In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. 56 (1980), 167-189, we associate with L a graded commutative algebra A(L). Solomon, Combinatorics and topology of complements of hyperplanes, Invent. T1 - On the algebra associated with a geometric lattice We claim that, for all i, H cell i ( B F ) (cid:27) H cell i ( B s F ) (9)To prove this we will show that B s is a double. As usual let B s and B s denote the deletion and restrictionof B with respect to the atom s. Note that these are all distinct.If the elements b ∨ L a for b ∈ A a are all are distinct (in L itself), then the Boolean cover e L a isprecisely the sub-Boolean B, and the result follows.Otherwise, there exist distinct atoms s = a ∨ B b and s ′ = a ∨ B b ′ of B that are mapped by f tothe same atom a ∨ L b = a ∨ L b ′ of L a. The atoms of the sub-Boolean B are the elements b ∨ B a where b ∈ A a. §§ § § Let B = B ( A ) be the Boolean lattice on the finite set A and let F be a sheaf on B. In this section we recall the basics we need (for homology rather thancohomology), restricting ourselves to the setting of Boolean lattices, and then reprove a theoremof Lusztig relating the homology of a lattice equipped with a sheaf to the cellular homology ofthe Boolean cover. In we define acellular cohomology that computes, for a large class of posets, the cohomology of a poset withcoe ffi cients in a sheaf. The ordinary singular homology of a space can be computed cellularly. Thisgives the desired isomorphism H ∗ ( L \ F ) (cid:27) H ∗ ( e L \ F ). The E -page of thesequence of Theorem 2 is thus zero except for the q = E p, = H p ( e L \ F ). Thus H q is the trivial sheaf when q > H = F. Let T ∗ ( P F ) be the chain complex whose n -chains are T n ( P F ) = L σ F ( x ), the sum is overthe non-degenerate chains σ = x n ∅, X ⊆ A. If P has a minimum, and F is any sheaf, then H ( P F ) = F ( ) and H i ( P F ) vanishes fori >. If rkL ≥ thenH i ( L \ Λ j F ) is trivial unless:– either x and P. Let F be the natural sheaf on L and Λ j F be the j-th exterior power of F. Let L be the intersection lattice of an essential hyperplane arrangement in a spaceV. Weconcentrate first on the case where the arrangement is essential, meaning that the intersection ofall the hyperplanes is trivial. F, where F is the natural sheaf and Λ j F is the j -th exterior power of F.The natural sheaf is not local.In this paper our principal object of interest is the sheaf homology of L with coe ffi cients inthe graded sheaf Λ There are various other sheaves that can be put on an intersection lattice –see – but they turn out to be what Yuzvinsky calls local sheaves, and so thehomology vanishes for general reasons. This generalises an old resultof Lusztig. We call this the natural sheaf, and in we showedthat the reduced sheaf homology is trivial in all degrees, except the top one, whose dimensionis related to the derivative of the characteristic polynomial of L.
The resulting sheaf homology H ∗ ( L \ F ), where L is theintersection lattice of a hyperplane arrangement and F is some interesting (naturally occuring)sheaf, then becomes worthy of investigation.Intersection lattices of hyperplanes arrangements come equipped with a canonical sheaf asthe elements of the lattice are vector spaces. Interest in ho-mology may be revived though by taking coe ffi cients in a more interesting local system, thatis to say, in a sheaf on the lattice. Thehomology of this lattice, with constant coe ffi cients, was first determined in, withQuillen showing that it has the homotopy type of a wedge of spheres. The combinatorics of a hyperplane arrangement is encapsulated by its intersection lattice. A number of tools are given for the cellular homology ofthese Boolean covers, including a deletion-restriction long exact sequence. The computationalmachinery we develop in this paper is quite di ff erent though: sheaf homology is lifted to what we call Booleancovers, where we instead compute homology cellularly. This builds on the results of our previous paper wherethis homology was computed for Λ F = F, itself a generalisation of an old result of Lusztig.
We compute the sheaf homology of the intersection lattice of a hyperplane arrangement with coe ffi cientsin the graded exterior sheaf Λ A ug Sheaf homology of hyperplane arrangements, Boolean covers andexterior powers
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