TITLE: Castelnuovo-Mumford regularity and exact sequence.
QUESTION [5 upvotes]: In a question on MathOverflow it is said that:

It is known that given a short exact sequence of finitely generated graded modules over a polynomial ring over a field:$$0 \to M'' \to M \to M' \to 0$$ then $\operatorname{reg}M\leq\max(\operatorname{reg}M',\operatorname{reg}M'').$

Unfortunately no proof is given there of the above result. Is this result obvious? Please help me to prove this.

REPLY [6 votes]: Yes, it is obvious. Take the long exact sequence of cohomology groups. Anytime, the cohomology of the two outer terms vanishes, the cohomology of the inner term will vanish, too.