals which is linearly compact (in any topology in which it is a topological ring) is the ring of all linear transformations of a vector space, and that a primitive Banach algebra is linearly compact only when it is finite dimensional. A ring E(F, A) of linear transformations of the vector space A over the division ring F is called i-transitive if to every set of less than K, elements aj of A, linearly independent over F, and any set of elements bj of A, in one-one correspondence with the aj, there exists a transformation oin E(F, A), such that ajo = bj for all j. Let K be an abstract ring and P an arbitrary subset thereof. The right ideal of all elements k in K which satisfies Pk = 0 shall be called a right annulet. Now let W= W(K) be the class of all right annulets which are cross-cuts of a finite number of maximal right annulets of K. By a W-coset is meant a coset of an ideal in the set W(K). If E(F, A) is any ring of linear transformations and S is a subspace of A, we denote by R(S) the totality of transformations oCE(F, A) satisfying So=0.