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Lower central series and free resolutions of hyperplane arrangements

Domenii publicaţii > Matematica + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: H.K. Schenck, A.I. Suciu

Editorial: Transactions of the American Mathematical Society, 354 (9), p.3409-3433, 2002.


If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $phi_k$, of the lower central series quotients of $pi_1(M)$ 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 $A$. 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 $A$ over the exterior algebra $E$.
From this analysis, we recover a formula of Falk for $phi_3$, and obtain a new formula for $phi_4$. 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 $A$ is Koszul iff the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b’_{i,i+1}$, of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i = 2$, then it is attained for all $i ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid sub-arrangements), we show that $b’_{i,i+1}$ is determined by the number of triangles and $K_4$ subgraphs in the graph.

Cuvinte cheie: lower central series, free resolution, hyperplane arrangement, change of rings spectral sequence, Koszul algebra, linear strand, graphic arrangements