1,141 research outputs found
Lower central series and free resolutions of hyperplane arrangements
If is the complement of a hyperplane arrangement, and A=H^*(M,\k) is
the cohomology ring of over a field of characteristic 0, then the ranks,
, of the lower central series quotients of can be computed
from the Betti numbers, b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i, of the linear
strand in a (minimal) free resolution of \k over . We use the
Cartan-Eilenberg change of rings spectral sequence to relate these numbers to
the graded Betti numbers, b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j, of a (minimal)
resolution of over the exterior algebra .
From this analysis, we recover a formula of Falk for , and obtain a
new formula for . The exact sequence of low degree terms in the
spectral sequence allows us to answer a question of Falk on graphic
arrangements, and also shows that for these arrangements, the algebra is
Koszul iff the arrangement is supersolvable. We also give combinatorial lower
bounds on the Betti numbers, , of the linear strand of the free
resolution of over ; if the lower bound is attained for , then it
is attained for all . For such arrangements, we compute the entire
linear strand of the resolution, and we prove that all components of the first
resonance variety of are local. For graphic arrangements (which do not
attain the lower bound, unless they have no braid sub-arrangements), we show
that is determined by the number of triangles and subgraphs
in the graph.Comment: 25 pages, to appear in Trans. Amer. Math. So
Resolutions and Cohomologies of Toric Sheaves. The affine case
We study equivariant resolutions and local cohomologies of toric sheaves for
affine toric varieties, where our focus is on the construction of new examples
of decomposable maximal Cohen-Macaulay modules of higher rank. A result of
Klyachko states that the category of reflexive toric sheaves is equivalent to
the category of vector spaces together with a certain family of filtrations.
Within this setting, we develop machinery which facilitates the construction of
minimal free resolutions for the smooth case as well as resolutions which are
acyclic with respect to local cohomology functors for the general case. We give
two main applications. First, over the polynomial ring, we determine in
explicit combinatorial terms the Z^n-graded Betti numbers and local cohomology
of reflexive modules whose associated filtrations form a hyperplane
arrangement. Second, for the non-smooth, simplicial case in dimension d >= 3,
we construct new examples of indecomposable maximal Cohen-Macaulay modules of
rank d - 1.Comment: 39 pages, requires packages ams*, enumerat
Characteristic varieties and Betti numbers of free abelian covers
The regular \Z^r-covers of a finite cell complex X are parameterized by the
Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording
when the Betti numbers b_1,..., b_i of the corresponding covers are finite
carves out certain subsets \Omega^i_r(X) of the Grassmannian.
We present here a method, essentially going back to Dwyer and Fried, for
computing these sets in terms of the jump loci for homology with coefficients
in rank 1 local systems on X. Using the exponential tangent cones to these jump
loci, we show that each \Omega-invariant is contained in the complement of a
union of Schubert varieties associated to an arrangement of linear subspaces in
H^1(X,\Q).
The theory can be made very explicit in the case when the characteristic
varieties of X are unions of translated tori. But even in this setting, the
\Omega-invariants are not necessarily open, not even when X is a smooth complex
projective variety. As an application, we discuss the geometric finiteness
properties of some classes of groups.Comment: 40 pages, 2 figures; accepted for publication in International
Mathematics Research Notice
- …