# on the local structure of schemes

Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:

$\Omega_{U}=E\oplus T$, where $E$ is a locally free sheaf of rank $d$ and $T$ just coherent.

Can we prove that locally for étale topology around $x$, $U$ is isomorphic to $\mathbb{A}^{d}\times Y$, for a certain scheme $Y$?

• In char. $p>0$, you can have $d>\dim X$, e.g. $X=\mathrm{Spec}\,(k[t]/(t^p))$ (here $\Omega$ is free of rank 1). – Laurent Moret-Bailly Jan 17 '15 at 14:10
• I found a positive result in characteristic $0$ if the singularities are isolated. – Jason Starr Dec 16 '16 at 16:00

Edit, 12/2016. The original post was long ago, but I had a reason to revisit this some months back when I was asked a similar question. I am including below the positive answer to this question that I found in a special case. There is a little setup required.

Flatness over $\mathbb{A}^r$. First there is a flatness lemma. Let $k$ be a commutative, unital ring (for instance, $k=\mathbb{Z}$). Let $A$ be a commutative, unital $k$-algebra that is Noetherian. Let $a\in A$ be an element. By the Noetherian hypothesis, the following chain of ideals stabilizes, $$\{0\}=\text{Ann(1)}\subset \text{Ann}(a^1)\subset \text{Ann}(a^2) \subset \dots \subset \text{Ann}(a^n) \subset \text{Ann}(a^{n+1}) \subset \dots$$ There exists a least nonnegative integer $n$ such that $\text{Ann}(a^n) = \text{Ann}(a^{n+1})$. Of course $n$ equals $0$ if and only if $a$ is a nonzerodivisor in $A$, and in that case $n$ is a zerodivisor in $A$ (since $n$ equals $0$). The following lemma proves that Moret-Bailly's examples above are definitely a positive characteristic phenomenon.

Lemma. If the integer $n$ is a nonzerodivisor in $A$, then for every $k$-derivation $\delta:A\to A$, $\delta(a)$ is a zerodivisor.

Proof. This is basically the same proof as in the existence of primary decompositions for Noetherian modules. Let $b$ be an element in $\text{Ann}(a^n)$ that is not in $\text{Ann}(a^{n-1})$. Since $a^nb$ equals $0$ in $A$, $-na^{n-1}b\delta(a)$ equals $a^n\delta(b)$. Since $a^nb$ equals $0$, $$a^{n+1}\delta(b) = -n(a^nb)\delta(a) = 0.$$ Since $\delta(b)$ is in $\text{Ann}(a^{n+1}),$ already $\delta(b)$ is in $\text{Ann}(a^n)$. Thus, $a^n\delta(b)$ equals $0$, i.e., $(-na^{n-1}b)\delta(a)$ equals $0$. By hypothesis, $a^{n-1}b$ is nonzero. Since also $n$ is a nonzerodivisor in $A$, $-na^{n-1}b$ is nonzero. Thus, $\delta(a)$ is a zerodivisor in $A$. QED

Let $r>0$ be an integer (the letter "d" is needed for differentials). Denote by $S_r$ the polynomial $k$-algebra $k[s_1,\dots,s_r]$.
Let $(a_1,\dots,a_r)\in A^r$ be a collection of elements, and let $f:S_r\to A$ be the unique $k$-algebra homomorphism with $f(s_i)=a_i$ for every $i=1,\dots,r$. There is an associated $A$-module homomorphism $f^*:\Omega_{S_r/k}\otimes_{S_r} A \to \Omega_{A/k}$.

Proposition. If $A$ contains $\mathbb{Q}$, and if there exists an $A$-linear retraction $\rho:\Omega_{A/k}\to \Omega_{S_r/k}\otimes_{S_r} A$ of $f^*$, then $(a_1,\dots,a_r)$ are a regular sequence. In that case, assuming also that $k$ is Noetherian and that $A$ is $k$-flat, then $f$ is a flat ring homomorphism.

Proof. This is proved by induction on $r$. First of all, composing the retraction $\rho$ with projection onto the summand $Ads_r$, there exists a $k$-derivation $\delta:A\to A$ with $\delta(a_r)$ equal to $1$. Thus, by the previous lemma, $a_r$ is a nonzerodivisor. When $r$ equals $1$, this proves that the sequence is a regular sequence. Assuming further that $k$ is Noetherian and $A$ is $k$-flat, by applying the Local Flatness Theorem, Theorem 22.2, p. 174 of Matsumura's "Commutative ring theory", also $k[t_1]\to A$ is flat. Thus, by way of induction, assume that $r>1$, and assume that the result is proved for $r-1$.

