Home
Hostname: page-component-99c86f546-zzcdp Total loading time: 0.265 Render date: 2021-12-03T08:23:19.735Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

# CHAPTER II - Repetitive algebras

Published online by Cambridge University Press:  28 January 2010

## Summary

t-categories

There is a rather useful notion in the theory of triangulated categories which was introduced by Beilinson, Bernstein, Deligne (1982).

A t-category is a triangulated category D endowed with two full subcategories D≤o and D≥o; which are closed under isomorphisms and such that for D≥o=n(D<Rsup>≥o) and Do=n(D≤o) the following three conditions are satisfied:

1. For X∈D≤o and Y∈D≤1 we have that Hom(X,Y) = 0.

2. D≤oD≤1 and D≥1D≤o.

3. For X ∈ V there is a triangle B'→X→Brdquo; →TB' such that B' ∈ D≤o and B” ∈ D≥1.

Under these conditions, we say that the pair (D≤o,D≥o) is a t-structure on D. Denote by H the full subcategory (D≤oD≥oH is called the heart of the t-structure. The following fact (which we will not use) is shown in Beilinson, Bernstein, Deligne (1982): the heart H of a t-structure is an abelian category.

We include an example. Let A be an abelian category and Db(A) be the derived category of A. Let D≤o(resp. D≥o) be the full subcategory of D (A) formed by the objects X· such that Hi(X·) = 0 for i>o (i<o respectively). The properties (1) and (2) are easily verified. For the third property we denote by τ≤ox· the subcomplex of X·=(Xi,diX) with (τ≤ox·)i= Xi for i<o, (τ≤ox·)o= ker doX and zero otherwise. Set τ≥1X·= X·/τ≥oX· Clearly τ≤oX·D≤o and τ≥1X·D≥1.The short exact sequence of complexes 0→τ≤oX·→X·→τ≥1X·→0 yields the required triangle.

A second example will be given in the next section.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 1988

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

### Purchase

Buy print or eBook[Opens in a new window]

# Send book to Kindle

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

• Repetitive algebras
• Book: Triangulated Categories in the Representation of Finite Dimensional Algebras
• Online publication: 28 January 2010
• Chapter DOI: https://doi.org/10.1017/CBO9780511629228.003
Available formats
×

# Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

• Repetitive algebras
• Book: Triangulated Categories in the Representation of Finite Dimensional Algebras
• Online publication: 28 January 2010
• Chapter DOI: https://doi.org/10.1017/CBO9780511629228.003
Available formats
×

# Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

• Repetitive algebras
• Book: Triangulated Categories in the Representation of Finite Dimensional Algebras
• Online publication: 28 January 2010
• Chapter DOI: https://doi.org/10.1017/CBO9780511629228.003
Available formats
×