## The Ziegler and Zariski spectra of some domestic string algebras (1999)

Venue: | Theory |

Citations: | 12 - 9 self |

### BibTeX

@ARTICLE{Burke99theziegler,

author = {Kevin Burke and Mike Prest},

title = {The Ziegler and Zariski spectra of some domestic string algebras},

journal = {Theory},

year = {1999},

volume = {5},

pages = {211--234}

}

### OpenURL

### Abstract

It was a conjecture of the second author that the Cantor-Bendixson rank of the Ziegler spectrum of a nite-dimensional algebra is either less than or equal to 2 or is undened. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary nite values greater than 2 are obtained. We give a complete description of the Ziegler and Gabriel-Zariski spectra of the simplest of these algebras. The conjecture has been independently refuted by Schroer [21] who, extending his work [20] on these algebras, computed their Krull-Gabriel dimension. 1 Indecomposable pure-injectives over domestic string algebras 12 Let R be a domestic string algebra over an arbitrary eld k. Modules will generally be left R-modules: the category of these we denote by R-Mod. 1 This work was done while the rst author was supported by EPSRC grant number GR/K19686. Both authors thank the EPSRC for this nancial support. 2 Primary: 16G20; Secondary: 03C60,...

### Citations

64 | Model Theory and Modules
- Prest
- 1988
(Show Context)
Citation Context ...ween CantorBendixson rank and Krull-Gabriel dimension). In the general case, it is a consequence of our descriptions of neighbourhood bases that the isolation condition, see [16], (\condition (^)"=-=; of [10]-=-), holds and so we get that the Krull-Gabriel dimension and Cantor-Bendixson ranks are equal. 2 We need some results from [1] which give the decomposition of certain pure-injective modules associated ... |

46 |
Auslander-Reiten sequences with few middle terms and applications to string algebras
- Butler, Ringel
- 1987
(Show Context)
Citation Context ...the category of abelian groups, having the property that for everysnite-dimensional band module N one has F (N) = 0 and for everysnitedimensional indecomposable string module N , dim k F (N) 1 (see [=-=3]-=- for string and band modules). Such a functor is locally simple in the terminology of [7]. We do not assert that such a functor F necessarily exists (though in all examples that we have looked at it d... |

19 |
Elementary duality of modules
- Herzog
- 1993
(Show Context)
Citation Context ...ect system of left modules M i , so then N 0 = Hom k (M; k). By part (a) all indecomposable summands of the pure-injective hull of M are in the Zieglerclosure of the M i and so, by elementary duality =-=[6-=-], N is in the Ziegler-closure of the N i . 2 So any innite set of modules of the form m n (nsxed) has the innite string module 1 n \in its Ziegler-closure", meaning that any indecomposable s... |

17 | Maps between representations of zero-relation algebras - Crawley-Boevey - 1989 |

15 | The spectrum of a module category
- Krause
- 2001
(Show Context)
Citation Context ...gebra and let N 2 Zg R be an indecomposable summand of the inverse limit of a set X of points of Zg R which are Homduals of left R-modules. Then N lies in the Ziegler-closure of X. Proof. For (a) see =-=[8]-=- or, in a general setting, [19]. Part (b) seems to be folklore - brie y, for, say, asnite-dimensional k-algebra, if N 0 is the inverse limit of an inverse system of modules N i each of which has the f... |

4 | Rings of de scalars and biendomorphism rings - Burke, Prest - 1997 |

3 |
Pure{injective hulls of expanding string modules
- Burke
- 1997
(Show Context)
Citation Context ...summand of the pure-injective hull of M(w). But, since R is domestic and hence every one-sided innite word is either expanding or contracting [17], such summands are completely described by [17] and [=-=1-=-] and are (apart from prufers and adics associated to bands, which we ignore since FN 6= 0) just the indecomposable pure-injectives described in [17]. Theorem 1.6 Let R be a domestic string algebra. S... |

2 |
In modules in the representation theory of algebras
- CrawleyBoevey
- 1998
(Show Context)
Citation Context ... a basis of open subsets for the Zariski topology on this set. In practice (see later), we specify such sets by a variety of means. For more on these topological spaces see, for instance, [16], [11], =-=[5]-=-. We also use D to denote the duality between the functor categories (mod-R; Ab) fp and (R-mod; Ab) fp . At this point we should also say something about the proofs in this section. Our original proof... |

2 |
Homological transfer from presented to in modules
- Lenzing
- 1983
(Show Context)
Citation Context ...snite-dimensional (string) modules N each of which is such that F (N ) 6= 0. Proof. We know that N is a direct summand of a direct product Q N of indecomposablesnite-dimensional modules (e.g. [9]). Denote the inclusion of N in Q N by j and the projections of Q N to its factors N by . Choose a 2 FN , a 6= 0. Set a = F ( j)(a) 2 FN : thus, since F commutes with products, a ... |

1 |
Locally simple objects
- Herzog
- 1998
(Show Context)
Citation Context ...module N one has F (N) = 0 and for everysnitedimensional indecomposable string module N , dim k F (N) 1 (see [3] for string and band modules). Such a functor is locally simple in the terminology of [=-=7]-=-. We do not assert that such a functor F necessarily exists (though in all examples that we have looked at it does). We make the further assumption that F is actually a subfunctor of the forgetful fun... |