# Functions with vanishing Kahler differentials along a subscheme

Question: Let $k$ be an algebraically closed field of characteristic zero.

Let $X=\mathrm{Spec}\!~R$ be a scheme of finite type over $k$. Suppose $I\subset R$ is an ideal and $f\in I$.

Assume that $df = 0$ in $\Omega_{R/k}\otimes_R R/I$.

Does it follow that there is some positive integer $k$ such that $f^k \in I^{k+1}?$

Motivation: Suppose that $X$ and $V(I)$ are both smooth, then we have the short exact sequence $$0 \to I/I^2 \to \Omega_{R/k}\otimes_R R/I \to \Omega_{(R/I)/k} \to 0.$$ Thus if $f\in I$ and $df \equiv 0$ mod $I$, then $f\in I^2$.

In general this fails. For example, we can take $R=k[x]$, $I=(x^m)$ and $f = x^{m+1}$. Then $f\not\in I^2$ if $m>1$. However $$f^{m}\in I^{m+1}.$$

Geometrically, the statement says that if a function vanishing on the subscheme also has vanishing differential, then the function is nilpotent on the normal cone.

A Quick Reduction and Some Partial Results: If the statement is true for some $R$, then it's also true for any quotient of $R$. Hence we may assume that $R$ is the polynomial ring $k[x_1,\dots,x_n]$. Under this hypothesis, the statement becomes

If $f\in I$ and $\partial f/\partial x_i\in I$ for $i=1,\dots,n$. Does it imply that $f^k\in I^{k+1}$ for sufficiently large $k$?

1. if $I=(g)$ is principal, then we can prove the statement by factorizing $g$ intro irreducible polynomials $$g=g_1^{a_1}\cdots g_r^{a_r}.$$
If we write $f=gh$. Then it's easy to show that $g_1\cdots g_r$ divides $h$. Hence we can take $k=\max \{a_i\}$.
2. if $I$ is a graded ideal (generated by monomials), then we may assume that $f$ is a monomial, say $f=x_1^{a_1}\cdots x_n^{a_n}\in I$ and set $k=a_1+\cdots +a_n$. Then we have $$f^{k} = \sum_{a_i\neq 0} \left(\frac{\partial f}{\partial x_i} \cdot \frac{x_i}{a_i} \right)^{a_i} \in I^k\cdot f\subset I^{k+1}.$$
3. a comment from S.Li below: Without loss of generality, we can simply take $I$ to be the ideal generated by $f$ and all the $\partial f/\partial x_i$.
• In the reduction case, the subscheme defined by $I$ is just the singular locus of the hypersurface defined by $f$, assuming $f$ is reduced. Commented Jan 20, 2017 at 19:00
• Since you know the result in the principal case, you can just pass to the blowing up of $I$. Commented Jan 21, 2017 at 7:32
• @JasonStarr : Thanks very much for the comment. I only know how to prove in the case $I$ is a principal ideal in the UFD $C[x_1,\dots,x_n]$. If I blow up along $I$, the resulting scheme is not factorial in general. I don't how to prove the statement in that case. Commented Jan 21, 2017 at 17:18
• You have already reduced to the case that the ambient scheme $X$ is smooth. You are assuming that the characteristic equals $0$. Thus, by Hironaka, there exists a projective, birational morphism $\nu:\widetilde{X}\to X$ such that $\widetilde{X}$ is smooth and such that the ideal sheaf $\nu^{-1}I\cdot \mathcal{O}_{\widetilde{X}}$ is locally principal, $\langle g_1^{a_1}\cdots g_r^{a_r} \rangle$, where $(g_1,\dots,g_r)$ is locally part of a regular system of parameters. Now cover $\widetilde{X}$ by finitely many open affines, and take $k$ to be the max of he $k$s for those open affines. Commented Jan 21, 2017 at 18:11
• @JasonStarr: I think that solves my problem! Thanks so much! I was hoping for an elementary proof at the beginning. But in retrospect it seems that the problem is related to multiplicities of the singularity any way. Commented Jan 22, 2017 at 4:27

I am writing up the comments above as an answer, partly because they involve a couple of fun lemmas about blowing up. I tried to find a proof that does not use resolution of singularities, but I could not find one.

