conormal sequence of quotient of locally complete intersection

Question

Let $V=\mathbb{A}_{\mathbb{C}}^n/G$ be an quotient singularity where $G=\mathbb{Z}/r\mathbb{Z}$ and $G$ acts on $\mathbb{A}^n$ by $(x_1,\ldots,x_n) \mapsto (\zeta^{a_1}x_1,\ldots,\zeta^{a_n}x_n)$ ($\zeta$: $r$-th root of unity) and $(a_1,\ldots,a_n)=1$.

Let $0 \neq f \in \mathbb{C}[z_1,\ldots,z_n]$ be a $G$-eigenfunction and $X=(f=0)/G \subset V$ be the Weil divisor of $V$. Let $I \subset \mathcal{O}_V$ be the ideal sheaf of $X$ in $V$. There is an exact sequence

$ I/I^2 \rightarrow \Omega^1_V|_X \rightarrow \Omega^1_X \rightarrow 0. $

I want to know whether the map $I/I^2 \rightarrow \Omega^1_V|_X$ is injective or not in general. I think this is well known if $r=1$ i.e. l.c.i. case.

Answer 1

Yes.I assume that $X$ is reduced scheme over algebraicall closed field $k$ of characteristic $0$. Sorry for the absense of explanations. The $r=1$ case I mentioned is Theorem D.2.7. in Sernesi's book on deformation of algebraic schemes.

Actually, I'm interested in whether $\text{Ext}^2(\Omega^1_X, \mathcal{O}_X) $ is the obstruction space for the deformation of such $X$. I thought that this is true if Theorem D.2.7. generalises to the case $f$ is the deformation of a hyperquotient singularity. I want to know the case when $X$ is 3-dimensional terminal singularity.

Answer 2

I believe that if $X$ is a normal variety over a perfect field $k$, there is an obstruction theory for $X$ with values in $\mathrm{Ext}^2(\Omega_X, \mathcal O_X)$ (one should not talk about "the" obstruction space).

Let $L_X$ be the cotangent complex of $X$ over $k$; the canonical obstruction theory of $X$ over $k$ has values in $\mathrm{Ext}^2(L_X, \mathcal O_X)$. If we denote by $\underline H^j(L_X)$ the $j^{\rm th}$ cohomology sheaf of $L_X$, we have $\underline H^j(L_X) = 0$ for $j > 0$, $\underline H^0(L_X) = \Omega_X$, while for $j < 0$ the sheaf $\underline H^j(L_X)$ is supported in the singular locus of $X$; hence $\mathrm{Ext}^i(\underline H^j(L_X), \mathcal O_X) = 0$ for $i =0 $ and $i = 1$, because $X$ is $S_2$ and the codimension of the singular locus is at least $2$ (because $L_X = \Omega_X$ when $X$ is smooth, and regular implies smooth over a perfect field).

I claim that the projection $L_X \to \Omega_X$ induces an isomorphism of $\mathrm{Ext}^2(\Omega_X, \mathcal O_X)$ with $\mathrm{Ext}^2(L_X, \mathcal O_X)$. This follows easily from the existence of the spectral sequence $$ E_2^{ij} = \mathrm{Ext}^i(\underline H^{-j}(L_X), \mathcal O_X) \Longrightarrow \mathrm{Ext}^{i+j}(L_X, \mathcal O_X) $$ together with the vanishing of the terms $E_2^{ij}$ for $j > 0$, $i=0$ or $1$.