**Short version:** 



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> Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
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**Longer version:**


Let $G$ be a commutative group, let $R$ be a $G$-graded commutative ring, and let $M$ be a $G$-graded $R$-module. The category of $G$-graded $R$-modules has enough projectives, and so we can define *the projective dimension* ${\rm pd}(M)$ of $M$ as the infimum of the lengths of all projective resolutions of $M$ *in the category of $G$-graded $R$-modules.*


Let $U(M)$ denote the ungraded $R$-module underlying $M$. Then, $U(M)$ also has a projective dimension ${\rm pd}(U(M))$, defined *in the category of (ungraded) $R$-modules.*


At several places in the literature one finds the statement that these two projective dimensions coincide, i.e., that $${\rm pd}(M)={\rm pd}(U(M)).$$ The reason for this is always given by the fact that a $G$-graded $R$-module is projective if and only if its underlying $R$-module is so. But this seems to yield only the inequality $${\rm pd}(M)\geq{\rm pd}(U(M)).$$


If we wish to show the converse, then we may consider a projective resolution $$0\rightarrow P\_n\rightarrow P\_{n-1}\rightarrow\cdots\rightarrow P\_1\rightarrow P\_0\rightarrow U(M)\rightarrow 0$$ in the category of $R$-modules and try to get from this a projective resolution of the same length in the category of $G$-graded $R$-modules. But as the $P\_i$ need not be obtained from $G$-graded $R$-modules it is not immediately clear how to proceed. So:



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> Is the above equality true? And if so, how do we prove it?
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*Note 1:* Of course this question (and hopefully also its answer) can be generalised to arbitrary coarsenings, but for the moment the forgetful functor $U$ will suffice.


*Note 2:* There seems to be something more general going on, for the same claim is found in the literature about the *weak dimension;* here, one should note that a $G$-graded $R$-module is flat if and only if its underlying $R$-module is so.



Projective dimension of graded modules
The other inequality follows from Schanuel's lemma.


For $M$ as is in your question, consider a truncated resolution of projective $G$-graded modules 
$$Q\_{n-1}\xrightarrow{f}Q\_{n-2}\to\dots\to Q\_1\to Q\_0\to M.$$
Then
$$U(Q\_{n-1})\xrightarrow{f}U(Q\_{n-2})\to\dots\to U(Q\_1)\to U(Q\_0)\to U(M)$$
is a truncated projective resolution of $U(M)$.
If $\mathrm{pd}\, U(M)\leq n$, then Schanuel's lemma implies that $U(\ker f)$ is projective. Thus, $\ker f$ is projective as a $G$-graded module and $\mathrm{pd}\, M\leq n$.


This argument works for any exact functor $U$ such that $X$ is projective if and only if $U(X)$ is projective (in the appropriate categories).


