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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.

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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.

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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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