Let Z —> Y be the normalization of an affine cuspidal curve over an algebraically closed field k. Let i : Z —> X be a closed immersion over Y with X smooth over Y.
Question: Does there exist a coherent module F on X, flat over Y, whose support is equal to Z set theoretically?
Answer: No in characteristic 0 and yes in characteristic p > 0.
To see that the answer is no in characteristic 0 you show that the map O_Y —> O_Z has an O_Y-linear section if you have F (and of course this isn’t possible for the normalization of the cuspidal curve). Namely, consider the map tau : O_Z —> O_Y which sends an element f of O_Z to the trace over O_Y of multiplication by f’ on F where f’ is any lift of f to O_X. You show that the choice of f’ doesn’t matter by checking at the generic point; the key fact is that the support condition tells us that f’ which vanish on Z give nilpotent operators on F. Finally, this gets us a section as tau(g) = rg for g in O_Y. Here r = rank_Y(F) > 0 which is invertible as we have char 0.
Remark: I think there doesn’t even exist a coherent F on X, flat over Y, such that the generic point of Z is an associated point of F. Exercise! [Edit on 2/12/22: Jason did the exercise and, uh, it isn’t true!]
To see that the answer is yes in characteristic p > 2, say Y is the spectrum of A = k[a, b]/(a^3 – b^2). Let X be the spectrum of A[t] and consider the closed subscheme, finite flat over Y, cut out by t^p – a^{(p – 3)/2}b. The reduction of this subscheme is isomorphic to Z. For p = 2 use t^2 – a. Cheers!
Regarding the exercise, for your presentation $k[a,b]/\langle a^3-b^2\rangle$ of the coordinate $A$ ring of $Y$, consider the closed subscheme of the spectrum of $A[t]$ with principal ideal generated by $\langle t^2-a \rangle$. The coordinate ring $F$ of this closed subscheme is $A$-flat (since it is the base change of the flat module $k[a][t]/\langle t^2-a \rangle$ over the ring $k[a]$), and the two generic points of the closed subscheme have closures with defining ideals $\langle t^2-a,t^3-b\rangle$ and $\langle t^2-a,t^3+b\rangle$ (I am assuming characteristic is not $2$, otherwise we are back in the setting of your observation). This time I spelled my name correctly in the reply form!
Of course! Thanks very much!
These kinds of “flat extensions of LCI morphisms on the generic fiber” come up when you try to get height bounds using the Chow variety (complete intersections have unobstructed deformations, so you can lift to characteristic 0).
Exactly! Do you remember the best bounds people have in char p? (Bounds depending on relative dimension of X over Y I think; I looked yesterday but I couldn’t find the paper I remember reading.)
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