Finite Corank Research Articles

Presentations for Subrings and Subalgebras of Finite CO-Rank

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

Generating subdirect products

We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence permutable varieties, which generalise previously known results for groups, and which apply to modules, rings, $K$-algebras and loops. For instance, if $C$ is a fiber product of $A$ and $B$ over a common quotient $D$, and if $A$, $B$ and $D$ are finitely presented, then $C$ is finitely generated. For subdirect products of more than two factors we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and $K$-algebras. In particular, let $C$ be a subdirect product of $K$-algebras $A_1,\dots,A_n$ for a Noetherian ring $K$ such that the projection of $C$ onto any $A_i\times A_j$ has finite co-rank in $A_i\times A_j$. Then $C$ is finitely generated (resp. finitely presented) if and only if all $A_i$ are finitely generated (resp. finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties.

Isometries of Grassmann spaces, II

Botelho, Jamison, and Molnár [1], and Gehér and Šemrl [4] have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space H. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set P∞(H) of all projections of infinite rank and infinite corank when H is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian P∞(H). However, the same method gives an alternative proof in the case of finite rank projections.

Isometric shifts on [formula omitted

For a linear isometry T :C 0(X)→C 0(Y) of finite corank, there is a cofinite subset Y 1 of Y such that Tf |Y 1 =h·f∘ϕ is a weighted composition operator and X is homeomorphic to a quotient space of Y 1 modulo a finite subset. When X=Y, such a T is called an isometric quasi- n-shift on C 0(X) . In this case, the action of T can be implemented as a shift on a tree-like structure, called a T-tree, in M(X) with exactly n joints. The T-tree is total in M(X) when T is a shift. With these tools, we can analyze the structure of T.

The manifold of finite rank projections in the algebra ℒ(H) of bounded linear operators

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V = ℒ( H). Using the algebraic structure of V, a torsionfree affine connection ∇ (that is invariant under the group of automorphisms of V) is defined on every connected component m of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0 ≤ r ≤ ∞). We prove that m admits a Riemann structure if and only if m consists of projections that have the same finite rank r or the same finite corank, and in that case ∇ is the Levi-Civita and the Kähler connection of m. Moreover, m turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed.

Some linear preserver problems onB(H) concerning rank and corank

As a continuation of the work of linear maps between operator algebras which preserve certain subsets of operators with finite rank, or finite corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a result on the form of linear maps which preserve the left ideals inB(H).

The elementary theory of free pseudo p-adically closed fields of finite corank

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

Free pseudo p-adically closed fields of finite corank

Let R, , . . . . iTC be p-adic closures of a field K. In this paper we investigate the behavior of the ‘field K, = KP’ n n K: n K(c,+ 1, . . . . cr ~,+,?I are automorphisms in the absolute Galois group G(K) of K which are chosen at random and where J?( CT~ + I ) . . . . ge + ,,,) is the fixed field of C e+ 17 ...9 gc+/n in the algebraic closure k of K. More precisely, the compact group G(K)‘+,’ is equipped with a unique normalized Haar measure (with respect to the Krull topology on G(K)). We prove :

Moshe Jarden and Ursel Kiehne. The elementary theory of algebraic fields of finite corank. Inventiones mathematicae, vol. 30 no. 3 (1975), pp. 275–294.

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An analogue of Čebotarev density theorem for fields of finite corank

An analogue of Čebotarev density theorem for fields of finite corank

Intersections of Conjugate Fields of Finite Corank over Hilbertian Fields

Intersections of Conjugate Fields of Finite Corank over Hilbertian Fields

The elementary theory of algebraic fields of finite corank

The elementary theory of algebraic fields of finite corank

Algebraic extensions of finite corank of hilbertian fields

We consider here a hilbertian fieldk and its Galois group (k s/k). For a natural numbere we prove that almost all (σ) ∈ (ks/k)e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ1, …, σe is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldk s (σ) of 〈σ〉, …, σe ink s, which containsk. Then the group (k s/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈ (k/k), τ≠1, of finite order such that [k s (σ):k s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk⊆K⊆k s (σ) such that 1<[k s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group (ks/k)e.