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Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true.
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To study $s$-homogeneous algebras, we introduce the category of quivers with $s$-homogeneous corelations and the category of $s$-homogeneous triples. We show that both of these categories are equivalent to the category of $s$-homogeneous algebras. We prove some properties of the elements of $s$-homogeneous triples and give some consequences for $s$-Koszul algebras. Then we discuss the relations between the $s$-Koszulity and the Hilbert series of $s$-homogeneous triples. We give some application of the obtained results to $s$-homogeneous algebras with simple zero component. We describe all $s$-Koszul algebras with one relation recovering the result of Berger and all $s$-Koszul algebras with one dimensional $s$-th component. We show that if the $s$-th Veronese ring of an $s$-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all $s$-homogeneous algebras with $s$-th Veronese rings ${bf k}langle x,yrangle/(xy,yx)$ and ${bf k}langle x,yrangle/(x^2,y^2)$. In particular, we show that all of these algebras are not $s$-Koszul while their $s$-homogeneous duals are $s$-Koszul.
The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostants weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostants partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so_4(C), so_5(C), sp_4(C), and the exceptional Lie algebra g_2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl_3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights lambda and mu of the Lie algebras considered.
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.
In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to injections between map-graded categories, we obtain Hochschild complexes with support. We revisit Kellers arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jollenbeck, Welker and Skoldberg. We then show how this result can be used to obtain models for algebras with a chosen Grobner basis, and briefly outline how to compute some classical homological invariants with it.
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