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Let X be a smooth and equidimensional algebraic stack over the field of complex numbers. Beilinson and Drinfeld calls such a stack good if $\forall n>0, Codim\{ x \in X \mid dim (Aut(x)) = n \} \geq n $ and very good if in the above equation the strict inequality holds. Since we are working with stacks, I am not able to understand the meaning of the above definition. How can we associate a substack of " points whose stabilizers have dimension n"? Or in the above definition are we considering the codimension in the topological space |X| associated to X?

Now assuming someone helps me with understanding the above definition, i would like to know how to prove the following remark due to Beilinson and Drinfeld:

X as above is very good if and only if $T^* (X^0) $ is dense in $T^* X $ where $ X^0$ is the largest DM substack of X

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    $\begingroup$ To define this locus as a substack, pass to a smooth cover of $X$ by affines. Then this dimension condition defines a locally closed subscheme, and you can descend this back to a substack. The codimension of the substack is the same as the codimension upstairs of this locally closed subscheme. $\endgroup$
    – dhy
    Commented Dec 27, 2017 at 14:14
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    $\begingroup$ It is maybe helpful to first think about the case where $X$ is $Y/G$, $Y$ a variety and $G$ a finite type algebraic group acting on $Y$. I believe this is how to define the cotangent bundle in this case: Let $Z$ be the locus in $T^*X$ of cotangent vectors at $x\in X$ which pair to zero with tangent vectors at $x$ coming from the infinitesimal action of $G$. Then $T^*X$ is just $Z/G.$ The remark you want to prove will follow because $Z$ is locally defined by $\operatorname{dim} G$ equations, which gives you a codimension bound on all components. $\endgroup$
    – dhy
    Commented Dec 27, 2017 at 14:20
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    $\begingroup$ Here is a slightly different formulation of what @dhy wrote. The cotangent complex of $X$ relative to $\mathbb{C}$ restricts on the small Zariski site of $Y$ to be equivalent to the $2$-term complex of locally free sheaves, $\Omega_Y \to \mathfrak{g}^\vee \otimes_{\mathbb{C}}\mathcal{O}_Y$, concentrated in degrees $1$ and $0$. The pullback to $Y$ of $T^*X$ is the relative Spec of the cokernel of the transpose of this homomorphism. $\endgroup$
    – Jason Starr
    Commented Dec 27, 2017 at 14:31
  • $\begingroup$ @dhy, Thank you .. your comments seem to suggest the following ( perhaps i am not thinking correctly) .. $T^*(X^0) $ is dense in $T^*(X) $ if and only if it intersects all the irreducible components .. in particular we must have $ codim (T^* X /T^*(X^0)) > 0$ .. the restriction of $T^*X $ to the substacks $X^m $ where the automorphism group has dimension $m$ are contained in this closed substack have codimension in $T^* X $ equal to $ codim_ X (X^m) - m $, which must be now non zero ... hence we get the very good condition ... am i thinking right? @ Jason Starr.. thanks for your comment. $\endgroup$
    – skeptic
    Commented Dec 28, 2017 at 6:14
  • $\begingroup$ @skeptic: Yep, that gives you one direction. To get the other direction, you need to make sure that you don't have any small (i.e. lower dimension than expected) extraneous irreducible components, which is why you need that your locus is locally defined by the right number of equation. $\endgroup$
    – dhy
    Commented Dec 28, 2017 at 11:32

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