超平面排列的面半群代数
The Face Semigroup Algebra of a Hyperplane Arrangement
摘要：
本文研究由超平面排列的面所张成的代数。计算了该代数的带关系的代数，证明了该代数是一个Koszul代数。证明了该代数只依赖于超平面排列的交格。构造了该代数的一个完整的本原正交幂等元系统，并确定了其它代数结构，包括：投射不可分解模的描述; Cartan不变量;单模的投射分解;霍赫希尔德同调和上同调;以及Koszul对偶代数.引入了偏序集上的一种新的上同调构造，并证明了将该构造应用于超平面排列的交格时，面半群代数与上同调代数同构。
英文摘要：
This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules; the Cartan invariants; projective resolutions of the simple modules; the Hochschild homology and cohomology; and the Koszul dual algebra. A new cohomology construction on posets is introduced and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.
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