A fresh perspective on canonical extensions for bounded lattices

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of thetheory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploscica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphismsthe Ploscica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Ploscica's paper. This leads to a construction of canonical extension valid for all bounded lattices,which is shown to be functorial, with the property that the canonical extension functor decomposes asthe composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors

A geometric model of tube categories

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A-double-infinity.Comment: 15 pages, 7 figures. Discussion of maximal rigid objects and triangulations at end of Section 3. Minor correction

Iwasawa theory and the Eisenstein ideal

In this paper, we relate three objects. The first is a particular value of a cup product in the cohomology of the Galois group of the maximal unramified outside p extension of a cyclotomic field containing the pth roots of unity. The second is an Iwasawa module over a nonabelian extension of the rationals, a subquotient of the maximal pro-p abelian unramified completely split at p extension of a certain pro-p Kummer extension of a cyclotomic field that contains all p-power roots of unity. The third is the quotient of an Eisenstein ideal in an ordinary Hecke algebra of Hida by the square of the Eisenstein ideal and the element given by the pth Hecke operator minus one. For the relationship between the latter two objects, we employ the work of Ohta, in which he considered a certain Galois action on an inverse limit of cohomology groups to reestablish the Main Conjecture (for p at least 5) in the spirit of the Mazur-Wiles proof. For the relationship between the former two objects, we construct an analogue to the global reciprocity map for extensions with restricted ramification. These relationships, and a computation in the Hecke algebra, allow us to prove an earlier conjecture of McCallum and the author on the surjectivity of a pairing formed from the cup product for p < 1000. We give one other application, determining the structure of Selmer groups of the modular representation considered by Ohta modulo the Eisenstein ideal.Comment: 37 page

Cluster combinatorics of d-cluster categories

We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly for $d-$cluster tilting objects, we prove the equivalences between $d-$cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of $d+1$ triangles of $d-$cluster tilting objects in [IY], we prove that any almost complete $d-$cluster tilting object has exactly $d+1$ complements, compute the extension groups between these complements, and study the middle terms of these $d+1$ triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established in [BMRRT] to $d-$cluster categories. They are applied to the Fomin-Reading's generalized cluster complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2], and to that of infinite root systems [Zh3].Comment: correted many typos according to the referee's comments, final version to appear in J. Algebr

Tilting chains of negative curves on rational surfaces

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of (-2)-curves, we obtain an equivalence with modules over a well known algebra.Comment: 13 page