2
$\begingroup$

Suppose we are given a positively graded $k$ algebra $\bigoplus_{i\ge 0}A_i$ such that $A_0$ has finite global dimension. Furthermore all $A_i$ are finite dimensional and $A$ is generated in degree 0 and 1 i.e. $A_1A_i=A_{i+1}$.

Now the question:

Does every finitely generated graded $A$ module $M$ admit a minimal graded projective resolution $...\rightarrow P^{i+1}\rightarrow P^i\rightarrow ...\rightarrow P^0\rightarrow M\rightarrow 0$ such that $P^{i+1}$ is mapped to $A_+P^i$ for all $i$ ?

I know that this is true if $A_0$ is semisimple but I found nothing about what happens in the case where $A_0$ has finite global dimension.

I would be deeply grateful for any information or literature.

Improve this question
$\endgroup$
7
  • $\begingroup$ This does not work even in the case where A_1 is zero, so that A is concentrated in degree zero. $\endgroup$
    – Mariano Suárez-Álvarez
    Oct 25, 2013 at 7:04
  • $\begingroup$ (If you redefine $A_+$ to be the sum of the things in positive degree and the Jacobson radical of $A_0$, on the other hand, you'll get essentially what you want. Straightforward changes to the usual proof should establish this; alternatively, the algebra is semi-perfect because this can be reduced immediately to the question of whether $A_0$ is semiperfect, which it is) $\endgroup$
    – Mariano Suárez-Álvarez
    Oct 25, 2013 at 7:19
  • $\begingroup$ What do you mean exactely...We take $\tilde{A}=A_+\oplus rad(A_0)$ and we can establish a graded projective resolution such that $P^{i+1}$ is mapped to $\tilde{A}P^i$ ? $\endgroup$
    – Aleksa
    Oct 25, 2013 at 7:37
  • $\begingroup$ Exactly. ${}{}{}$ $\endgroup$
    – Mariano Suárez-Álvarez
    Oct 25, 2013 at 7:39
  • $\begingroup$ Where can I find this stuff on semi-perfect algabras? I even did not find the usual proof...:) $\endgroup$
    – Aleksa
    Oct 25, 2013 at 7:43

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.