Castelnuovo-Mumford regularity and exact sequence. 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.

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.