2003/05/31 by Eva Maria Feichtner, Eva María Feichtner, Sergey Yuzvinsky · 117 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Basis (linear algebra) #Cohomology #Combinatorics #Commutative Algebra and Its Applications #Geometry #Graded ring #Hyperplane #Ideal (ethics) #Lattice (music) #Mathematics #Monomial #Pure mathematics #Toric variety #Variety (cybernetics) #math.AG #math.CO
paper · pdf · doi:10.1007/s00222-003-0327-2
published in Inventiones mathematicae 155(3), 515-536 (Springer Science+Business Media) · 23 pages, 7 figures, final revision with minor changes, to appear in Invent. Math
arxiv created 2003/08/03 · openalex publication_date 2004/02/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study a graded algebra D=D(L,G) defined by a finite lattice L and a subset G in L, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi. Our main result is a representation of D, for an arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we construct from L and G. We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Groebner basis of the relation ideal of D and a monomial basis of D over Z.