I would like to know if there is a notion of normal bundle to a quotient map. In one specific case, let $G$ be a finite group acting on an algebraic variety $V$. Denote by the map $\pi:V \rightarrow V/G$ the quotient map. Is it possible to define the normal bundle to $\pi$ having good properties. For example, I would like to get the projection formula. If $V/G$ is a substack of $X$ of co-dimension $r,$ do we get a vector bundle of rank $r$ on $V$ as the normal bundle to the composition $V \rightarrow V/G \rightarrow X$? I am interested in the case where $V/G$ is a Deligne-Mumford stack.

The question may not be stated correctly and clearly. I apologize about it.

I would like to know if there is a notion of normal bundle to a quotient map. In one specific case, let $G$ be a finite group acting on an algebraic variety $V$. Denote by the map $\pi:V \rightarrow V/G$ the quotient map. Is it possible to define the normal bundle to $\pi$ having good properties. For example, I would like to get the projection formula. If $V/G$ is a substack of $X$ of co-dimension $r,$ do we get a vector bundle of rank $r$ on $V$ as the normal bundle to the composition $V \rightarrow V/G \rightarrow X$? I am interested in the case where $V/G$ is a Deligne-Mumford stack.

I would like to know the conditions under which the equality $$i^*(i_*(x))=x.c_r(N)$$ for $x\in A^*(V)$ holds.

Here, $N$ is the desired normal bundle. What is the condition under which the Chern classes of this bundle exist?

The question may not be stated correctly and clearly. I apologize about it.

I would like to know if there is a notion of normal bundle to a quotient map. In one specific case, let $G$ be a finite group acting on an algebraic variety $V$. Denote by the map $\pi:V \rightarrow V/G$ the quotient map. Is it possible to define the normal bundle to $\pi$ having good properties. For example, I would like to get the projection formula. If $V/G$ is a substack of $X$ of co-dimension $r,$ do we get a vector bundle of rank $r$ on $V$ as the normal bundle to the composition $V \rightarrow V/G \rightarrow X$?

The question may not be stated correctly and clearly. I apologize about it.

I would like to know if there is a notion of normal bundle to a quotient map. In one specific case, let $G$ be a finite group acting on an algebraic variety $V$. Denote by the map $\pi:V \rightarrow V/G$ the quotient map. Is it possible to define the normal bundle to $\pi$ having good properties. For example, I would like to get the projection formula. If $V/G$ is a substack of $X$ of co-dimension $r,$ do we get a vector bundle of rank $r$ on $V$ as the normal bundle to the composition $V \rightarrow V/G \rightarrow X$? I am interested in the case where $V/G$ is a Deligne-Mumford stack.

The question may not be stated correctly and clearly. I apologize about it.