Form the quotient ring $\overline{A}=A/\langle a_r \rangle$ and the image sequence $(\overline{a}_1,\dots,\overline{a}_{r-1})\in \overline{A}^{r-1}$. Form the projection onto the first $r-1$ summands, $$\Omega_{S_r/k}\otimes_{S_r} A \to \Omega_{S_{r-1}/k}\otimes_{S_{r-1}} A \to \Omega_{S_{r-1}/k}\otimes_{S_{r-1}}\overline{A}.$$ Using the fundamental exact sequence, $$\overline{A}\xrightarrow{d(a_r)} \Omega_{A/k}\otimes_A \overline{A} \to \Omega_{\overline{A}/k} \to 0,$$ the projection above factors through a retraction, $$\overline{\rho}:\Omega_{\overline{A}/k} \to \Omega_{S_{r-1}/k}\otimes_{S_{r-1}}\overline{A}.$$ Thus, by the induction hypothesis, $(\overline{a}_1,\dots,\overline{a}_{r-1})$ is a regular sequence in $\overline{A}$. Thus, by definition, also $(a_1,\dots,a_r)$ is a regular sequence in $A^r$. Assuming further that $k$ is Noetherian and $A$ is flat, then $A/\langle a_r\rangle$ is flat over $S_r/\langle s_r \rangle = S_{r-1}$. Thus, by the Local Flatness Criterion once more, $A$ is flat over $S_r$. So the propostion is proved by induction on $r$. QED

For any rank $r$, free summand of $\Omega_{A/k}$, since $\Omega_{A/k}$ is generated by symbols $da$ for $a\in A$, at least up to passing to a finite Zariski open cover of $\text{Spec}(A)$, we can assume that there exist elements $a_1,\dots,a_r$ such that the retractions of $d(a_1),\dots,d(a_r)$ form a basis for the summand. Let $f$ be the associated ring homomorphism. By the proposition, $f$ is flat.

That is enough to conclude the local product structure, at least when the fibers of $f$ have isolated singularities (as in my proposed counterexample).

Hypothesis. Assume that $k$ is a field containing $\mathbb{Q}$, assume that $f$ is locally finitely presented, and assume that the closed fibers of $f$ have only isolated singularities.

Here is the e-mail I sent to a colleague about this situation.

Dear ***,

As soon as you left, I wrote out the computation for 4 concurrent lines, and I saw that you are (of course) correct: the extension class is zero if and only if the cross-ratio of the four lines is constant. Then I had to teach my class. I just finished, and I remembered another reference. That reference quickly leads to a proof of the formal local splitting whenever you have vanishing of the Ext class, at least in characteristic 0, at least for germs of isolated singularities. The reference is in the following book of Michael Artin.

Lectures on Deformations of Singularities
Volume 54 of mathematics: Tata Institute of Fundamental Research lectures
Michael Artin, C. S. Seshadri, Allen Tannenbaum
Tata Institute of Fundamental Research, 1976
http://www.math.tifr.res.in/~publ/ln/tifr54.pdf

In Theorem 8.1, Artin establishes that there is a versal deformation space $\text{Spf}(R)$ for a germ of an isolated singularity $(X,0)$ with $X$ locally finitely presented over a field. Earlier, on pp. 26, 27, in Theorem 6.2 and Proposition 6.1, Artin proves that for a flat, finitely presented morphism,

$$f : X \to \text{Spec}(A),$$

with a section,

$$s: \text{Spec}(A) \to X,$$

that is a family of isolated singularities of $f$, for the induced morphism to the versal deformation space,

$$g : \text{Spec}(A)^{\wedge} \to \text{Spf}(R),$$

the induced derivative is given by the map from the global Ext^1 you were describing to a natural target space T^1 introduced by Schlessinger. In particular, if your Ext class is zero, the derivative is zero.

Of course a nonzero morphism can have a zero derivative at one point. However, by generic smoothness / Sard's Theorem, in characteristic $0$ a $k$-morphism from a smooth $k$-scheme to an arbitrary (locally finite type) $k$-scheme is constant if the derivative map is everywhere zero. Perhaps you can find a counterexample in positive characteristic $p$ by considering the family of concurrent lines, $\text{Zero}(xy(y-x)(y-t^p x))$ in $\mathbb{A}^2 \times \mathbb{A}^1$ with coordinates $((x,y),t)$, where $\mathbb{A}^1$ is the base of the deformation.

For non-isolated singularities, I am not sure what is the best way to proceed. Often there is no versal deformation space (at least not as the formal completion of a finite type scheme at a point), but perhaps the proofs of Theorem 6.2 and Proposition 6.1 still apply. You may be able to find something in the book on deformations of singularities by Greuel, Lossen, and Shustin.

Jason Starr, April 2016.

Original answer. I am adding below my original (wrong) answer, as well as the edit I made right after posting the answer. I believe this is false. There are equisingular families that are not étale locally, nor even formally locally, trivial. Let $(z,w)$ be coordinates in $\mathbb{A}^2$. Let $t$ be a coordinate on some dense Zariski open subset $V$ of $\mathbb{A}^1$. Consider the zero scheme $X$ of $zw(w-z)(w-tz)$ in $\mathbb{A}^2\times V$. I believe that $\Omega_X$ will have a direct sum decomposition as above for $V$ a suitable open subset. However, I very much doubt there is any étale decomposition of $X$. If there were, then the fiber of the projection to $Y$ would correspond to the zero locus of $(z,w)$. However, the formal completion of $X$ along this zero scheme is not a product, not even after making a further étale base change of $V$.

Edit. I just did the computation for my polynomial above. There is not a direct sum decomposition of $\Omega_X$. I was wrongly assuming that the existence of a direct sum decomposition would be preserved under small deformations. Thus, deforming from a trivial family to an equisingular family would give a counterexample. However, because the Ext group is nonzero, existence of a direct sum decomposition is not preserved under small deformations.