What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:14:47Z http://mathoverflow.net/feeds/question/63969 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63969/what-is-an-explicit-example-of-a-variety-x-which-is-finite-over-spec-f-p-but-whic What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p? David Zureick-Brown 2011-05-05T02:59:02Z 2011-05-05T17:02:58Z <p>What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?</p> http://mathoverflow.net/questions/63969/what-is-an-explicit-example-of-a-variety-x-which-is-finite-over-spec-f-p-but-whic/63984#63984 Answer by Olivier for What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p? Olivier 2011-05-05T11:42:43Z 2011-05-05T11:42:43Z <p>Am I missing something or is this the classical question of Serre? A class of examples is given in Exemples de variétés projectives en caractéristique p non relevables en caractéristique zéro. Proc. Nat. Acad. Sci. U.S.A. 47 1961 108–109. </p> <p>They come from the quotient of some complete intersections by some finite groups, but if you read closely the proof (by the theory of the étale fundamental group, the impossibility of constructing a lift is reduced to the impossibility of constructing some group representations), you see that the ideas are in fact quite general. In order to give a non vacuous answer, let me also draw your attention to the letter of Serre in the appendix of the document illusie_trieste.pdf on Luc Illusie's website.</p> http://mathoverflow.net/questions/63969/what-is-an-explicit-example-of-a-variety-x-which-is-finite-over-spec-f-p-but-whic/64026#64026 Answer by David Zureick-Brown for What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p? David Zureick-Brown 2011-05-05T17:02:58Z 2011-05-05T17:02:58Z <p>I received the following very explicit answer via private communication: the algebra</p> <p>\$\$ A = \mathbb{F}_p[x_1,\ldots,x_6]/(x_1^p,\cdots, x_6^p, x_1x_2 + x_3x_4 + x_5x_6) \$\$</p> <p>does not lift to a finite flat \$\mathbb{Z}_p\$ algebra. (I am still working out the details of why this does not lift.) This is exampe 3.2(4) of Berthelot-Ogus.</p>