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Let $R$ be a commutative ring with unity, $I$ be an ideal and $a\in R$ be an element in $R$. We have the following short exact sequence:$$0\rightarrow R/(I:a)\rightarrow R/I\rightarrow R/(I+(a))\rightarrow 0$$ where the injection is multiplication by $a$, and the surjection is the canonical one. Moreover, it is known that whenever we have a short exact sequence of finitely generated graded modules over a polynomial ring over a field: $$0\rightarrow M''\rightarrow M\rightarrow M'\rightarrow 0$$ we can bound CM regularity as $\mathop{\rm reg}M\leq\max(\mathop{\rm reg} M'',\mathop{\rm reg}M')$.

In particular, if we let $R=\mathbb{C}[x]$, $I=(x^2)$ and $a=x$, we have $(I:a)=I+(a)=(x)$ homogeneous with $$\mathop{\rm reg}(R/I)=\mathop{\rm reg}(\mathbb{C}[x]/(x^2))=1$$ and $$\mathop{\rm reg}(R/(I:a))=\mathop{\rm reg}(R/(I+(a)))=\mathop{\rm reg(\mathbb{C}[x]/(x))}=0.$$ This seems to contradict the estimation of the regularity. Where can be the problem?

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Of course, after I have been thinking about this for three weeks now, I find the answer right when I post the question. I am sorry.

The problem is that the exact sequence above is not a graded exact sequence. We need to shift the grading, and then the estimation is fine.

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