Cluster structures for 2-Calabi-Yau categories and unipotent groups

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised, especially Chapter III replaces the old Chapter III and I

Tame concealed algebras and cluster quivers of minimal infinite type

In this paper we explain how and why the list of Happel-Vossieck of tame concealed algebras is closely related to the list of A. Seven of minimal infinite cluster quivers.Comment: 16 pages, new version with an additional section on cluster-tilted algebras of minimal infinite typ

The amalgamated duplication of a ring along a multiplicative-canonical ideal

After recalling briefly the main properties of the amalgamated duplication of a ring $R$ along an ideal $I$, denoted by R\JoinI, we restrict our attention to the study of the properties of R\JoinI, when $I$ is a multiplicative canonical ideal of $R$ \cite{hhp}. In particular, we study when every regular fractional ideal of $R\Join I$ is divisorial

Bounded derived categories of very simple manifolds

An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical terms. This is a partial generalization of an impressive result due to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2) and many typos are correcte

Boundary manifolds of projective hypersurfaces

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.Comment: 31 pages; accepted for publication in Advances in Mathematic

Spike-based coupling between single neurons and populations across rat sensory cortices, perirhinal cortex, and hippocampus

Cortical computations require coordination of neuronal activity within and across multiple areas. We characterized spiking relationships within and between areas by quantifying coupling of single neurons to population firing patterns. Single-neuron population coupling (SNPC) was investigated using ensemble recordings from hippocampal CA1 region and somatosensory, visual, and perirhinal cortices. Within-area coupling was heterogeneous across structures, with area CA1 showing higher levels than neocortical regions. In contrast to known anatomical connectivity, between-area coupling showed strong firing coherence of sensory neocortices with CA1, but less with perirhinal cortex. Cells in sensory neocortices and CA1 showed positive correlations between within- and between-area coupling; these were weaker for perirhinal cortex. All four areas harbored broadcasting cells, connecting to multiple external areas, which was uncorrelated to within-area coupling strength. When examining correlations between SNPC and spatial coding, we found that, if such correlations were significant, they were negative. This result was consistent with an overall preservation of SNPC across different brain states, suggesting a strong dependence on intrinsic network connectivity. Overall, SNPC offers an important window on cell-to-population synchronization in multi-area networks. Instead of pointing to specific information-coding functions, our results indicate a primary function of SNPC in dynamically organizing communication in systems composed of multiple, interconnected areas

Neural correlates of object identity and reward outcome in the sensory cortical-hippocampal hierarchy:coding of motivational information in perirhinal cortex

Neural circuits support behavioral adaptations by integrating sensory and motor information with reward and error-driven learning signals, but it remains poorly understood how these signals are distributed across different levels of the corticohippocampal hierarchy. We trained rats on a multisensory object-recognition task and compared visual and tactile responses of simultaneously recorded neuronal ensembles in somatosensory cortex, secondary visual cortex, perirhinal cortex, and hippocampus. The sensory regions primarily represented unisensory information, whereas hippocampus was modulated by both vision and touch. Surprisingly, the sensory cortices and the hippocampus coded object-specific information, whereas the perirhinal cortex did not. Instead, perirhinal cortical neurons signaled trial outcome upon reward-based feedback. A majority of outcome-related perirhinal cells responded to a negative outcome (reward omission), whereas a minority of other cells coded positive outcome (reward delivery). Our results highlight a distributed neural coding of multisensory variables in the cortico-hippocampal hierarchy. Notably, the perirhinal cortex emerges as a crucial region for conveying motivational outcomes, whereas distinct functions related to object identity are observed in the sensory cortices and hippocampus

Tilted algebras and short chains of modules

We provide an affirmative answer for the question raised almost twenty years ago concerning the characterization of tilted artin algebras by the existence of a sincere finitely generated module which is not the middle of a short chain

Cycle-finite module categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited