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Register a new userDerived Langlands V:The simplicial and Hopflike categories
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Authors
Victor Snaith
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This is the fifth article in the Derived Langlands series which consists of one monograph and four articles. In this article I describe the Hopf algebra and Positive Selfadjoint Hopfalgebra (PSH) aspects to classification of a number of new classes of presentations and admissibility which have appeared earlier in the series. The paper begins with a very estensive. partly hypothetical, of the synthesis of the entire series. Many of the proofs and ideas in this series are intended to be suggestive rather than the finished definitive product for extenuating circumstances explained therein.
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This is a sequel to the authors book Derived Langlands which introduced an embedding of the category of admissible representations of a locally p-adic group in to the derived category of the monomial category of the group. This article gives a reformulation in terms of the hyperHecke algebra and relates this viewpoint to a number of topics, including the Bernstein centre of the category of admissible representations.
This sequel to Derived Langlands II studies some PSH algebras and their numerical invariants, which generalise the epsilon factors of the local Langlands Programme. It also describes a conjectural Hopf algebra structure on the sum of the hyperHecke algebras of products of the general linear groups over a $p$-adic local field or a finite field.
This is Part IV of a thematic series currently consisting of a monograph and four essays. This essay examines the form of induced representations of locally p-adic Lie groups G which is appropriate for the abelian category of ${mathcal M}_{c}(G)$-admissible representations. In my non-expert manner, I prove the analogue of Jacquets Theorem in this category. The final section consists of observations and questions related to this and other concepts introduced in the course of this series.
In this paper, we investigate properties of the bounded derived category of finite dimensional modules over a gentle or skew-gentle algebra. We show that the Rouquier dimension of the derived category of such an algebra is at most one. Using this result, we prove that the Rouquier dimension of an arbitrary tame projective curve is equal to one, too. Finally, we elaborate the classification of indecomposable objects of the (possibly unbounded) homotopy category of projective modules of a gentle algebra.
There are many ways to present model categories, each with a different point of view. Here wed like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras, it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) Were going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well.
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