Jet closures and the local isomorphism problem

TL;DR

This paper investigates whether isomorphisms of all local jet schemes imply the original morphism is an isomorphism, introducing jet-based closures to analyze the problem and providing partial solutions with both negative and positive results.

Contribution

It introduces jet scheme-based closure operations and relates them to the local isomorphism problem, offering new insights and partial resolutions.

Findings

Negative answer in general cases

Positive results in specific geometric situations

New closure operations among ideals

Abstract

If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.

Equations

$$X_m^x:={\pi }_m^{-1}(x)$$

$$Z\mapsto {\mathrm{Hom}}_k(Z\times \mathrm{Spec}k[t]/(t^{m+1}),X).$$

$$Z\mapsto {\mathrm{Hom}}_k(Z\hat{\times }\mathrm{Spf}k[[t]],X).$$

$$X_m^x:={\pi }_m^{-1}(x).$$

$$\varphi :(Y,y)\to (X,x)$$

$${\varphi }_m^{\mathrm{loc}}:Y_m^y\to X_m^x$$

$${\mathfrak{a}}_m:=(D_i(f)\mid f\in \mathfrak{a},0\le i<m+1)\subset R_m$$

$${\mathfrak{a}}^{m\text{mathchar}45\mathrm{jc}}:=\{f\in R\mid (f{)}_m\subset {\mathfrak{a}}_m(\text{mod}\mathfrak{m}{R}_m)\}.$$

$${\mathfrak{a}}^{\mathrm{ac}}:={\mathfrak{a}}^{\infty \text{mathchar}45\mathrm{jc}}$$

$${\mu }_m:X_m\hat{\times }\mathrm{Spf}k[t]/(t^{m+1})\to X$$

$${\mu }_m^{\ast }:R\to R_m[t]/(t^{m+1}),f\mapsto \sum _{i=0}^m{D}_i(f)t^i.$$

$${\lambda }_m^{\ast }:R\to (R_m/\mathfrak{m}{R}_m)[t]/(t^{m+1}).$$

$${(0)}^{m\text{mathchar}45\mathrm{jc}}=\ker {\lambda }_m^{\ast }.$$

$${(0)}^{m\text{mathchar}45\mathrm{jc}}=\{f\in R\mid (f{)}_m=0(\text{mod}\mathfrak{m}{R}_m)\}=\ker {\lambda }_m^{\ast }.$$

$${\mathfrak{a}}^{m\text{mathchar}45\mathrm{jc}}=\{f\in R\mid V(f{)}_m^o\supset V(\mathfrak{a}{)}_m^o\}.$$

$${(0)}^{m\text{mathchar}45\mathrm{jc}}=\{f\in R\mid V(f{)}_m^o=X_m^o\},$$

$$\frac{R}{{\mathfrak{a}}^{m\text{mathchar}45\mathrm{jc}}}\cong \frac{R/\mathfrak{a}}{{((0)R/\mathfrak{a})}^{m\text{mathchar}45\mathrm{jc}}}.$$

$$\mathfrak{a}\subset {\mathfrak{a}}^{m\text{mathchar}45\mathrm{jc}}={({\mathfrak{a}}^{m\text{mathchar}45\mathrm{jc}})}^{m\text{mathchar}45\mathrm{jc}}.$$

$$x_1^2,2x_2{x}_1+y_1^2,2x_3{x}_1+x_2^2+3y_2{y}_1^2.$$

$$\bigcap _{m\in \mathbb{N}}{\mathfrak{a}}^{m\text{mathchar}45\mathrm{jc}}={\mathfrak{a}}^{\mathrm{ac}}.$$

$$\bigcap _{m\in \mathbb{N}}\ker ({\lambda }_m^{\ast })=\ker ({\lambda }_{\infty }^{\ast }).$$

$${\gamma }^{\ast }:R\to A[s]/(s^{m+1}).$$

$$A[s]/(s^{m+1})\to\big{(}A[s]/(s^{m+1})\big{)}[[t]],\quad s\mapsto st,$$

$$\alpha^{*}\colon R\to\big{(}A[s]/(s^{m+1})\big{)}[[t]].$$

$${\mathfrak{a}}^{m\text{mathchar}45\mathrm{jsc}}:=\{f\in R\mid (f{)}_m\subset \sqrt{{\mathfrak{a}}_m}(\text{mod}\mathfrak{m}{R}_m)\}.$$

