TITLE: Injective dimension of graded-injective modules
QUESTION [13 upvotes]: In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra":

Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with $A_0 = k$ a field). A graded module $I$ which is injective in the category of left graded modules has injective dimension at most $1$ in the category of left modules.

There is a proof of this fact for commutative graded algebras in this paper by Fossum and Foxby, but I don't really see how to transfer this to the non-commutative setting. Can anyone provide any pointers or a reference to a proof? Thanks in advance.
PS: In a later paper, Yekutieli and Zhang state that the only proof they know of this fact is "quite involved", which eased my anxiety at being unable to solve a pleasant excercise in the area I'm supposed to be PhD-ing in...

REPLY [3 votes]: Here is a proof of Van den Bergh's theorem, just posted on the arxiv:
http://arxiv.org/abs/1407.5916 .
My proof is pretty much self contained, and very detailed. The proof uses the same strategy as the original proof of Van den Bergh (which was very similar to the one given above by Ralph), namely passing through the Rees ring. But it is somewehat more conceptual, in that it isolates the precise "reason" for the dimension jump. This is possible since I work with derived categories (and not with the more cumbersome method of spectral sequences).