Seminar Representation Theory, WS 2018/19

Time and place: Thursdays 10-12 in room V4-116

Organisers: Prof. Dr. William Crawley-Boevey, Prof. Dr. Henning Krause

The seminar is mainly devoted to an introduction to weighted projective lines and their connections to representation theory.

Schedule of Talks

  • 18 October (in U2-232): William Crawley-Boevey: Coherent sheaves on $\mathbb{P}^1$, part 1: lecture notes
  • 25 October: Julia Sauter: Coherent sheaves on $\mathbb{P}^1$, part 2
  • 08 November: Shiquan Ruan: Weighted projective spaces and weighted projective lines [GL1, §1]
  • 15 November: Minghui Zhao: Serre duality and Riemann-Roch theorem [GL1, §2]
  • 22 November: Yuta Kimura: Tilting from sheaves to modules, Sheaves and modules over canonical algebras [GL1, §§3–4]
  • 13 December: Andrew Hubery: Hereditary algebras
  • 20 December: Baptiste Rognerud: Morita and derived equivalences for Gendo-symmetric algebras [FHK, LR]
  • 17 January: Andrew Hubery: Coherent sheaves on weighted projective lines via periodic functors, part 1
  • 24 January: Andrew Hubery: Coherent sheaves on weighted projective lines via periodic functors, part 2
  • 31 January: Shiquan Ruan: Ladders of recollements via p-cycle construction

$\DeclareMathOperator{\Aut}{Aut}$ $\DeclareMathOperator{\coh}{coh}$ $\DeclareMathOperator{\mod}{mod}$ $\DeclareMathOperator{\vect}{vect}$

Abstract

Shiquan Ruan: Ladders of recollements via p-cycle construction
We investigate ladders of recollements for the bounded derived categories of coherent sheaves over exceptional curves via p-cycle construction. As applications, we classify recollements (by explicit functors) for the categories of coherent sheaves over weighted projective lines, and give an explicit description of recollements for the stable category of vector bundles.

Main Reference

[GL1] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras, in: Singularities, Representation of Algebras, and Vector Bundles (LNM 1273, Springer 1987), 265–297, doi:10.1007/BFb0078849
(the first paper introducing weighted projective line, using group graded coordinate ring and Serre construction; via a canonical tilting sheaf, derived equivalent to the canonical algebra)