$${\mathfrak{a}}^{\mathrm{asc}}:={\mathfrak{a}}^{\infty \text{mathchar}45\mathrm{jsc}}$$

$${(0)}^{m\text{mathchar}45\mathrm{jsc}}=\ker {\nu }_m^{\ast },$$

$${\nu }_m^{\ast }:R\to (R_m/\mathfrak{m}{R}_m{)}_{\mathrm{red}}[t]/(t^{m+1})$$

$$\mathfrak{a}\subset {\mathfrak{a}}^{m\text{mathchar}45\mathrm{jsc}}={({\mathfrak{a}}^{m\text{mathchar}45\mathrm{jsc}})}^{m\text{mathchar}45\mathrm{jsc}},$$

$${\mathfrak{a}}^{m\text{mathchar}45\mathrm{jsc}}=\{f\in R\mid (V(f{)}_m^o{)}_{\mathrm{red}}\supset (V(\mathfrak{a}{)}_m^o{)}_{\mathrm{red}}\}.$$

Full text

The research of the first author was partially supported by NSF Grant DMS-1402907 and NSF FRG Grant DMS-1265285. The research of the second author was partially supported by NSF Grant DMS-1501085. The research of the third author was partially supported by JSPS grant (C), 16K05089.

1. Introduction

In [Nas95], Nash related singularities of algebraic varieties to the geometry of arc spaces. Further studies of the structure of arc spaces were carried out by Denef and Loeser in connection to motivic integration [DL99], and their results were later applied to study singularities of pairs by Mustaţă [Mus02]. Several applications in birational geometry have been obtained since. In this paper, we investigate new connections of arc spaces in commutative ring theory.

Let $X$ be a scheme defined over a field $k$. For a nonegative integer $m$, let $X_{m}$ denote the $m$-jet scheme of $X$; when $m=\infty$, we have the arc space $X_{\infty}$, which we also call the $\infty$-jet scheme of $X$. Given a point $x\in X$, for every $m$ we denote by

[TABLE]

the fiber over $x$ of the natural projection $\pi_{m}\colon X_{m}\to X$. We call $X_{m}^{x}$ the scheme of local $m$-jets of $X$.

Let now $\phi\colon(Y,y)\to(X,x)$ be a morphism of germs of schemes over $k$. Suppose that for every $m\geq 0$ (including $m=\infty$) the induced morphism $\phi_{m}\colon Y_{m}\to X_{m}$ maps the scheme of local $m$-jets $Y_{m}^{y}$ of $Y$ isomorphically to the scheme of local $m$-jets $X_{m}^{x}$ of $X$. Does it follows that $\phi$ is an isomorphism? What if we assume that $\phi$ is a closed immersion?

We will refer to these questions as the local isomorphism problem and the embedded local isomorphism problem. Focusing on the local ring $\mathcal{O}_{X,x}$, the latter can be interpreted as asking whether there is a nonzero ideal in this ring which defines a subscheme of $X$ with same local jet schemes at $x$. This questions leads us to discover a new aspect to commutative ring theory by means of jet schemes.

Given any local $k$-algebra $R$, we introduce and study various closure operations among the ideals of $R$ which we call $m$-jet closures. These are defined for every $m\geq 0$, including $m=\infty$ where the operation is also called arc closure. For every $m$, the $m$-jet closure of an ideal $\mathfrak{a}\subset R$ is the largest ideal defining a scheme in $\operatorname{Spec}R$ whose scheme of local $m$-jets is equal to the one of the scheme defined by $\mathfrak{a}$. We also introduce a notion of $m$-jet support closure, where we put an analogous condition on the reduced scheme of local $m$-jets. For ideals of regular local rings, the intersection of all $m$-jet support closures of an ideal, for $m$ finite, is the same as the integral closure, but the two differ in general, with the first one giving a tighter closure operation.

One of the properties we establish is that the intersection of all $m$-jet closures of an ideal, for $m$ finite, is equal to the arc closure of the ideal. Using this, we prove that the embedded local isomorphism problem has a positive answer for a germ $(X,x)$ if and only if the zero ideal of $\mathcal{O}_{X,x}$ is arc closed.

