In the paper "A proof of the strong no loop conjecture"* there is the following remark: "The preceding theorem establishes the strong no loop conjecture for finite dimensional elementary algebras and hence for finite dimensional algebras over an algebraically closed field."

I'd like to understand why this implication is true. An algebra here is called elementary if all simple modules are 1-dimensional over the ground field. Not all algebras over an algebraically closed field are elementary (consider matrix algebras), so there is something sophisticated going on.

About the strong no loop conjecture: It claims that when you have a finite dimensional algebra $A$ and a simple $A$-module $S$ which has finite projective dimension, then already Ext$^1_A(S, S) = 0$.

I'd like to know if the following reasoning is correct:

Let $A$ be a finite dimensional algebra over the algebraically closed field $k$. By a theorem of Gabriel, there is a Morita equivalence between $A$ and some algebra $kQ/I$, where $Q$ is a quiver and $I$ is an admissible ideal. $kQ/I$ is elementary and so the no loop conjecture holds over this algebra. The Morita equivalence has the following properties:

(i) It is exact and additive

(ii) it sends simples to simples

(iii) It sends projectives to projectives

(I think (ii) and (iii) follow from (i))

(iv) As short exact sequences correspond via the equivalence, Ext$^1_A(S,S) = 0$ if and only if Ext$^1_{kQ/I}(S',S') = 0$ for the corresponding simple $S'$.

If all of this is true then I guess the strong no loop conjecture follows immediately also for $A$. What do you think?

*Authors: Kiyoshi Igusa, Shiping Liu, Charles Paquette; appeared in Advances in Mathematics Volume 228, Issue 5, 1 December 2011, Pages 2731-2742

  • $\begingroup$ Indeed, all that is true. A Morita equivalence induces isomoprhisms on all Exts (and, in fact, passes on to a triangulated equivalence of derived categories) $\endgroup$ – Mariano Suárez-Álvarez Jan 29 '17 at 10:16
  • $\begingroup$ (When you cite a paper, please provide a complete bibliographical reference, including obviously the author...) $\endgroup$ – Mariano Suárez-Álvarez Jan 29 '17 at 10:18
  • $\begingroup$ Thank you very much! I now provided a bibliographical reference, I hope it's fine now. $\endgroup$ – Leon Lang Jan 29 '17 at 11:49

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