Further References

[GL2] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273–343, doi:10.1016/0021-8693(91)90107-J
(using perpendicular category strategy to connect weighted projective lines of different weight types; describe the existence of adjoint functors, link to epimorphism of rings)
[Ha] D. Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), 381–398, doi:10.1007/s002220100135
(algebraically closed field $k$; only two kinds of hereditary abelian $k$-categories with tilting objects: $\mod A$ and $\coh \mathbb{X}$; using perpendicular category and one-point extension strategies)
[HR] D. Happel and I. Reiten, Hereditary abelian categories with tilting object over arbitrary base fields, J. Algebra 256 (2002), 414–432, doi:10.1016/S0021-8693(02)00088-1
(arbitrary commutative Artin ring; any hereditary abelian category with tilting object is derived equivalent to $\mod H$, where $H$ is a hereditary algebra or a squid algebra)
[KLM] D. Kussin, H. Lenzing and H. Meltzer, Triangle singularities, ADE-chains and weighted projective lines, Adv. Math. 237 (2013), 194–251, doi:10.1016/j.aim.2013.01.006
(focus on weight type $(a,b,c)$; introduce the Frobenius structure on $\vect \mathbb{X}$ with indecomposable projective-injectives all line bundles; construct a tilting object on the stable category $\underline{\vect} \mathbb{X}$ with endomorphism algebra $kA_{a-1}\otimes kA_{b-1}\otimes kA_{c-1}$; fractional Calabi–Yau property on $\underline{\vect} \mathbb{X}$; related to ADE-chains and Happel–Seidel symmetry on Nakayama algebras)
[Le1] H. Lenzing, A K-theoretic study of canonical algebras, in: Representation theory of algebras (CMS Conf. Proc. 18, 1996), 433–454
(arbitrary base field; study the Grothendieck groups of canonical algebras via abstract bilinear lattices; mutations with an exceptional object or with a tube are introduced, describe automorphism group of a bilinear lattice; give the Riemann–Roch formula; classify bilinear lattices)
[Le2] H. Lenzing, Representations of finite-dimensional algebras and singularity theory, in: Trends in ring theory (CMS Conf. Proc. 22, 1998), 71–97
(generalize weighted projective lines to exceptional curves, from algebraically closed field to arbitrary field; p-cycle construction approach to the category of coherent sheaves, related to (quasi) parabolic structure; many important examples and applications are stated)
[Le3] H. Lenzing, Hereditary categories, in: Handbook of tilting theory (London Math. Soc. Lecture Note Ser. 332, 2007), 105–146, doi:10.1017/CBO9780511735134.006
(algebraically closed field; axiomatic introduction of $\coh \mathbb{X}$, as a certain hereditary abelian category with tilting object; many examples of hereditary abelian categories are collected)
[Le4] H. Lenzing, Weighted projective lines and applications, in: Representations of algebras and related topics (EMS Ser. Congr. Rep., 2011), 153–187, doi:10.4171/101-1/5
(focus on weight triple case; using stability and Euler character to the describe the AR-stucture of $\coh \mathbb{X}$ trichotomy; different exact structures on $\vect \mathbb{X}$ are introduced, related to triangle singularity, Kleinian singularity and Fuchsian singularity; tilting objects on stable category of $\vect \mathbb{X}$ are studied; connection with the flag of invariant subspaces for nilpotent operators)
[Le5] H. Lenzing, Weighted projective lines and Riemann surfaces, in: Proceedings of the 49th Symposium on Ring Theory and Representation Theory (Symp. Ring Theory Represent. Theory Organ. Comm., 2017), 67–79
(complex field; provide an equivarient version for the category of coherent sheaves over a smooth projective curve from compact Riemann surfaces; using orbifold fundamental group and Fuchsian group)
[LdP1] H. Lenzing and J.A. de la Peña, Concealed-canonical algebras and separating tubular families, Proc. London Math. Soc. 78 (1999), 513–540, doi:10.1112/S0024611599001872
(describe concealed-canonical algebra by the existence of a separating tubular family; a new approach to $\coh \mathbb{X}$ from the module category of concealed-canonical algebras, using tubular mutation in the derived category)
[LdP2] H. Lenzing and J.A. de la Peña, Extended canonical algebras and Fuchsian singularities, Math. Z. 268 (2011), 143–167, doi:10.1007/s00209-010-0663-z
(introduce a new kind of Frobenius structure on $\vect$ with indecomposable projective-injectives $\tau^{\mathbb{Z}}\mathcal{O}$; the stable category has a tilting object, whose endomorphism algebra is a one-point extension of canonical algebra; link to Fuchsian singularity)
[LM1] H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Representations of Algebras (CMS Conf. Proc. 14, 1993), 313–337
(using tubular mutation to classify the indecomposable vector bundles over a weighted projective line of tubular type)
[LM2] H. Lenzing and H. Meltzer, The automorphism group of the derived category for a weighted projective line, Comm. Algebra 28 (2000), 1685–1700, doi:10.1080/00927870008826922
(the automorphism group $\Aut(D^b(\coh \mathbb{X}))$ is described, mainly including suspension shifts, grading shifts and tubular mutations)

Additional References

[FHK] M. Fang, W. Hu and S. Koenig, Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions, arXiv:1607.03513
[LR] M. Linckelmann and B. Rognerud, On Morita and derived equivalences for cohomological Mackey algebras, Math. Z. 289 (2018), 39–50, doi:10.1007/s00209-017-1942-8