This result prompts us to investigate which local $k$-algebras have the property that their zero ideals are arc closed. We find an example of a non-Noetherian ring whose zero ideal is not arc closed, and this implies that in general the above questions have a negative answer, even assuming that $\phi$ is a closed embedding. In the positive direction, we prove that the embedded local isomorphism problem has a positive answer for several classes of schemes, among which are all varieties and homogeneous schemes over a field.

Acknowledgements

The local isomorphism problem was posed by Herwig Hauser in discussion with the third author, when she visited him at University of Innsbruck in 2007. We are very grateful to Herwig Hauser for giving us the motivation to study jet closures. We also thank Kei-ichi Watanabe for useful discussions, Hiraku Kawanoue for pointing out a mistake in a preliminary version of the paper, and the referee for useful comments.

2. The local isomorphism problem

We work over a field $k$. In this paper, ${\mathbb{N}}$ denotes the set of nonnegative integers.

Let $X$ be an arbitrary scheme over $k$. For every $m\in{\mathbb{N}}$, we define the $m$-jet scheme of $X$ to be the scheme $X_{m}$ representing the functor from $k$-schemes to sets given by

[TABLE]

A point of $X_{m}$ is called an $m$-jet of $X$; it corresponds to a map $\operatorname{Spec}K[t](t^{m+1})\to X$ where $K$ is the residue field of the point. We will also consider $A$-valued $m$-jets for any $k$-algebra $A$; these are maps $\operatorname{Spec}A[t](t^{m+1})\to X$, and correspond to the $A$-valued points of $X_{m}$.

Truncations $k[t]/(t^{p+1})\to k[t]/(t^{m+1})$, defined for $p>m$, induce natural projections $X_{p}\to X_{m}$ which are affine morphisms. By taking the inverse limit, we define the arc space (or $\infty$-jet scheme) $X_{\infty}=\operatorname{\varprojlim}_{m}X_{m}$. This is the scheme representing the functor from $k$-schemes to sets given by

[TABLE]

A point of $X_{\infty}$ is called an arc (or $\infty$-jet) of $X$. It can be equivalently viewed as a map $\operatorname{Spf}K[[t]]\to X$ or a map $\operatorname{Spec}K[[t]]\to X$, where $K$ is the residue field of the point. Just like for jets, we will also consider $A$-valued arcs for any $k$-algebra $A$, which are maps $\operatorname{Spec}A[[t]]\to X$. If $X$ is affine (or quasi-compact and quasi-separated, see [Bha16]), then $A$-valued arcs correspond to $A$-valued points of $X_{\infty}$.

We denote by $\psi_{m}\colon X_{\infty}\to X_{m}$, $\pi_{m}\colon X_{m}\to X$, and $\pi=\pi_{\infty}\colon X_{\infty}\to X$ the natural projections. For ease of notation, it is often convenient to let $m$ range in ${\mathbb{N}}\cup\{\infty\}$ and denote $A[[t]]$, for a $k$-algebra $A$, also by $A[t]/(t^{\infty})$.

For more on jet schemes and arc spaces, we refer to [IK03, Voj07, EM09].

Given a point $x\in X$, for every $m\in{\mathbb{N}}\cup\{\infty\}$ we denote

[TABLE]

Definition 2.1**.**

We call $X_{m}^{x}$ the scheme of local $m$-jets of $X$ at $x$.

Roughly speaking, we would like to determine how much of the local structure of the scheme $X$ at the point $x$ is encoded in the schemes of local jets $X_{m}^{x}$.

A related question asks how much of the geometry of a scheme $X$ is encoded in its jet schemes $X_{m}$ for $m\geq 1$ (including $m=\infty$). This question was addressed in [IW10], where it is shown that the answer is both positive and negative depending on how one interprets the question. On the one hand, an example is given of two non-isomorphic schemes $X$ and $Y$ whose jet schemes $X_{m}$ and $Y_{m}$ are isomorphic for all $m\geq 1$ in a compatible way with respect to the truncation morphisms $X_{m+1}\to X_{m}$ and $Y_{m+1}\to Y_{m}$. On the other hand, if the isomorphisms $Y_{m}\to X_{m}$ are induced by a given morphism $\phi\colon Y\to X$, then it is proved that $\phi$ must be an isomorphism.

In view of the negative example in [IW10], we consider the following setting. Let

[TABLE]

is a $k$-morphism of germs of $k$-schemes. For every $m\in{\mathbb{N}}\cup\{\infty\}$, there is an induced morphism of scheme of local $m$-jets

[TABLE]

given by restriction of the natural map $\phi_{m}\colon Y_{m}\to X_{m}$.

