Resolutions of ideals of subspace arrangements. (English) Zbl 1514.13011
Summary: Given a collection of \(t\) subspaces in an \(n\)-dimensional vector space \(W\) we can associate to them \(t\) linear ideals in the symmetric algebra \(\mathcal{S}(W^*)\). A. Conca and J. Herzog [Collect. Math. 54, No. 2, 137–152 (2003; Zbl 1074.13004)] showed that the Castelnuovo-Mumford regularity of the product of \(t\) linear ideals is equal to \(t\). H. Derksen and J. Sidman [Adv. Math. 172, No. 2, 151–157 (2002; Zbl 1040.13009)] showed that the Castelnuovo-Mumford regularity of the intersection of \(t\) linear ideals is at most \(t\). We show that analogous results hold when we work over the exterior algebra \(\bigwedge (W^*)\) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor \(\Omega\) from the category of polynomial functors to itself. The functor \(\Omega\) transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13P20 | Computational homological algebra |
| 16E05 | Syzygies, resolutions, complexes in associative algebras |
| 20C32 | Representations of infinite symmetric groups |
| 14N20 | Configurations and arrangements of linear subspaces |
Keywords:
subspace arrangement; exterior algebra; equivariant resolution; Castelnuovo-Mumford regularityReferences:
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