Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-vl2kb Total loading time: 0.16 Render date: 2021-12-06T21:57:32.670Z 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 IV - Piecewise hereditary algebras

Published online by Cambridge University Press:  28 January 2010

Get access

Summary

Piecewise hereditary algebras

We call a finite-dimensional k-algebra A piecevise hereditary if Db(A) is triangle-equivalent to Db(kΔ) for some finite quiver Δ without oriented cycle. We recall that Δ is uniquely determined up to the relation ∼ introduced in I. 5.7. In this section we present some general facts about piecewise hereditary algebras. But first we need to recall some elementary facts for hereditary finite-dimensional k-algebras.

LEMMA. Let B be a hereditary finite-dimensional k-algebra and let X1,X2,X3 be B-modules. Suppose that f: X1 → X2 is subjective and that g: X2→ X3 is injective. Then there exists a module Y and linear maps h1: X1 → Y and h2: Y → X3 such that

is exact.

Proof. Consider the following exact sequence

Since B is hereditary, Ext1B(X3/X2,f) is surjective. Let

be a preimage of (*) in Ext1B(X3/X2,X1). Then we obtain the following commutative diagram of exact sequences

with h1 injective and h2 subjective. By construction we have that

is exact.

LEMMA. Let B be a hereditary finite-dimensional k-algebra and let X,Y be indecomposable B-modules. Suppose that Ext1B(Y,X)= 0. Then 0 ≠h ∈ HomB(X,Y) is either injective or surjective.

Proof. Let 0 ≠h ∈ HomB(X,Y). Let X → Z → Y be a factorization of h with f surjective and g injective. By 1.2 we obtain an exact sequence

By assumption this sequence splits. By Krull-Schmidt we infer that Z is isomorphic to X or isomorphic to Y.

We also note the following immediate consequence for an indecomposable module X over a hereditary finite-dimensional k-algebraB.

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.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your 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.

  • Piecewise hereditary algebras
  • Dieter Happel
  • Book: Triangulated Categories in the Representation of Finite Dimensional Algebras
  • Online publication: 28 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511629228.005
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.

  • Piecewise hereditary algebras
  • Dieter Happel
  • Book: Triangulated Categories in the Representation of Finite Dimensional Algebras
  • Online publication: 28 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511629228.005
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.

  • Piecewise hereditary algebras
  • Dieter Happel
  • Book: Triangulated Categories in the Representation of Finite Dimensional Algebras
  • Online publication: 28 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511629228.005
Available formats
×