Problem 2.2** (Local Isomorphism Problem).**

With the above notation, if $\phi_{m}^{\mathrm{loc}}$ is an isomorphism for every $m\in{\mathbb{N}}\cup\{\infty\}$, does it follow that $\phi$ is an isomorphism of germs?

Problem 2.3** (Embedded Local Isomorphism Problem).**

With the above notation, assume that $\phi$ is a closed immersion. If $\phi_{m}^{\mathrm{loc}}$ is an isomorphism for every $m\in{\mathbb{N}}\cup\{\infty\}$, does it follow that $\phi$ is an isomorphism of germs?

Remark 2.4*.*

Taking $m=0$, we see that the assumptions in either Problem 2.2 or Problem 2.3 include the condition that $\phi$ induces an isomorphism between the residue fields $k(x)$ of $x$ and $k(y)$ of $y$. We will identity these fields, and denote them by $K$. Note that, for all $m$, the maps $\phi_{m}^{\mathrm{loc}}\colon Y_{m}(y)\to X_{m}(x)$ are naturally $K$-morphisms (and hence $K$-isomorphisms under those assumptions).

Definition 2.5**.**

We say that a germ $(X,x)$ has the embedded local isomorphism property if Problem 2.3 has a positive answer for every closed immersion $\phi\colon(Y,y)\to(X,x)$. We also say that a germ $(X,x)$ has the local isomorphism property for a certain class of schemes if Problem 2.2 has a positive answer for every morphism $\phi\colon(Y,y)\to(X,x)$ with $(Y,y)$ in that class.

Proposition 2.6**.**

Assume that $(X,x)$ has the embedded local isomorphism property. Then $(X,x)$ has the local isomorphism property for all locally Noetherian $k$-scheme germs $(Y,y)$.

The proposition is an immediate consequence of the following property.

Lemma 2.7**.**

Let $\phi\colon(Y,y)\to(X,x)$ be a $k$-morphism of germs inducing an isomorphism of the residue fields at $x$ and $y$ (which we identify and denote by $K$). Assume that $Y$ is locally Noetherian at $y$ and $\phi_{1}^{\mathrm{loc}}\colon Y_{1}^{y}\to X_{1}^{x}$ is a $K$-isomorphism. Then $\phi$ is a closed immersion of germs.

Proof.

By hypothesis, the differential of $\phi$ induces an isomorphism of $K$-vector spaces between the Zariski tangent spaces $\mathfrak{m}_{Y,y}/\mathfrak{m}_{Y,y}^{2}$ of $Y$ at $y$ and $\mathfrak{m}_{X,x}/\mathfrak{m}_{X,x}^{2}$ of $X$ at $x$. This implies that the homomorphism $\phi^{*}\colon\mathcal{O}_{X,x}\to\mathcal{O}_{Y,y}$ corresponding to $\phi$ induces a a surjection $\mathfrak{m}_{X,x}\to\mathfrak{m}_{Y,y}$. Indeed, we have $\mathfrak{m}_{Y,y}=\phi^{*}(\mathfrak{m}_{X,x})+\mathfrak{m}_{Y,y}^{2}$ by the isomorphism of the Zariski tangent spaces, and therefore we have $\mathfrak{m}_{Y,y}^{i}=\phi^{*}(\mathfrak{m}_{X,x}^{i})+\mathfrak{m}_{Y,y}^{i+1}$ for $i\geq 1$. By successive substitutions, we obtain $\mathfrak{m}_{Y,y}=\phi^{*}(\mathfrak{m}_{X,x})+\mathfrak{m}_{Y,y}^{n}$ for all $n\geq 1$. Since $Y$ is Noetherian at $y$, this implies that $\mathfrak{m}_{Y,y}=\phi^{*}(\mathfrak{m}_{X,x})$. By this and the isomorphism between the residue fields, we conclude that $\phi^{*}\colon\mathcal{O}_{X,x}\to\mathcal{O}_{Y,y}$ is surjective. ∎

When restricting to the embedded setting, we often denote the germ by $(X,o)$. Later in Section 5, we shall reinterpret the embedded local isomorphism problem from an algebraic point of view, relating it to certain “jet theoretic” notions of closure of ideals in local rings.

