Topological and geometric aspects of the Ziegler spectrum

Mike Prest Department of Mathematics University of Manchester Manchester M13 9PL UK

The (right) Ziegler spectrum of a ring R is a topological space which was introduced by Ziegler in his pivotal paper [18] on the model theory of modules. This space Zg_{R} may be defined purely in terms of the category Mod-R of (right) R-modules. The points of the space are the (isomorphism classes of) indecomposable pure-injective (=algebraically compact) R-modules. The topology may be defined in a number of equivalent ways: in terms of the model theory of modules (as Ziegler did); in terms of morphisms in the category of R-modules; in terms of the associated category (R−mod, Ab) of additive functors from the category of finitely presented left R-modules to that of abelian groups.

For the purposes of Ziegler’s paper, the central properties of this space are that its closed sets correspond to complete theories of modules and that isolation of points of the space may be described by properties of certain types (the notion of ”type” is one of the basic concepts of model theory).

Understanding the Ziegler spectrum of a ring may, with good justification, be regarded as the key to understanding most aspects of the model theory of modules over that ring.

If R is an artin algebra then every indecomposable module of finite length is a point of the Ziegler spectrum. So the ubiquity of wild algebras warns us that, in general, we cannot expect anything like an explicit description of the Ziegler spectrum. But we may hope that, at least for tame algebras, we may describe the Ziegler spectrum and, in the limited cases that have been thoroughly investigated so far, this hope has been borne out.

Typically, the Ziegler spectrum has many infinite-length points. Indeed, over an artin algebra R, the finite length indecomposables are precisely the isolated points of the space and hence, if R is not of finite representation type, then, since the space is compact (meaning just that every open cover has a finite subcover: Zg_{R} is seldom Hausdorff), there must be an infinite length point. In fact, over an artin algebra, the subset of Zg_{R} consisting of points of finite length is an open dense subset and so all infinite-length indecomposable pure-injectives are, in the topological sense, limits of finitelength indecomposable modules.

The connection between finite and infinite length modules also goes the other way. Crawley-Boevey describes an indecomposable module as generic if it is an infinite length point of Zg_{R} which is of finite length over its endomorphism ring. He showed, for finite-dimensional algebras over algebraically closed fields, that the existence of generic modules implies the existence of one-parameter families of finite-length points (see [4] for a precise statement). Since one can deduce the existence of generic modules from the existence of infinite-length points satisfying a rather weak finiteness condition, one sees that the infinite-length points of the spectrum are intimately related to families of finite-length points.

All this suggests that, beside the original model-theoretic motivation, there are strong representation-theoretic motivations for investigating the Ziegler spectrum.

The ”weak finiteness condition” referred to above is the possession of what Garavaglia [7] called elementary Krull dimension, equivalently the possession of what is called rank in [18], equivalently m-dimension in [10] (again, equivalently, N ∈ Zg_{R} has these iff the functor N ⊗ − has Krull-Gabriel dimension in the category (R−mod, Ab)). This is closely related to (conjecturally equivalent to) the point having Cantor-Bendixson rank in the space Zg_{R}.

This topological, Cantor-Bendixson, rank, which is defined by inductively throwing away all the isolated points of a space, seems to be the right measure of the topological complexity of the space Zg_{R}. We will discuss the general significance of this rank and we will present some examples where it has been computed ([3],[12],[16], [17]), with the aim of illustrating how useful it is to know the Cantor-Bendixson layering of Zg_{R}.

In the second part of my talk I will discuss another topology on the underlying set of the Ziegler spectrum. This is obtained by taking, for a basis of open sets, the complements of the compact open sets of the Ziegler topology [11], [13]. It may reasonably be called the Gabriel-Zariski topology of R since if we follow Gabriel [6] in formulating the definition of the Zariski topology of a commutative noetherian ring in terms of the category of modules and then apply this definition to the category (R−mod, Ab), we obtain this space (up to the identification of an indecomposable pureinjective module N with the corresponding indecomposable injective functor N ⊗ −).

This space comes equipped with a sheaf of rings, the sheaf of rings of locally definable scalars [14], which generalises the structure sheaf of a commutative noetherian ring. The rings which occur in this sheaf may be defined [2] via model theory (as rings of definable functions), module theory (as biendomorphism rings) or via functor categories (as endomorphism rings of localised functors). The centre of this sheaf is a sheaf of local rings.

We will present some examples of this ringed space and, in particular, we will compare the sheaf of locally definable scalars of a tame hereditary algebra with the sheaves discussed in [1], [8] and [5].

References

[1] D. Baer, W. Geigle and H. Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra, 15 (1987), 425-457.

[2] K. Burke and M. Prest, Rings of definable scalars and biendomorphism rings, pp. 188-201 in ”Model Theory of Groups and Automorphism Groups”, London Math. Soc. Lecture Note Ser., Vol. 244, Cambridge University Press, 1997.

[3] K. Burke and M. Prest, Ziegler spectra of some domestic string algebras, in preparation.

[4] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc., 63 (1991), 241-265.

[5] W. Crawley-Boevey, Regular modules for tame hereditary algebras, Proc. London Math. Soc., 62 (1991), 490-508.

[6] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France, 90 (1962), 323-448.

[7] S. Garavaglia, Dimension and rank in the model theory of modules, University of Michigan, East Lansing, preprint, 1979.

[8] H. Lenzing, Curve singularities arising from the representation theory of tame hereditary algebras, pp. 199-231 in Representation Theory I: Finite Dimensional Algebras, Lecture Notes in Mathematics, Vol. 1177, Springer- Verlag, 1986.

[9] M. Prest, Model theory and representation type of algebras, pp. 219-260 in ”Logic Colloquium ’86”, North-Holland, 1988.

[10] M. Prest. Model Theory and Modules, London Math. Soc. Lecture Notes Ser., Vol. 130. Cambridge University Press, 1988.

[11] M. Prest, Remarks on elementary duality, Ann. Pure Applied Logic, 62 (1993), 183-205.

[12] M. Prest, Ziegler spectra of tame hereditary algebras, J. Algebra, to appear.

[13] M. Prest, The Zariski spectrum of the category of finitely presented modules, University of Manchester, preprint, 1997.

[14] M. Prest, The sheaf of locally definable scalars over a ring, University of Manchester, preprint, 1988.

[15] C. M. Ringel, Some algebraically compact modules I, pp. 419-439 in Abelian Groups and Modules, A. Facchini and C. Menini (Eds.), Kluwer, 1995.

[16] C. M. Ringel, The Ziegler spectrum of a tame hereditary algebra, Colloq. Math., 76 (1998), 105-115.

[17] J. Schroer, On the Krull-Gabriel dimension of an algebra, Universität Bielefeld, preprint, 1998.

[18] M. Ziegler, Model theory of modules, Ann. Pure Applied Logic, 26 (1984), 149-213.