Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M\_{\ge d}:=\oplus\_{\ell \ge d} M\_\ell$ the sub-module of $M$. Is $M\_{\ge d}$ again a projective $A$-module?



Projective modules and gradings
A counter example would be $A = k[x, y]$, graded by the total degree, and $M = A$, $d = 1$.


The module $M\_1$ is then the ideal of $A$ generated by $x$ and $y$. It is not projective.


Because:


Consider the surjection $A^2 \rightarrow M\_1$, sending $(u, v)$ to $xu + yv$. It's an exercise to show that this map doesn't have a section.