3. Jet closures

Throughout this section, let $(R,\mathfrak{m})$ be a local algebra over a field $k$, and let $X=\operatorname{Spec}R$. We denote by $o\in X$ the closed point.

For every $m\in{\mathbb{N}}\cup\{\infty\}$, let $R_{m}$ be the ring of regular functions of the $m$-jet scheme $X_{m}$, so that $X_{m}=\operatorname{Spec}R_{m}$. For an ideal $\mathfrak{a}\subset R$ and $m\in{\mathbb{N}}\cup\{\infty\}$, we denote

[TABLE]

the ideal generated by the Hasse–Schmidt derivations $D_{i}(f)$ of the elements $f$ of $I$. If $V(\mathfrak{a})\subset X$ is the subscheme defined by $\mathfrak{a}$, then its $m$-jet scheme $V(\mathfrak{a})_{m}$ is the subscheme of $X_{m}$ defined by $\mathfrak{a}_{m}$.

Definition 3.1**.**

For any ideal $\mathfrak{a}\subset R$ and any $m\in{\mathbb{N}}\cup\{\infty\}$, we define the $m$-jet closure of $\mathfrak{a}$ to be

[TABLE]

For $m=\infty$, we also call the ideal

[TABLE]

the arc closure of $\mathfrak{a}$.

Unless otherwise stated, in the following statements we let $m$ be an arbitrary element of ${\mathbb{N}}\cup\{\infty\}$ We start with the following property which implies that the $m$-jet closure an ideal $\mathfrak{a}$ is intrinsic, that is, only depends on the quotient ring $R/\mathfrak{a}$ and hence on the scheme $\operatorname{Spec}R/\mathfrak{a}$, and not by the embedding of $\operatorname{Spec}R/\mathfrak{a}$ in $\operatorname{Spec}R$.

Lemma 3.2**.**

The $m$-jet closure ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ of an ideal $\mathfrak{a}\subset R$ is the inverse image via the quotient map $R\to R/\mathfrak{a}$ of the $m$-jet closure of the zero ideal of $R/\mathfrak{a}$.

Proof.

Let $f\in R$ be an element and let $\overline{f}$ denote the class of $f$ in $R/\mathfrak{a}$. Note that the image of $(f)_{m}$ in $R_{m}/\mathfrak{a}_{m}$ is equal to the ideal $(\overline{f})_{m}$. The condition that $(f)_{m}\subset\mathfrak{a}_{m}$ modulo $\mathfrak{m}R_{m}$ is equivalent to the condition that $(\overline{f})_{m}=0$ modulo $\mathfrak{m}R_{m}/\mathfrak{a}_{m}$. It follows that $f\in{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ if and only if $\overline{f}$ is in the $m$-jet closure of the zero ideal of $R/\mathfrak{a}$. ∎

In view of this fact, in order to study properties of $m$-jet closures we can reduce to the case of the zero ideal $(0)\subset R$, after replacing $R$ by $R/\mathfrak{a}$. In this special case, the $m$-jet closure can be expressed in terms of the universal $m$-jet homomorphism.

Recall that the universal $m$-jet

[TABLE]

is defined by the ring homomorphism

[TABLE]

We denote by $\lambda_{m}$ the restriction of $\mu_{m}$ to the scheme of local $m$-jets $X_{m}^{o}$, and consider the corresponding ring homomorphism

[TABLE]

Lemma 3.3**.**

We have

[TABLE]

Proof.

It suffices to observe that, by definition,

[TABLE]

∎

The next property provides a geometric characterization of $m$-jet closures.

Proposition 3.4**.**

The $m$-jet closure ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ of an ideal $\mathfrak{a}\subset R$ is the largest ideal $\mathfrak{b}\subset R$ such that $V(\mathfrak{b})_{m}^{o}=V(\mathfrak{a})_{m}^{o}$.

Proof.

By 3.2, it suffices to prove the proposition for the zero ideal $(0)$ of $R$. The fact that ${(0)}^{m\mathchar 45\relax\mathrm{jc}}$ is an ideal is clear by 3.3. The second assertion follows by the geometric reinterpretation of the definition of $m$-jet closure:

[TABLE]

In the case of the zero ideal, we have

[TABLE]

and this implies the assertion. ∎

Remark 3.5*.*

Now that we know that the $m$-jet closure of an ideal is itself an ideal, we can rephrase their intrinsic property by saying that, for every ideal $\mathfrak{a}\subset R$, there is a canonical isomorphism of $k$-algebras

[TABLE]

The next corollary implies that the $m$-jet closure is indeed a closure operation.

