Propriedades cohomológicas de álgebras de incidência de posets

Abstract

In the context of a finite poset, we present an algorithmic construction, called \textit{cohomological graph}.The vertices, edges, and cycles of this graph play a crucial role in determining the projective resolution of the simple modules associated with the incidence algebra of the poset.The works of Cibils [C89] and Gerstenhaber and Schack [GS83] have established that simplicial cohomology corresponds to the Hochschild cohomology of incidence algebras.The latter can be determined through the $\Ext$ functor. Consequently, we deduce that the $i$-cycles, specific cycles within the cohomological graph, play a pivotal role in determining the rank of Hochschild cohomology groups associated with the incidence algebra. These Hochschild cohomology groups are organized into classes of equivalence based on cycles within the cohomological graph. This method provides algorithms for calculating simplicial cohomology and singular cohomology of finite $T_0$ topological spaces. Expanding upon the investigations of Marcos and Moreira [MM21] and De La Peña and Saorin [DS01], the cohomological graph provides a criterion for the simply connected nature of the incidence algebra. Additionally, the cohomological graph offers a method for calculating the Euler characteristic and Möbius function of the poset. Through the extraction of information from $i$-cycles to the poset, we identify certain subposets associated with the cohomology groups of the algebra. This identification leads to the development of two reduction methods, termed the \textit{minimal cycles method}. In essence, certain subposet can be removed without altering the Hochschild cohomology of the incidence algebra. Our methods introduce reductions that distinguish themselves from other established techniques [S66], [B10], [O99]. Furthermore, the integration of our methods with existing approaches yields more significant reductions.