Direct sum

Abstract: This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Abstract: It is shown that an arbitrary symmetric tensor ψ ab (or ψ ab ) of any weight can be covariantly decomposed on a Riemannian manifold (M,g) into a unique sum of transverse‐traceless, longitudinal, and pure trace parts. The summands involve only linear operators and are mutually orthogonal in the global scalar product on (M,g). Each summand transforms separately into itself if the decomposition is carried out properly in a conformally related space (M,g). The decomposition is therefore determined by a conformal equivalence class of Riemannian manifolds. This property makes the decomposition ideally suited to the initial‐value problem of general relativity, which becomes, as a result, a well‐defined system of elliptic equations. Three of the four initial‐value equations are linear and determine the decomposition of a symmetric tensor. The fourth equation is quasilinear and determines the conformal factor. The decomposition applied to the space of symmetric tensors on (M,g) can be written in terms of a direct sum of orthogonal linear spaces and gives a framework for treating and classifying deformations of Riemannian manifolds pertinent to the theory of gravitation and to pure geometry.

Abstract: A. Linear Ultrametric Analysis and Valuation Theory.- 1. Norms and Valuations.- 1.1. Semi-normed and normed groups.- 1.1.1. Ultrametric functions.- 1.1.2. Filtrations.- 1.1.3. Semi-normed and normed groups. Ultrametric topology.- 1.1.4. Distance.- 1.1.5. Strictly closed subgroups.- 1.1.6. Quotient groups.- 1.1.7. Completions.- 1.1.8. Convergent series.- 1.1.9. Strict homomorphisms and completions.- 1.2. Semi-normed and normed rings.- 1.2.1. Semi-normed and normed rings.- 1.2.2. Power-multiplicative and multiplicative elements.- 1.2.3. The category and the functor A ? A~.- 1.2.4. Topologically nilpotent elements and complete normed rings.- 1.2.5. Power-bounded elements.- 1.3. Power-multiplicative semi-norms.- 1.3.1. Definition and elementary properties.- 1.3.2. Smoothing procedures for semi-norms.- 1.3.3. Standard examples of norms and semi-norms.- 1.4. Strictly convergent power series.- 1.4.1. Definition and structure of A?X?.- 1.4.2. Structure of A?X??.- 1.4.3. Bounded homomorphisms of A?X?.- 1.5. Non-Archimedean valuations.- 1.5.1. Valued rings.- 1.5.2. Examples.- 1.5.3. The Gauss-Lemma.- 1.5.4. Spectral value of monic polynomials.- 1.5.5. Formal power series in countably many indeterminates.- 1.6. Discrete valuation rings.- 1.6.1. Definition. Elementary properties.- 1.6.2. The example of F. K. Schmidt.- 1.7. Bald and discrete B-rings.- 1.7.1. B-rings.- 1.7.2. Bald rings.- 1.8. Quasi-Noetherian B-rings.- 1.8.1. Definition and characterization.- 1.8.2. Construction of quasi-Noetherian rings.- 2. Normed modules and normed vector spaces.- 2.1. Normed and faithfully normed modules.- 2.1.1. Definition.- 2.1.2. Submodules and quotient modules.- 2.1.3. Modules of fractions. Completions.- 2.1.4. Ramification index.- 2.1.5. Direct sum. Bounded and restricted direct product.- 2.1.6. The module L(L, M) of bounded A-linear maps.- 2.1.7. Complete tensor products.- 2.1.8. Continuity and boundedness.- 2.1.9. Density condition.- 2.1.10. The functor M ? M~. Residue degree.- 2.2. Examples of normed and faithfully normed A-modules.- 2.2.1. The module An.- 2.2.2. The modules A(I)A(?)c(A) and b(A).- 2.2.3. Structure of L(cI(A), M).- 2.2.4. The ring A [Y1, Y2, ...] of formal power series.- 2.2.5. b-separable modules.- 2.2.6. The functor M ? T(M).- 2.3. Weakly cartesian spaces.- 2.3.1. Elementary properties of normed spaces.- 2.3.2. Weakly cartesian spaces.- 2.3.3. Properties of weakly cartesian spaces.- 2.3.4. Weakly cartesian spaces and tame modules.- 2.4. Cartesian spaces.- 2.4.1. Cartesian spaces of finite dimension.- 2.4.2. Finite-dimensional cartesian spaces and strictly closed subspaces.