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Let $f : X \to \operatorname{Spec}(R)$ be a flat and projective morphism of schemes over $\mathbb{C}$, where $R$ is a DVR. Assume additionally that the fibers of $f$ are Gorenstein and seminormal (in particular reduced). Let $i : X_0 \to X$ denote the closed immersion of the special fiber.

Is it then the case that the morphism of tangent sheaves $i^*\mathcal{T}_{X / R} \to \mathcal{T}_{X_0 / \mathbb{C}}$ is an isomorphism?

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    $\begingroup$ I recommend that you do the computation for yourself when $R$ equals $\mathbb{C}[[t]]$ and when $X$ equals $\text{Spec}(A)$ for $A=R[u,v]/\langle uv-t \rangle$. $\endgroup$
    – Jason Starr
    Jul 18, 2018 at 20:26
  • $\begingroup$ Thanks for the counter-example. I'm in the situation where I need the cokernel $K$ (which can be thought of as the $t$-torsion elements of $\mathcal{T}^1_{X / R}$) of the above morphism to satisfy $H^1(X,K) = 0$. Any insight into this problem would be greatly appreciated. $\endgroup$
    – Mellon
    Jul 19, 2018 at 18:55
  • $\begingroup$ If I'm not mistaken, $K$ is supported at the singular locus of $X_0$, hence $H^1(X,K)$ vanishes if $X$ is fibered in curves. In higher dimension, I'm not so sure. $\endgroup$
    – Mellon
    Jul 19, 2018 at 20:18
  • $\begingroup$ Yes, the cokernel is supported on the singular locus of $X_0$. $\endgroup$
    – Jason Starr
    Jul 19, 2018 at 20:45

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