We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projectiveinjective object is Geiss-Leclerc-Schröer's category Sub Q k , which categorifies the coordinate ring of the big cell in this Grassmannian.
We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the Riedtmann-Zwara theorem for module varieties. Applications to tilting complexes are given, in particular that any twoterm tilting complex is determined by its graded module structure.
In an earlier paper we defined a relation ≤Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space [Formula: see text] for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤Δ is always a partial order.
Abstract. Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category D b (A) of finitely generated Amodules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an. In this paper we define study the type of singularity at every degeneration in the bounded derived categrory.
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