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Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations.

Is it true that the I-adic completion of A has finite homological dimension?

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    $\begingroup$ Do you really mean the $I$-adic completion of $A$, or do you mean the $I$-adic completion of $c[Q]$? (The ideal $I$ "becomes zero" in the factor algebra $A$.) $\endgroup$
    – Manny Reyes
    Commented Jul 31, 2013 at 14:44
  • $\begingroup$ To continue Manny's question, another thing you might mean is: the completion of A with respect to the ideal of all arrows. That is the completion which I have seen used (e.g., in representations of quivers with potential). $\endgroup$
    – Hugh Thomas
    Commented Aug 15, 2013 at 4:53
  • $\begingroup$ You're right: I meant the I-adic completion of c[Q]. $\endgroup$
    – user41521
    Commented Oct 18, 2013 at 18:47

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