Let $X$ be a scheme, and let $\mathcal{J}$ be a quasi-coherent sheaf of ideals. Then the blowing up, $\nu:Z\to X$, is final among morphisms to $X$ such that the inverse image ideal sheaf $\nu^{-1}\mathcal{J}\cdot \mathcal{O}_Z$ is everywhere locally principal and generated by a nonzerodivisor (this condition is automatically true for the unique morphism from the empty scheme). There is also a construction of $Z$ as $$Z=\underline{\text{Proj}}_X\ \bigoplus_{n \geq 0} \mathcal{J}^n .$$ In particular, if $\mathcal{J}$ is everywhere locally finitely generated, then $\nu$ is proper. In that case, for every $n\geq 0$, there is a natural homomorphism of locally finitely generated, quasi-coherent $\mathcal{O}_X$-modules, $$\alpha_n:\mathcal{J}^n \to \nu_*(\nu^{-1}(\mathcal{J}^n)\cdot \mathcal{O}_Z).$$ It need not be the case that every $\alpha_n$ is an isomorphism, e.g., for $X=\text{Spec}\ k[s,t]$ and $\mathcal{J} = (\langle s^d, s^{d-1}t,st^{d-1},t^d \rangle)^\widetilde{\ \ \ }$, then $\alpha_n$ is an isomorphism precisely for $n\geq d-2$.

Lemma 1. If $X$ is Noetherian and $\mathcal{J}$ is coherent, then there exists $n_0\geq 0$ such that for all $n\geq n_0$, $\alpha_n:\mathcal{J}^n\to \nu_*(\nu^{-1}\mathcal{J}^n\cdot \mathcal{O}_Z)$ is an isomorphism.

The proof is basically Exercise II.5.9 from Hartshorne's "Algebraic Geometry".

Now let $X$ be an integral, Noetherian scheme, let $\mathcal{I}$ and $\mathcal{J}$ be nonzero coherent ideal sheaves, and let $\mu:Y\to X$, resp. $\nu:Z\to X$, be the blowing up of $X$ along the ideal sheaf $\mathcal{I}$, resp. $\mathcal{Z}$. Each blowing up a dominant morphism of integral schemes such that the pullback of the ideal sheaf is locally principal, and universal among all such morphisms. Then both the blowing up of $Y$ along $\mu^{-1}\mathcal{J}\cdot \mathcal{O}_Y$ and the blowing up of $Z$ along $\nu^{-1}\mathcal{I}\cdot \mathcal{O}_Z$ are universal among dominant morphisms of integral schemes to $X$ such that the pullback ideal sheaves of both $\mathcal{I}$ and $\mathcal{J}$ are locally principal. Thus, there is a canonical isomorphism of these two blowings up; call the common scheme $W$. So there is a commutative diagram $$\begin{array}{lcr} W & \xrightarrow{\widetilde{\nu}} & Y \\ \widetilde{\mu}\downarrow & & \downarrow \mu \\ Z & \xrightarrow{\nu} & X \end{array}$$ where every morphism is a blowing up. In fact, this is the blowing up of $\mathcal{I}\cdot \mathcal{J}$, cf. http://stacks.math.columbia.edu/tag/085Y Denote the composite morphism from $W$ to $X$ by $\lambda:W\to X$. Denote by $\widetilde{\mathcal{I}}$, resp. $\widetilde{\mathcal{J}}$, the inverse image ideal sheaf $\nu^{-1}\mathcal{I}\cdot \mathcal{O}_Z$, resp. $\mu^{-1}\mathcal{J}\cdot \mathcal{O}_Y$. Since $\mu^{-1}\mathcal{I}\cdot \mathcal{O}_Y$ is already locally principal generated by a nonzerodivisor, the natural homomorphism, $$\beta_n:\mu^{-1}\mathcal{I}^n\cdot \mathcal{O}_Y \to \widetilde{\nu}_*(\widetilde{\nu}^{-1}(\mu^{-1}\mathcal{I}_n\cdot \mathcal{O}_Y)\cdot \mathcal{O}_W),$$ is an isomorphism for every $n\geq 0$. Then $\lambda^{-1}\mathcal{I}\cdot \mathcal{O}_W$ equals $\nu^{-1}(\mu^{-1}\mathcal{I}\cdot \mathcal{O}_Y)\cdot \mathcal{O}_W$. Thus, by Lemma 1 above, for every $n\geq n_0$, the natural map, $$\mathcal{I}^n \to \lambda_*(\lambda^{-1}\mathcal{I}^n\cdot \mathcal{O}_W),$$ is an isomorphism. Again by Lemma 1 above, for every $n\geq n_1$, $$\widetilde{\mathcal{I}}^n \to \widetilde{\mu}_*(\widetilde{\mu}^{-1}\widetilde{\mathcal{I}}^n\cdot \mathcal{O}_W),$$ is an isomorphism. Altogether, this proves the following.

