Fundamental domains of cluster categories inside module categories

CAPPA, JUAN ÁNGEL; PLATZECK, MARÍA INÉS; REITEN, IDUN

JOURNAL OF ALGEBRA

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2012 vol. 2012 p. 234 - 249

0021-8693

Abstract. Let H be a finite dimensional hereditary algebra over an algebraically closed field, and let CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. field, and let CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. field, and let CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. Let H be a finite dimensional hereditary algebra over an algebraically closed field, and let CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category mod �� of finitely generated modules over a suitable tilted algebra ��. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. quiver of) an arbitrary cluster-tilted algebra. . Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.