# The intersection complex and the Cohen-Macaulay property

Let $\Delta:Y\rightarrow X$ a closed immersion of $k$-schemes of finite type and equidimensionnal.

We assume that $\Delta^{*}[-d]IC_{X}=IC_{Y}$, if $X$ is Cohen-Macaulay, does it imply that $Y$ is Cohen-Macaulay?

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Let assume also that both X and Y are equidimensionnal, and that a resolution of singularities for X gives by base change a resolution of singularities for Y. –  prochet Apr 15 '13 at 21:30
Could you explain what it means by $IC_{-}$? Are they canonical bundle of $Y$ and $X$ respectively? Also what is $d$? –  Youngsu Apr 16 '13 at 9:44
d is the codimension, IC is the intersection complex. –  prochet Apr 16 '13 at 20:50