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I am reading a few texts on Castelnuovo-Mumford regularity. If I understand correctly, almost all of them say:

If $I$ is a homogeneous ideal in $k[X_0,...,X_n]$ where $k$ is algebraically closed and $S$ is a generating set of $I$, then the regularity of $I$ is at least equal to the maximal degree of any element in $S$. Is this correct? Is there a good reference for this or is it obvious?

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By definition $\operatorname{reg}(R/I)=\max\{j-i\mid\beta_{ij}(R/I)\ne 0\}$. This is clearly greater or equal to $\max\{j\mid\beta_{0j}(R/I)\ne 0\}$, and the last one represents the maximal degree of a homogeneous generator of $I$.

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