Corollary 3.6**.**

For any ideal $\mathfrak{a}\subset R$ we have

[TABLE]

Proof.

Both the inclusion $\mathfrak{a}\subset{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ and the equality ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}={({\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}})}^{m\mathchar 45\relax\mathrm{jc}}$ are immediate consequences of 3.4. ∎

Definition 3.7**.**

We say that an ideal $\mathfrak{a}\subset R$ is $m$-jet closed if $\mathfrak{a}={\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$. When $m=\infty$ (where the condition can be written as $\mathfrak{a}={\mathfrak{a}}^{\mathrm{ac}}$), we also say that $\mathfrak{a}$ is arc closed.

Corollary 3.8**.**

For two ideals $\mathfrak{a},\mathfrak{b}\subset R$ we have ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}={\mathfrak{b}}^{m\mathchar 45\relax\mathrm{jc}}$ if and only if $V(\mathfrak{a})_{m}^{o}=V(\mathfrak{b})_{m}^{o}$.

Proof.

This is also an immediate consequences of 3.4. ∎

Proposition 3.9**.**

For any ideal $\mathfrak{a}\subset R$ and any $m\in{\mathbb{N}}$, we have $\mathfrak{a}+\mathfrak{m}^{m+1}\subset{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$.

Proof.

By 3.2, it suffices to prove the case where $\mathfrak{a}=(0)\subset R$. Recall that ${(0)}^{m\mathchar 45\relax\mathrm{jc}}=\ker\lambda_{m}^{*}$ by 3.3. Since $\lambda_{m}^{*}(\mathfrak{m})\subset(t)$, we have $\lambda_{m}^{*}(\mathfrak{m}^{m+1})\subset(t^{m+1})=0$, and therefore $\mathfrak{m}^{m+1}\subset\ker\lambda_{m}^{*}={(0)}^{m\mathchar 45\relax\mathrm{jc}}$. ∎

Remark 3.10*.*

3.9 implies in particular that even in nice situations (e.g., assuming that $R$ is a Noetherian $k$-algebra) the $m$-jet closure operation on ideals is, for finite $m$, a nontrivial operation. For instance, if $\mathfrak{a}$ is not $\mathfrak{m}$-primary, then $\mathfrak{a}\subsetneq{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ for all $m\in{\mathbb{N}}$.

The following is an example where $\mathfrak{a}+\mathfrak{m}^{m+1}\neq{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$.

Example 3.11**.**

Let $\mathfrak{a}=(x^{2}+y^{3})\subset R=k[[x,y]]$, where $k$ is a field of characteristic $\neq 2,3$. A direct computation shows that ${\mathfrak{a}}^{{4}\mathchar 45\relax\mathrm{jc}}=(x^{2}+y^{3},xy^{3})$, and hence $\mathfrak{a}+\mathfrak{m}^{5}\neq{\mathfrak{a}}^{{4}\mathchar 45\relax\mathrm{jc}}$. To see this, we introduce the coordinates $x_{i}:=D_{i}(x)$ and $y_{i}:=D_{i}(y)$ on the jet schemes of $X=\operatorname{Spec}R$, where $(D_{i})_{i\geq 0}$ is the sequence of universal Hasse–Schmidt derivations. Denoting by $o\in X$ the closed point, we have $X_{4}^{o}=\operatorname{Spec}k[x_{i},y_{i}\mid 1\leq i\leq 4]$, and the ideal defining $V(\mathfrak{a})_{4}^{o}$ in there is generated by the three elements

[TABLE]

One can check that $x_{1}y_{1}^{3}$ is in this ideal, and this implies that $xy^{3}\in{\mathfrak{a}}^{{4}\mathchar 45\relax\mathrm{jc}}$. On the other hand, $xy^{3}\not\in\mathfrak{a}+\mathfrak{m}^{5}$.

Recall that we defined the arc closure to be ${\mathfrak{a}}^{\mathrm{ac}}:={\mathfrak{a}}^{{\infty}\mathchar 45\relax\mathrm{jc}}$. The following proposition shows how the arc closure compares to the other $m$-jet closures.

Proposition 3.12**.**

For any ideal $\mathfrak{a}\subset R$, we have

[TABLE]

Proof.

