Compact corigid objects in triangulated categories and co-t-structures

David Pauksztello

Open Mathematics (2008)

  • Volume: 6, Issue: 1, page 25-42
  • ISSN: 2391-5455

Abstract

top
In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, C , of a triangulated category, 𝒯 , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on 𝒯 whose heart is equivalent to Mod(End( C )op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, 𝒮 , of a triangulated category, 𝒯 , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End( 𝒮 )op), and hence an abelian subcategory of 𝒯 .

How to cite

top

David Pauksztello. "Compact corigid objects in triangulated categories and co-t-structures." Open Mathematics 6.1 (2008): 25-42. <http://eudml.org/doc/269400>.

@article{DavidPauksztello2008,
abstract = {In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, \[ C \] , of a triangulated category, \[ \mathcal \{T\} \] , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on \[ \mathcal \{T\} \] whose heart is equivalent to Mod(End(\[ C \] )op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, \[ \mathcal \{S\} \] , of a triangulated category, \[ \mathcal \{T\} \] , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End(\[ \mathcal \{S\} \] )op), and hence an abelian subcategory of \[ \mathcal \{T\} \] .},
author = {David Pauksztello},
journal = {Open Mathematics},
keywords = {Triangulated category; rigid and corigid object; t-structure; co-t-structure; cochain DGA; triangulated category; rigid object; corigid object},
language = {eng},
number = {1},
pages = {25-42},
title = {Compact corigid objects in triangulated categories and co-t-structures},
url = {http://eudml.org/doc/269400},
volume = {6},
year = {2008},
}

TY - JOUR
AU - David Pauksztello
TI - Compact corigid objects in triangulated categories and co-t-structures
JO - Open Mathematics
PY - 2008
VL - 6
IS - 1
SP - 25
EP - 42
AB - In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, \[ C \] , of a triangulated category, \[ \mathcal {T} \] , which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on \[ \mathcal {T} \] whose heart is equivalent to Mod(End(\[ C \] )op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, \[ \mathcal {S} \] , of a triangulated category, \[ \mathcal {T} \] , induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End(\[ \mathcal {S} \] )op), and hence an abelian subcategory of \[ \mathcal {T} \] .
LA - eng
KW - Triangulated category; rigid and corigid object; t-structure; co-t-structure; cochain DGA; triangulated category; rigid object; corigid object
UR - http://eudml.org/doc/269400
ER -

References

top
  1. [1] Aldrich S.T., García Rozas J.R., Exact and semisimple differential graded algebras, Comm. Algebra, 2002, 30, 1053–1075 http://dx.doi.org/10.1080/00927870209342371 Zbl1011.16009
  2. [2] Avramov L., Halperin S., Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra algebraic topology and their interactions (Stockholm, 1983), 1–27, Lecture Notes in Math. 1183, Springer, Berlin, 1986 
  3. [3] Beilinson A.A., Bernstein J., Deligne P., Faisceaux pervers, Astérique, 100, Soc. Math. France, Paris, 1982 (in French) 
  4. [4] Beligiannis A., Reiten I., Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc., 2007, 188, no. 883 Zbl1124.18005
  5. [5] Van den Bergh M., A remark on a theorem by Deligne, Proc. Amer. Math. Soc., 2004, 132, 2857–2858 http://dx.doi.org/10.1090/S0002-9939-04-07334-4 
  6. [6] Bernstein J., Lunts V, Eguivariant sheaves and functors, Lecture Notes in Math. 1578, Springer-Verlag, Berlin Heidelberg, 1994 Zbl0808.14038
  7. [7] Bondarko M.V., Weight structures for triangulated categories: weight filtrations, weight spectral sequences and weight complexes, applications to motives and to the stable homotopy category, preprint available at http://arxiv.org/abs/0704.4003v1 
  8. [8] Enochs E., Injective and flat covers, envelopes and resolvents, Israel J. Math., 1981, 39, 189–209 http://dx.doi.org/10.1007/BF02760849 Zbl0464.16019
  9. [9] Félix Y., Halperin S., Thomas J-C., Rational homotopy theory, Graduate Texts in Mathematics 205, Springer, New York, 2001 Zbl0961.55002
  10. [10] Frankild A., Jørgensen P., Homological identities for differential graded algebras, J. Algebra, 2003, 265, 114–135 http://dx.doi.org/10.1016/S0021-8693(03)00025-5 Zbl1041.16005
  11. [11] Hoshino M., Kato Y., Miyachi J., On t-structures and torsion theories induced by compact objects, J. Pure Appl. Algebra, 2002, 167, 15–35 http://dx.doi.org/10.1016/S0022-4049(01)00012-3 Zbl1006.18011
  12. [12] Iyama O., Yoshino Y., Mutations in triangulated categories and rigid Cohen-Macaulay modules, preprint available at http://arxiv.org/abs/math/0607736 Zbl1140.18007
  13. [13] Kashiwara M., Schapira P., Sheaves on manifolds, A Series of Comprehensive Studies in Mathematics 292, Springer-Verlag, Berlin Heidelberg, 1990 Zbl0709.18001
  14. [14] MacLane S., Categories for the working mathematician, Springer, New York, 1971 Zbl0232.18001
  15. [15] Neeman A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., 1996, 9, 205–236 http://dx.doi.org/10.1090/S0894-0347-96-00174-9 Zbl0864.14008
  16. [16] Neeman A., Triangulated categories, Annals of Mathematics Studies, Princeton University Press, Princeton and Oxford, 2001 Zbl0974.18008
  17. [17] Rotman J.J., An introduction to algebraic topology, Graduate Texts in Mathematics 119, Springer-Verlag, New York, 1988 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.