- 2.4.3. Cartesian spaces of arbitrary dimension.- 2.4.4. Normed vector spaces over a spherically complete field.- 2.5. Strictly cartesian spaces.- 2.5.1. Finite-dimensional strictly cartesian spaces.- 2.5.2. Strictly cartesian spaces of arbitrary dimension.- 2.6. Weakly cartesian spaces of countable dimension.- 2.6.1. Weakly cartesian bases.- 2.6.2. Existence of weakly cartesian bases. Fundamental theorem.- 2.7. Normed vector spaces of countable type. The Lifting Theorem.- 2.7.1. Spaces of countable type.- 2.7.2. Schauder bases. Orthogonality and orthonormality.- 2.7.3. The Lifting Theorem.- 2.7.4. Proof of the Lifting Theorem.- 2.7.5. Applications.- 2.8. Banach spaces.- 2.8.1. Definition. Fundamental theorem.- 2.8.2. Banach spaces of countable type.- 3. Extensions of norms and valuations.- 3.1. Normed and faithfully normed algebras.- 3.1.1. A-algebra norms.- 3.1.2. Spectral values and power-multiplicative norms.- 3.1.3. Residue degree and ramification index.- 3.1.4. Dedekind's Lemma and a Finiteness Lemma.- 3.1.5. Power-multiplicative and faithful A-algebra norms.- 3.2. Algebraic field extensions. Spectral norm and valuations.- 3.2.1. Spectral norm on algebraic field extensions.- 3.2.2. Spectral norm on reduced integral K-algebras.- 3.2.3. Spectral norm and field polynomials.- 3.2.4. Spectral norm and valuations.- 3.3. Classical valuation theory.- 3.3.1. Spectral norm and completions.- 3.3.2. Construction of inequivalent valuations.- 3.3.3. Construction of power-multiplicative algebra norms.- 3.3.4. Hensel's Lemma.- 3.4. Properties of the spectral valuation.- 3.4.1. Continuity of roots.- 3.4.2. Krasner's Lemma.- 3.4.3. Example, p-adic numbers.- 3.5. Weakly stable fields.- 3.5.1. Weakly cartesian fields.- 3.5.2. Weakly stable fields.- 3.5.3. Criterion for weak stability.- 3.5.4. Weak stability and Japaneseness.- 3.6. Stable fields.- 3.6.1. Definition.- 3.6.2. Criteria for stability.- 3.7. Banach algebras.- 3.7.1. Definition and examples.- 3.7.2. Finiteness and completeness of modules over a Banach algebra.- 3.7.3. The category A.- 3.7.4. Finite homomorphisms.- 3.7.5. Continuity of homomorphisms.- 3.8. Function algebras.- 3.8.1. The supremum semi-norm on k-algebras.- 3.8.2. The supremum semi-norm on k-Banach algebras.- 3.8.3. Banach function algebras.- 4 (Appendix to Part A). Tame modules and Japanese rings.- 4.1. Tame modules.- 4.2. A Theorem of Dedekind.- 4.3. Japanese rings. First criterion for Japaneseness.- 4.4. Tameness and Japaneseness.- B. Affinoid algebras.- 5. Strictly convergent power series.- 5.1. Definition and elementary properties of Tn and T?n.- 5.1.1. Description of Tn.- 5.1.2. The Gauss norm is a valuation and T?n is a polynomial ring over k?.- 5.1.3. Going up and down between Tn and T?n.- 5.1.4. Tn as a function algebra.- 5.2. Weierstrass-Ruckert theory for Tn.- 5.2.1. Weierstrass Division Theorem.- 5.2.2. Weierstrass Preparation Theorem.- 5.2.3. Weierstrass polynomials and Weierstrass Finiteness Theorem.- 5.2.4. Generation of distinguished power series.- 5.2.5. Ruckert's theory.- 5.2.6. Applications of Ruckert's theory for Tn.- 5.2.7. Finite Tn-modules.- 5.3. Stability of Q(Tn).- 5.3.1. Weak stability.- 5.3.2. The Stability Theorem. Reductions.- 5.3.3. Stability of k(X) if |k|is divisible.- 5.3.4. Completion of the proof for arbitrary |k|.- 6. Affinoid algebras and Finiteness Theorems.- 6.1. Elementary properties of affinoid algebras.- 6.1.1. The category of k-affinoid algebras.- 6.1.2. Noether normalization.- 6.1.3. Continuity of homomorphisms.- 6.1.4. Examples. Generalized rings of fractions.- 6.1.5. Further examples. Convergent power series on general polydiscs.- 6.2. The spectrum of a k-affinoid algebra and the supremum semi-norm.- 6.2.1. The supremum semi-norm.- 6.2.2. Integral homomorphisms.- 6.2.3. Power-bounded and topologically nilpotent elements.- 6.2.4. Reduced k-affinoid algebras are Banach function algebras.