According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}\_{r+2}$, where $r \geq 2$. 


I don't know how to obtain this statement. The Happel-Rickard-Schofield theorem entails that the previous statement is equivalent to saying that $C(2,2,r)$ is an iterated tilted algebra of type $\tilde{D}\_{r+2}$. 


But I also don't know how to obtain that $C(2,2,r)$ is indeed an iterated tilted algebra of type $\tilde{D}\_{r+2}$. 


I've been trying to read Ringel's book in order to get an answer but I'm facing some difficulties. Does someone have suggestions of books or articles which can help me?


canonical algebra: [Introduction to coherent sheaves on weighted projective lines](http://arxiv.org/pdf/0911.4473.pdf)


derived equivalent, etc: [Hochschild cohomology of piecewise hereditary algebras](http://matwbn.icm.edu.pl/ksiazki/cm/cm78/cm7825.pdf)



> 
> **Theorem**: Let $A$ and $B$ be finite-dimensional algebras over a field $k$, with $A$ hereditary. The following are equivalent:
> 
> 
> * $B$ is an iterated tilted algebra of type $A$,
> * $A$ and $B$ are tilting equivalent,
> * $B$ is pieceise hereditary of type $A$.
> 
> 
> 



A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)
This result is attributed to Thomas Hübner in Thm. 3.5 of Lenzing's ICRA 2010 proceedings "Weighted projective lines and applications", available [here](http://www.ems-ph.org/books/show_abstract.php?proj_nr=135&vol=1&rank=5 "here") with subscription and [here](http://books.google.co.jp/books?id=WrgIgVpJnrcC&lpg=PA153&dq=Weighted%20projective%20lines%20and%20applications&hl=es&pg=PA162#v=onepage&q&f=false) via Google Books. Note that the result quoted there with proof.


Also, on the page following the theorem (p. 163) there is a table describing the correspondence between weight types for a weighted projective line and extended Dynkin types.


In Thm. 3.5 it is shown moreover that there exists a tilting bundle T whose endomorphism algebra is a path algebra of extended Dynkin type D. From this follows that C(2,2,r) is concealed as in your definition in the comments, since it is also the endomorphism algebra of a tilting bundle T in cohX(2,2,r).


(I couldn't find MO's policy regarding links, please edit if necessary)


