Birge Huisgen-Zimmermann

The geometry of uniserial representations of finite dimensional algebras I

It is shown that, given any finite dimensional, split basic algebra λ = KΓI (where Γ is a quiver and I an admissible ideal in the path algebra KΓ), there is a finite list of affine algebraic varieties, the points of which correspond in a natural fashion to the isomorphism types of uniserial left Λ-modules, and the geometry of which faithfully reflects the constraints met in constructing such modules. A constructive coordinatized access to these varieties is given, as well as to the accompanying natural surjections from the varieties onto families of uniserial modules with fixed composition series. The fibres of these maps are explored, one of the results being a simple algorithm to resolve the isomorphism problem for uniserial modules. Moreover, new invariants measuring the complexity of the uniserial representation theory are derived from the geometric viewpoint. Finally, it is proved that each affine algebraic variety arises as a variety of uniserial modules over a suitable finite dimensional algebra, in a setting where the points are in one-one correspondence with the isomorphism classes of uniserial modules.

The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type

A description is given of those sequences

The geometry of finite dimensional algebras with vanishing radical square

{\Bbb S}= (S(0),S(1),\dots,S(l))

DIRECT PRODUCTS OF MODULES AND THE PURE SEMISIMPLICITY CONJECTURE

of simple modules over a finite dimensional algebra for which there are only finitely many uniserial modules with consecutive composition factors

DIRECT PRODUCTS OF MODULES AND THE PURE SEMISIMPLICITY CONJECTURE. PART II

S(0),\dots,S(l)

Orbit closures and rational surfaces

. Necessary and sufficient conditions for an algebra to permit only a finite number of isomorphism types of uniserial modules are derived. The main tools in this investigation are the affine algebraic varieties parametrizing the uniserial modules with composition series

Geometry of chain complexes and outer automorphisms under derived equivalence

{\Bbb S}

Varieties of uniserial representations IV. Kinship to geometric quotients

.

The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness

Abstract Let Λ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical J vanishes. We determine the irreducible components of the module variety Rep d ( Λ ) for any dimension vector d. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when Λ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops. For any algebra Λ with J 2 = 0 , our criteria for identifying the components of Rep d ( Λ ) permit us to characterize the modules parametrized by the individual irreducible components. Focusing on such a component, we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on the one hand and their stably equivalent hereditary counterparts on the other. The bridge links certain closed subvarieties of Grassmannians parametrizing the modules with fixed top over the two types of algebras. By way of this connection, we transfer results of Kac and Schofield from the hereditary case to algebras of Loewy length 2. Finally, we use the transit of information to show that any algebra of Loewy length 2 which enjoys the dense orbit property in the sense of Chindris, Kinser and Weyman has finite representation type.

Classifying representations by way of Grassmannians

It is shown that, if R is either an Artin algebra or a commutative noetherian domain of Krull dimension 1, then infinite direct products of R-modules resist direct sum decomposition as follows: If is a family of non-isomorphic, finitely generated, indecomposable R-modules, then is not a direct sum of finitely generated modules. The bearing of this direct product condition on the pure semisimplicity problem is discussed.