Abstract
AbstractWe introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.
Highlights
In continuing the studies of [24], we introduce a notion of hypersurface support for infinite-dimensional modules over a “noncommutative complete intersection”
By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation Q → R by a Noetherian algebra Q which is of finite global dimension
In the sibling project [23], we use hypersurface support for infinitedimensional modules to classify thick ideals in certain tensor triangulated categories associated to finite-dimensional Hopf algebras
Summary
In continuing the studies of [24], we introduce a notion of hypersurface support for infinite-dimensional modules over a “noncommutative complete intersection”. In the sibling project [23], we use hypersurface support for infinitedimensional modules to classify thick ideals in certain tensor triangulated categories associated to finite-dimensional Hopf algebras. For an R-module M , either finite-dimensional or infinite-dimensional, we say that M is supported at such a point c if the base change MK is of infinite projective dimension over Qc, and we define the hypersurface support of M as supphPyp(M ) :=. Note that a deformation parametrized by a smooth, i.e. finite type, algebra Z can be completed to produce a formally smooth analog This is technically convenient, because working with local rings is convenient. In Proposition 3.1 and Theorem 5.4 below, we deal with these fundamental issues (see Proposition 6.7)
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