By 3.2, we reduce to prove the formula when $\mathfrak{a}$ is the zero ideal of $R$, where by 3.3 the proposition is equivalent to the assertion that

[TABLE]

To show one inclusion, let $f\in R$ be an element such that $\lambda_{\infty}^{*}(f)\neq 0$ in $(R_{\infty}/\mathfrak{m}R_{\infty})[[t]]$. Then the image $\lambda_{\infty}^{*}(f)$ in $(R_{\infty}/\mathfrak{m}R_{\infty})[t]/(t^{m+1})$ is nonzero for all sufficiently large integers $m$. Since, for any such $m$, the resulting map $R\to(R_{\infty}/\mathfrak{m}R_{\infty})[t]/(t^{m+1})$ factors through $(R_{m}/\mathfrak{m}R_{m})[t]/(t^{m+1})$ by the universality of $\lambda_{m}^{*}$, it follows that $\lambda_{m}^{*}(f)\neq 0$, and hence we have $f\not\in\bigcap_{m\in{\mathbb{N}}}\ker(\lambda_{m}^{*})$.

For the reverse inclusion, assume that $f\not\in\ker(\lambda_{m}^{*})$ for some $m\in{\mathbb{N}}$. Then there exists an $A$-valued $m$-jet $\gamma\colon\operatorname{Spec}A[s]/(s^{m+1})\to\operatorname{Spec}R$, for some $k$-algebra $A$, such that $f$ is not in the kernel of the homomorphism

[TABLE]

By further composing with the homomorphism

[TABLE]

which is injective, we obtain an $A[s]/(s^{m+1})$-valued arc \alpha\colon\operatorname{Spec}\big{(}A[s]/(s^{m+1})\big{)}[[t]]\to\operatorname{Spec}R, and $f$ is not in the kernel of

[TABLE]

This implies that $f\not\in\ker(\lambda_{\infty}^{*})$. ∎

4. Jet support closures

As in the previous section, let $(R,\mathfrak{m})$ be a local algebra over a field $k$, let $X=\operatorname{Spec}R$, and let $o\in X$ be the closed point. By looking at the local jet schemes of $(X,o)$ with their reduced structure, we introduce the following variant of the $m$-jet closure.

Definition 4.1**.**

For any ideal $\mathfrak{a}\subset R$ and any $m\in{\mathbb{N}}\cup\{\infty\}$, we define the $m$-jet support closure of $\mathfrak{a}$ to be

[TABLE]

For $m=\infty$, we also call the ideal

[TABLE]

the arc support closure of $\mathfrak{a}$.

The $m$-jet support closure operation satisfies many analogous properties of the $m$-jet closure operation studied in the previous section that can be proven in a similar way.

For instance, the $m$-jet support closure is intrinsic. That is, for any ideal $\mathfrak{a}\subset R$, the $m$-jet support closure ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ is the inverse images via the quotient map $R\to R/\mathfrak{a}$ of the $m$-jet support closure of the zero ideal of $R/\mathfrak{a}$.

Furthermore, the $m$-jet support closure of the zero ideal of $R$ is given by

[TABLE]

where

[TABLE]

is the ring homomorphism associated with the restriction $\nu_{m}$ of the universal $m$-jet morphism $\mu_{m}$ to the reduced scheme of local $m$-jets $(X_{m}^{o})_{\mathrm{red}}$.

It follows from these two properties that for any ideal $\mathfrak{a}\subset R$ the $m$-jet support closure ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jsc}}$ is an ideal. In general, we have

[TABLE]

which says that the $m$-jet support closure is a closure operation among the ideals of $R$.

Definition 4.2**.**

We say that an ideal $\mathfrak{a}\subset R$ is $m$-jet support closed if $\mathfrak{a}={\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jsc}}$. When $m=\infty$ (where the condition can be written as $\mathfrak{a}={\mathfrak{a}}^{\mathrm{asc}}$), we also say that $\mathfrak{a}$ is arc support closed.

The following comparison between $m$-jet support closure and $m$-jet closure is clear from the definitions.

Proposition 4.3**.**

For every ideal $\mathfrak{a}\subset R$, we have ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}\subset{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jsc}}$.

Geometrically, we have

[TABLE]

In the case of the zero ideal, we have

[TABLE]

Using these facts, the same argument as in the proof of 3.4 gives the following property.

