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Let $S=k[x_1,\dots,x_n]$ and $J\subset I$ two squarefree monomial ideals in $S$ such that $I$ is generated in degree $d$, and $J$ is generated in degree $d+1$. Let $$I=(f_1,\ldots,f_r),\quad I'=(f_1,\ldots,f_e),e\leq r$$ and $J'=I'∩J$. If we have $\operatorname{depth}(I'/J')=d+1$, and$$\operatorname{depth}(S/I)=\operatorname{depth}(S/J)=\operatorname{depth}(S/I')=\operatorname{depth}(S/J')=\operatorname{depth}(S/(I'+J))=d,$$ is it possible to have $\operatorname{depth}(I/J)>d+1$ ?

I think that it isn't true, but I don't see why. (Here $\operatorname{depth}(I/J)$ is defined from Depth Lemma.)

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Please explain what do you mean by "generated in degree $d$". I also don't understand why do you feel the need to say that $\text{depth}\, I/J$ is defined by depth lemma. – user26857 May 21 '12 at 23:52
I is a square free monomial ideal, and the generating monomials have degree = d. I said it's defined by depth lemma so there wouldn't be missunderstandings between notations – Andrei May 24 '12 at 11:11

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