- 6.3. The reduction functor A ? A?.- 6.3.1. Monomorphisms, isometries and epimorphisms.- 6.3.2. Finiteness of homomorphisms.- 6.3.3. Applications to group operations.- 6.3.4. Finiteness of the reduction functor A ? A?.- 6.3.5. Summary.- 6.4. The functor A ? A.- 6.4.1. Finiteness Theorems.- 6.4.2. Epimorphisms and isomorphisms.- 6.4.3. Residue norm and supremum norm. Distinguished k-affinoid algebras and epimorphisms.- C. Rigid analytic geometry.- 7. Local theory of affinoid varieties.- 7.1. Affinoid varieties.- 7.1.1. Max Tn and the unit ball Bn(ka).- 7.1.2. Affinoid sets. Hilbert's Nullstellensatz.- 7.1.3. Closed subspaces of Max Tn.- 7.1.4. Affinoid maps. The category of affinoid varieties.- 7.1.5. The reduction functor.- 7.2. Affinoid subdomains.- 7.2.1. The canonical topology on Sp A.- 7.2.2. The universal property defining affinoid subdomains.- 7.2.3. Examples of open affinoid subdomains.- 7.2.4. Transitivity properties.- 7.2.5. The Openness Theorem.- 7.2.6. Affinoid subdomains and reduction.- 7.3. Immersions of affinoid varieties.- 7.3.1. Ideal-adic topologies.- 7.3.2. Germs of affinoid functions.- 7.3.3. Locally closed immersions.- 7.3.4. Runge immersions.- 7.3.5. Main theorem for locally closed immersions.- 8. ?ech cohomology of affinoid varieties.- 8.1. Cech cohomology with values in a presheaf.- 8.1.1. Cohomology of complexes.- 8.1.2. Cohomology of double complexes.- 8.1.3. ?ech cohomology.- 8.1.4. A Comparison Theorem for Cech cohomology.- 8.2. Tate's Acyclicity Theorem.- 8.2.1. Statement of the theorem.- 8.2.2. Affinoid coverings.- 8.2.3. Proof of the Acyclicity Theorem for Laurent coverings.- 9. Rigid analytic varieties.- 9.1. Grothendieck topologies.- 9.1.1. 6r-topological spaces.- 9.1.2. Enhancing procedures for G-topologies.- 9.1.3. Pasting of (G-topological spaces.- 9.1.4. G-topologies on affinoid varieties.- 9.2. Sheaf theory.- 9.2.1. Presheaves and sheaves on G-topological spaces.- 9.2.2. Sheafification of presheaves.- 9.2.3. Extension of sheaves.- 9.3. Analytic varieties. Definitions and constructions.- 9.3.1. Locally G-ringed spaces and analytic varieties.- 9.3.2. Pasting of analytic varieties.- 9.3.3. Pasting of analytic maps.- 9.3.4. Some basic examples.- 9.3.5. Fibre products.- 9.3.6. Extension of the ground field.- 9.4. Coherent modules.- 9.4.1. -modules.- 9.4.2. Associated modules.- 9.4.3. It-coherent modules.- 9.4.4. Finite morphisms.- 9.5. Closed analytic subvarieties.- 9.5.1. Coherent ideals. The nilradical.- 9.5.2. Analytic subsets.- 9.5.3. Closed immersions of analytic varieties.- 9.6. Separated and proper morphisms.- 9.6.1. Separated morphisms.- 9.6.2. Proper morphisms.- 9.6.3. The Direct Image Theorem and the Theorem on Formal Functions.- 9.7. An application to elliptic curves.- 9.7.1. Families of annuli.- 9.7.2. Affinoid subdomains of the unit disc.- 9.7.3. Tate's elliptic curves.- Glossary of Notations.

Abstract: This article identifies some of the structural properties of the matrix triple $(C,A,B)$ which remain invariant under various transformation groups. The paper begins with a brief account of a recent result which states that the controllable space of $(A,B)$ can be decomposed into a direct sum of singly-generated controllability subspaces, the dimension of each subspace being determined by one of the controllability indices of $(A,B)$. In certain instances the component subspaces of this decomposition can be chosen so that their C-images also decompose system output space in a special way; matrix triples $(C,A,B)$ for which such a decomposition is possible are called prime. If $\bar {\mathcal{C}}$ is an appropriately defined group of system-coordinate and state-feedback transformations acting on prime triples, then the controllability indices of $(A,B)$ determine a complete orbital invariant under $\bar {\mathcal{C}}$.By imbedding $\bar {\mathcal{C}}$ in a richer transformation group $\mathcal{C}^ * $, whi...