MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be a affine smooth scheme finite type over $A/pA$, where $A$ a complete DVR and $chk=p>0$.

I know that since $H^{2}=0=H^{1}$, we have a unique lifting to $A/p^{2}$. In algebraic schemes case, we can obtain a trivial deformation by fiber product. But in this case, for example, $A$ is a p-adic number ring, What is the trivial deformation over $A/p^{2}$ ?

PS: I added some smooth conditions to $X$.

share|cite|improve this question
There might not exist a lifting of $X$ to $A/p^2$, let alone a canonical one. – Angelo Apr 6 '13 at 9:05
I'm confused Angelo. The question says $X$ is affine and the obstruction to such a lifting lies in $H^2$ which vanishes, so doesn't such a lift always exist? – Matt Apr 6 '13 at 16:15
@Matt: Assuming the scheme is finite type over $k$, the obstruction most naturally lives in an Ext group of the cotangent complex. If $X$ is smooth, this reduces to $H^2$, which will be zero as you say. However, this need not be the case if $X$ is singular. – Jason Starr Apr 6 '13 at 16:22
Ah, thanks! I've clearly been indoctrinated by only thinking about liftability of smooth varieties. – Matt Apr 6 '13 at 19:40

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.