Lemma 2. For an integral, Noetherian scheme $X$, for a nonzero coherent sheaf $\mathcal{J}$ with blowing up $\nu:Z\to X$, for every coherent sheaf $\mathcal{I}$, there exists an integer $n_0\geq 0$ such that for every $n\geq n_0$, the natural homomorphism $\mathcal{I}^n\to \nu_*(\nu^{-1}\mathcal{I}^n\cdot \mathcal{O}_Z)$ is an isomorphism.

Thus, in your setting, for every $m\geq n_0-1$, to prove that $f^m$ is a global section of $\mathcal{I}^{m+1}$, it suffices to prove that $\nu^*f^m$ is a global section of $\nu^{-1}(\mathcal{I}^{m+1})$. Here I am assuming that $X$ is a smooth, finite type $k$-scheme, where $k$ is an algebraically closed field of characteristic $0$. By strong resolution of singularities, there exists a blowing up $\nu:Z\to X$ such that $\nu^{-1}\mathcal{I}\cdot \mathcal{O}_Z$ is everywhere locally principal generated by an element of the form $g_1^{a_1}\cdots g_r^{a_r}$, where $(g_1,\dots,g_r)$ is part of a regular system of parameters locally. By your proof, locally, $\nu^*f^m$ is a section of $\nu^{-1}\mathcal{I}^{m+1}\cdot \mathcal{O}_Z$ for every $m\geq \max(a_1,\dots,a_r)$.

Since the quasi-compact scheme $Z$ is covered by finitely many open affines on which $\nu^{-1}\mathcal{I}\cdot \mathcal{O}_Z$ is locally generated by $g_1^{a_1}\cdots g_r^{a_r}$, it follows that there exists $m_1$ such that for every $m\geq m_1$, $\nu^* f^m$ is a global section of $\nu^{-1} \mathcal{I}^{m+1}\cdot \mathcal{O}_Z$. Thus, by the previous paragraph, there exists $n_1=\max(n_0-1,m_1)$ such that for every $m\geq n_1$, $f^m$ is a section of $I^{m+1}$.

• Thanks very much for clarifying the technical details. I have the feeling that for a large class of maps $f:X \to Y$, we can test whether two ideal $\mathcal I, \mathcal J$ on $Y$ contain each other by looking at $f^{-1}\mathcal I \cdot \mathcal O_X$ and $f^{-1}\mathcal J \cdot \mathcal O_X$. This is obviously true for faithfully flat maps. Maybe it's true for any surjective maps between smooth varieties, or may be at least for blowing of a smooth variety along a smooth subvariety? Maybe this is only an illusion and it's fails for very nice cases? Commented Jan 23, 2017 at 8:45
• If $f:X\to Y$ is the blowing up of the ideal $\langle s,t\rangle$ in $Y=\text{Spec}\ k[s,t]$, then for $\mathcal{I}=(\langle s^d,s^{d-1}t, t^d \rangle)^{\widetilde{\ \ \ }}$ and for $\mathcal{J} = (\langle s^d, st^{d-1},t^d \rangle)^{\widetilde{\ \ \ }}$, then $f^{-1}\mathcal{I}\cdot \mathcal{O}_X$ equals $f^{-1}\mathcal{J}\cdot \mathcal{O}_X$, even though neither of $\mathcal{I}$ nor $\mathcal{J}$ is contained in the other. However, $\mathcal{I}^n$ does equal $\mathcal{J}^n$ for $n\gg 0$. Commented Jan 23, 2017 at 9:15
• I see. It's similar to the fact the two graded module can give the same sheaf on the projective space, as you have mentioned. Thanks! Commented Jan 24, 2017 at 8:28