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Preprojective Algebras, Singularity Categories and Orthogonal Decompositions Read chapter Read first chapter

2013 | OriginalPaper | Chapter

The Fundamental Group of a Morphism in a Triangulated Category

Author : Ragnar-Olaf Buchweitz

Published in: Algebras, Quivers and Representations

Publisher: Springer Berlin Heidelberg

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Abstract

We introduce the fundamental group of a morphism in a triangulated category and show that the groupoid of distinguished triangles containing a given extension of objects from an abelian category is equivalent to the Quillen groupoid of the corresponding extension category as studied by Retakh (Uspekhi Mat. Nauk 41:179–180, 1986), Neeman–Retakh (Compositio Math. 102:203–242, 1996), and Schwede (J. Reine Angew. Math. 498:153–172, 1998).

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previous chapter (Contravariant) Koszul Duality for DG Algebras
next chapter On Hochschild Cohomology for Selfinjective Special Biserial Algebras
Footnotes
1
We will throughout assume that all categories are (essentially) small, thus, ignoring any potential set-theoretic issues.
 
2
J.P. May shows in [8, 9] that TR3 can be deduced from the remaining axioms. See also Balmer’s discussion [1] of this point in his review of [8].
 
Literature
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P. Balmer, Review of [J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163, no. 1 (2001), 34–73]. MR1867203 (2002k:18019)
2.
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V. S. Retakh, Homotopy properties of categories of extensions. Usp. Mat. Nauk 41, no. 6(252) (1986), 179–180 (in Russian). English translation: Russian Math. Surveys 41, no. 6 (1986), 217–218. MR0890505 (88f:18020). MathSciNetMATH
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J. Rickard, Review of [A. Neeman, Some new axioms for triangulated categories. J. Algebra 139, no. 1 (1991), 221–255]. MR1106349 (92b:18011).
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J.-L. Verdier, Des catégories dérivées des catégories abéliennes. With a preface by Luc Illusie. Edited and with a note by Georges Maltsiniotis, Astérisque 239 (1996), xii+253 pp., 1997. MR1453167 (98c:18007)
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C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38. Cambridge University Press, Cambridge, 1994. MR1269324 (95f:18001). MATHCrossRef
Metadata
Title
The Fundamental Group of a Morphism in a Triangulated Category
Author
Ragnar-Olaf Buchweitz
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Algebras, Quivers and Representations

Print ISBN: 978-3-642-39484-3

Electronic ISBN: 978-3-642-39485-0

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-39485-0_3

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