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I have a question about the concept restriction to a deformation Harthsorne deals with in its book Deformation Theory. He is not very clear about it.


If $X$ is a scheme over $k$ and $A$ is an Artinian ring over $k$, Hartshorne defines a deformation $X'$ of $X$ over $A$ as a scheme with a closed immersion $X \hookrightarrow X'$ such that the induced map $X \to X'\times_A k$ is an isomorphism. In the proof of theorem $5.3$ and in exercise $5.7$ he talks about restrictions of deformations to affine patches $U \subseteq X$. He does not explain what he means by that. What are these restrictions? I would like to have an explanation or a reference. Thank you in advance.


Here I will quickly review my thoughts about this. We had two ideas:

  • We have an closed immersion of schemes $X \hookrightarrow X'$, hence an embedding of topological spaces. We can apply this to the subset $U \subseteq X$, but it is not obvious how to make a scheme out of this.
  • If it were possible, we would like to define the restriction $U '$ to be a fibered product, $X' \times_A U$ but the arrows don't point in the right direction to make this work.

This is the way Hartshorne talks about restrictions of deformations:

Let $X$ be a scheme over $k$, and let $X'$ be a deformation of $X$ over the dual numbers. For each open affine subset $U_i \subseteq X$, the restriction of $X'$ to $U_i$ is a deformation of $U_i$, so determines an element $\alpha_i$ in $T^1(U, \mathcal{O}_U)$.

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    $\begingroup$ Doesn't it just mean the restriction of the morphism $X\to X'$ to an open affine $U\subset X$? I don't have my copy of Hartshrone nearby, but for any morphism of schemes $f:X \to Y$ you can define a restriction. You basically mention this in your first bullet - where does it break down? $\endgroup$ Jul 17, 2017 at 17:57
  • $\begingroup$ @ Derek Allums, thank you so much for your answer! I agree you can define a restriction of the morphism $X \to X'$, but that is not what Hartshorne talks about. I will add a quote to make this more clear. $\endgroup$ Jul 17, 2017 at 19:17
  • $\begingroup$ Right, but again I'm confused why you can't just restrict your morphism. As you give it, a deformation of $X$ over $A$ is the following data: $X'$, and a closed embedding $X \hookrightarrow X'$ such that the induced map is an isomprhism. So, you have two things: a space and a morphism. I would guess that the restriction of $X'$ to $U_i \subset X$ is simply the restriction of your morphism to $U_i$, or am I still not understanding you or missing something? $\endgroup$ Jul 17, 2017 at 21:03
  • $\begingroup$ @Derek Allums, this restriction is not necessarily a closed immersion right? ( I confused immersion and embedding by the way.) $\endgroup$ Jul 18, 2017 at 10:09
  • $\begingroup$ Correct - since $U_i$ is open. $\endgroup$ Jul 18, 2017 at 13:51

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After asking around, the restriction of a deformation is simply the restriction of the morphism as usual plus the condition that we restrict the image of the morphism to the induced scheme structure on the open. This uses that any open subscheme $U \subset X$ can be endowed with an obvious scheme structure namely $(U,\mathcal{O}_X|_U)$ and the fact that a deformation is an homeomorphism [isomorphism on the underlying space].

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