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Abstract We 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.more » « less
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Abstract We consider finite-dimensional Hopf algebras $$u$$ that admit a smooth deformation $$U\to u$$ by a Noetherian Hopf algebra $$U$$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $$1$$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $$u$$, via the singularity categories of the hypersurfaces $U/(f)$ associated with functions $$f$$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $$A$$.more » « less
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We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ⩾ 3 p\geqslant 3 , starting with a definition of a π \pi -point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π \pi -points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k [ t , τ ] / ( t p − τ 2 ) k[t,\tau ]/(t^p-\tau ^2) , where t t has even degree and τ \tau has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.more » « less
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