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Let $mathcal{M}$ be a small $n$-abelian category. We show that the category of finitely presented functors $mod$-$mathcal{M}$ modulo the subcategory of effaceable functors $mod_0$-$mathcal{M}$ has an $n$-cluster tilting subcategory which is equivalent to $mathcal{M}$. This gives a higher-dimensional version of Auslanders formula.
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Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. In this article, we give an $n$-exangulated version of Auslanders defect and Auslander-Reiten duality formula. Moreover, we also give a classification of substructures (=closed subbifunctors) of a given skeletally small $n$-exangulated category by using the category of defects.
The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $mathcal{E}$ are reflected in $operatorname{mod_{mathsf{adm}}}(mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ as a subcategory of $operatorname{mod}(mathcal{E})$ when $mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $operatorname{mod_{mathsf{adm}}}(mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $mathcal{C}$ and certain resolving subcategories of $operatorname{mod}(mathcal{C})$.
We provide the first formulae for the weights of all simple highest weight modules over Kac-Moody algebras. For generic highest weights, we present a formula for the weights of simple modules similar to the Weyl-Kac character formula. For the remaining highest weights, the formula fails in a striking way, suggesting the existence of multiplicity-free Macdonald identities for affine root systems.
In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac-Moody algebra at the critical level. The periodic Kazhdan-Lusztig polynomials can be computed by using another family of polynomials, called the periodic $R$-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic $R$-polynomials in terms of the doubled Bruhat graph associated to a finite Weyl group and a finite root system.
Let $SsubsetPs^r$ ($rgeq 5$) be a nondegenerate, irreducible, smooth, complex, projective surface of degree $d$. Let $delta_S$ be the number of double points of a general projection of $S$ to $Ps^4$. In the present paper we prove that $ delta_Sleq{binom {d-2} {2}}$, with equality if and only if $S$ is a rational scroll. Extensions to higher dimensions are discussed.
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