Let $S=k[x_1,\dots,x_n]$ be a standard graded polynomial ring and $I\subset S$ a homogeneous ideal. Denote $\mathrm{maxdeg}(I)$ the maximal degree of an element in a minimal system of homogeneous generators of $I$. I read that $$\mathrm{maxdeg(I)}\le\mathrm{reg}(I),$$ where $\mathrm{reg}(I)$ is the regularity of $I$, but I cannot understand why. Can someone please help me? Thank you very much in advance!

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    $\begingroup$ What definition of regularity are you working with? $\endgroup$ – Zach Teitler Jul 3 '17 at 2:44
  • $\begingroup$ @ZachTeitler The height of the Betti Table. I know there is another definition in terms of local cohomology, but I never studied it. Is it easier to answer by using it? $\endgroup$ – Miles Eagle Jul 3 '17 at 9:53

If regularity is defined as the height of the Betti table, then the greatest degree of a generator is the greatest row number of a nonzero entry in the $1$st column (where rows and columns are counted starting from $0$). So the height of the table is at least as great as that.


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