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Register a new userHypersurface support and prime ideal spectra for stable categories
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We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in type $A$, Drinfeld doubles of height 1 Borels in finite characteristic, and rings of functions on finite group schemes over a perfect field. We then identify the prime ideal (Balmer) spectra for these stable categories. In the curious case of functions on a finite group scheme $G$, the spectrum of the category is identified not with the spectrum of cohomology, but with the quotient of the spectrum of cohomology by the adjoint action of the subgroup of connected components $pi_0(G)$ in $G$.
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Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $pi$-point style rank variety for the Drinfeld double, identify $pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.
We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshinos subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitzs triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitzs triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras
We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation $Qto R$ by a Noetherian algebra $Q$ which is of finite global dimension. We show that hypersurface support defines a support theory for the big singularity category $Sing(R)$, and that the support of an object in $Sing(R)$ 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 a companion piece, we employ hypersurface support, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.
We describe graded commutative Gorenstein algebras ${mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)cong{mathcal E}_{n}(p)$, where $mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently constructed in cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,Coulembier}. We investigate the combinatorics of these algebras, and the relationship with Mincs partition function, as well as possible actions of the Steenrod operations on them. Evidence for the conjecture includes a large number of computations for small values of $n$. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in ${mathcal E}_n(p)$ with a homogeneous system of parameters in $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)$. These parameters have degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1le i le n$. This at least shows that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)$ is a finitely generated graded commutative algebra with the same Krull dimension as ${mathcal E}_n(p)$. For $p=2$ we also show that $mathrm{Ext}^bullet_{mathsf{Ver}_{2^{n+1}}}(1,1)$ has the expected rank $2^{n(n-1)/2}$ as a module over the subalgebra of parameters.
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $tau^2=id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.
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