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Since you asked for reference, here are some references that may be helpful (the question is local):

1) (EDITED: Thanks to BCnrd for keeping me honest here!) If $(R,m)\to (S,n)$ is a map of Cohen-Macaulay local rings of same dimensions, $N$ a finite $S$-module which is $R$-free R$-free flat then $${\rm{depth}}_ S N ={\rm{depth}} R = {\rm{depth}} S.$$ This follows from Prop 1.2.16 and Theorem A.11 of Bruns-Herzog "Cohen-Macaulay rings". Namely, take$M=R$in both results, one get from A.11 that$\dim_SN = \dim_RN + \dim_SN/mN$, so$\dim_SN/mN=0$, then 1.2.16 gives${\rm{depth}}_SN = {\rm{depth}} R$. 2) If$S$is regular, then f.g modules with maximal depth are free. This is well-known. It follows from Auslander-Buchsbaum formula, as Boyarsky pointed out. EDIT: Just to be clear, it is enought to assume:$f$is finite,$X$regular, and$Y$Cohen-Macaulay scheme (certainly true if$Y$also regular or smooth, but is a much weaker condition). For example the map induced by$k[x,y]/(xy) \to k[x]$by killing$y$works. 4 added 264 characters in body Since you asked for reference, here are the some references you need that may be helpful (the question is local): 1) (EDITED: Thanks to BCnrd for keeping me honest here!) If$R\to S$(R,m)\to (S,n)$ is a finite, local map of Cohen-Macaulay local rings of same dimensions, $N$ a finite $S$-module which is $R$-free then $${\rm{depth}}_ S N ={\rm{depth}} R = {\rm{depth}} S.$$

This follows from Prop 1.2.16 and Theorem A.11 of Bruns-Herzog "Cohen-Macaulay rings" (rings". Namely, take $M=R$).M=R$in both results, one get from A.11 that$\dim_SN = \dim_RN + \dim_SN/mN$, so$\dim_SN/mN=0$, then 1.2.16 gives${\rm{depth}}_SN = {\rm{depth}} R$. 2) If$S$is regular, then f.g modules with maximal depth are free. This is well-known. It follows from Auslander-Buchsbaum formula, as Boyarsky pointed out. EDIT: Just to be clear, the condition you need it is enought to assume:$f$is finite,$X$regular, and$Y$Cohen-Macaulay scheme (certainly true if$Y$also regular or smooth, but is a much weaker condition). For example the map induced by$k[x,y]/(xy) \to k[x]$works)by killing$y$works. 3 added 5 characters in body Since you asked for reference, here are the references you need (the question is local): 1) If$R\to S$is a finite, local map of Cohen-Macaulay local rings,$N$a finite$S$-module which is$R$-free then$\text{depth}_SN ${\rm{depth}}_ S N =\text{depth} {\rm{depth}} R = \text{depth} S$.{\rm{depth}} S.

This follows from Prop 1.2.16 of Bruns-Herzog "Cohen-Macaulay rings" (take $M=R$).

2) If $S$ is regular, then f.g modules with maximal depth are free.

This is well-known. It follows from Auslander-Buchsbaum formula, as Boyarsky pointed out.

EDIT: Just to be clear, the condition you need is: $f$ is finite, $X$ regular, and $Y$ Cohen-Macaulay scheme (certainly true if $Y$ also regular or smooth, but is a much weaker condition). For example the map induced by $k[x,y]/(xy) \to k[x]$ works).

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