Helmut Röhrl

Flat and coherent functors

The main objective of this paper is to characterize flat group valued functors. We obtain the following theorem, announced in [;7: Let X be a small additive category with dual X0 and S an object in [x0, AB], the category of all additive functors from X to the category Ab of Abelian groups. Then S is flat, i.e., the functor S @r : [X, AB] + AB is exact if and only if the fiber X/S of the Yoneda embedding X --+ [x0, AB] over S is filtered from above, or if and only if S is a filtered direct limit of representable functors. There are several other equivalent statements, and it is, mutatis mutandis, enough to aSsume X preadditive. A similar theorem has been obtained by B. Stenstrom in [II]. He proves that a functor is flat if and only if it is a filtered direct limit of projective (instead of representable) functors. For Abelian X the result was obtained by J. Fisher [5J; in this particular case “flat” means “left exact,” and a short proof is possible. Our result is a generalization of the well-known characterization of flat modules by means of generators and relations, and has applications in the study of the exactness of the direct limit functor [7], and in the singular homology theory of sheaves [S]. Using the above characterization of flat functora we show in analogy to the results of S. U. Chase [4] on coherent rings that the category [X0, AB] is locally coherent, i.e., has a family of coherent generators, if and only if

Separated totally convex spaces

In this paper we define for every totally convex space a suitable topology, the radial topology. We prove that a morphism in the category TCsep of separated totally convex spaces is an epimorphism if and only if its image is dense in the radial topology, and that TCsep is the full subcategory of TC generated by its Hausdorff objects. These results remain valid for finitely totally convex spaces when the radial topology is replaced by the distance-radial topology.

Holomorphic fiber bundles over Riemann surfaces

For the purpose of this paper a fiber bundle F—>X over a Riemann surface X is meant to be a fiber bundle in the sense of N. Steenrod [62] where the base space is X, the fiber a complex space, the structure group G a complex Lie group that acts as a complex transformation group on the fiber, and the transition functions g%j{x) are holomorphic mappings into G. Correspondingly, cross-sections are assumed to be holomorphic cross-sections. We shall use freely the notations of [62], Whenever we report about families of fiber bundles we mean holomorphic families of fiber bundles; the basic notations concerning families of fiber bundles can be found in [30 ] and shall also be used freely. Triviality of bundles resp. families of bundles is always supposed to be holomorphic triviality.

On the zeros of polynomials over arbitrary finite dimensional algebras

In [3] it was shown that a polynomial of degree n with coefficients in an associative division algebra, which is d-dimensional over its center, has either infinitely many or at most nd zeros. In this paper we raise the same question for arbitrary m-ary F-algebras A which are d-dimensional over the algebraically closed field F. Our main result states that in the affine space of m-ary algebras of dimension d there is a non-empty Zariski-open set whose elements A have the following property: in the space of polynomial of precise degree n with coefficients in A there is a non-empty Zariski-open set whose elements have precisely nd zeros. It is shown that all simple algebras, all semi-simple associative algebras, all semisimple Jordan algebras (char F≠2), all semi-simple Lie algebras (char F=0), and the generic algebra possess this property.

A theorem on non-associative algebras and its application to differential equations

AbstractFor a given field F, the set of F-algebras (resp. commutative F-algebras) of arity n≥2 and F-dimension m can be identified with the mn+1 (resp. m(m+n−1n)) dimensional F-affine space S of structure coefficients. We show: If F is algebraically closed, then there exists an affine subvariety A of S with A≠S, which is defined over the prime field of F, such that all F-algebras corresponding to the points of S-A posses precisely nm−1 idempotent elements ≠0 and fail to have nil potent elements ≠0. This implies for a system of ordinary differential equations

Convexity theories IV. Klein-Hilbert parts in convex modules

\left( * \right)\dot X_i = D_i \left( {X_l ,..,X_m } \right),i = l,..,m,

Left linear theories — A generalization of module theory

with Di(Xi,...,Xm)∈ℂ[X1,...,Xm] homogeneous polynomials of degree n: If the coefficients of the polynomials Di, i=1,...,m, are algebraically independent over the field of rationals, then (*) possesses precisely nm−1 ray solutions and fails to have a critical point other than the origin.

Convexity theories 0. Foundations

Let R be an associative, commutative, unital ring. By a R-algebra we mean a unital R-module A together with a R-module homomorphism μ: ⊗RnA→A (n≥2). We raise the question whether such an algebra possesses either an idempotent or a nilpotent element. In section 1 an affirmative answer is obtained in case R=k is an algebraically closed field and dimkA<∞, as well as in case R=ℝ, dimℝS<∞, and n≡0(2). Section 2 deals with the case of reduced rings R and R-algebras which are finitely generated and projective as R-modules. In section 3 we show that the “generic” algebra over an integral domain D fails to have nilpotent elements in any integral domain extending its base ring Dn,m, and thus acquires an idempotent element in some integral domain extending Dn,m.