Proposition 4.4**.**

The $m$-jet support closure ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jsc}}$ of an ideal $\mathfrak{a}\subset R$ is the largest ideal $\mathfrak{b}\subset R$ such that $(V(\mathfrak{b})_{m}^{o})_{\mathrm{red}}=(V(\mathfrak{a})_{m}^{o})_{\mathrm{red}}$.

Corollary 4.5**.**

For two ideals $\mathfrak{a},\mathfrak{b}\subset R$, we have ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jsc}}={\mathfrak{b}}^{m\mathchar 45\relax\mathrm{jsc}}$ if and only if $(V(\mathfrak{a})_{m}^{o})_{\mathrm{red}}=(V(\mathfrak{b})_{m}^{o})_{\mathrm{red}}$.

There is one proof from the previous section that does not have an analogue for $m$-jet support closure, and that is the proof of 3.12. For this reason, it is convenient to give the following definition.

Definition 4.6**.**

The jet support closure of an ideal $\mathfrak{a}\subset R$ is the ideal

[TABLE]

The fact that this is a closure operation is proven in the next proposition.

Proposition 4.7**.**

For every ideal $\mathfrak{a}\subset R$, we have

[TABLE]

Proof.

The inclusion $\mathfrak{a}\subset{\mathfrak{a}}^{\mathrm{jsc}}$ is clear from the definition. Regarding the equality, we have

[TABLE]

and since ${\mathfrak{a}}^{\mathrm{jsc}}\subset{({\mathfrak{a}}^{\mathrm{jsc}})}^{\mathrm{jsc}}$, the two ideals are the same. ∎

Definition 4.8**.**

We say that an ideal $\mathfrak{a}\subset R$ is jet support closed if $\mathfrak{a}={\mathfrak{a}}^{\mathrm{jsc}}$.

Since all $m$-jet support closures are intrinsic, it follows that so is the jet support closure. In particular, we have the following property.

Proposition 4.9**.**

An ideal $\mathfrak{a}\subset R$ is jet support closed if and only if the zero ideal of $R/\mathfrak{a}$ is jet support closed.

An adaptation of the first part of the proof of 3.12 gives the following comparison between jet support closure and arc support closure. We do not know whether the reverse inclusion holds.

Proposition 4.10**.**

For any ideal $\mathfrak{a}\subset R$, we have ${\mathfrak{a}}^{\mathrm{jsc}}\subset{\mathfrak{a}}^{\mathrm{asc}}$.

The jet support closure compares to the arc closure as follows.

Proposition 4.11**.**

For any ideal $\mathfrak{a}\subset R$, we have ${\mathfrak{a}}^{\mathrm{ac}}\subset{\mathfrak{a}}^{\mathrm{jsc}}$.

Proof.

By 4.3, we have

[TABLE]

and hence the assertion follows from 3.12. ∎

Proposition 4.12**.**

Let $R$ be a local integral domain essentially of finite type over a field $k$. Let $\mathfrak{a}\subset R$ be an ideal, and denote by $\overline{\mathfrak{a}}$ its integral closure.

1. (a)

There is an inclusion ${\mathfrak{a}}^{\mathrm{jsc}}\subset\overline{\mathfrak{a}}$. 2. (b)

If $R$ is regular, then ${\mathfrak{a}}^{\mathrm{jsc}}=\overline{\mathfrak{a}}$.

Proof.

First, we claim that

[TABLE]

for every arc $\alpha\in X_{\infty}^{o}$. Suppose this is not the case for some $\alpha$. Since $\mathfrak{a}\subset{\mathfrak{a}}^{\mathrm{jsc}}$, we must have $\operatorname{ord}_{\alpha}({\mathfrak{a}}^{\mathrm{jsc}})<\operatorname{ord}_{\alpha}(\mathfrak{a})$. Setting $m=\operatorname{ord}_{\alpha}({\mathfrak{a}}^{\mathrm{jsc}})$, this means that there is an element $f\in{\mathfrak{a}}^{\mathrm{jsc}}$ such that $\operatorname{ord}_{\alpha}(f)=m<\operatorname{ord}_{\alpha}(\mathfrak{a})$. This implies that $\psi_{m}(\alpha)\in(V(\mathfrak{a})_{m-1}^{o})_{\mathrm{red}}\setminus(V(f)_{m-1}^{o})_{\mathrm{red}}$. The contradiction then follows by 4.4, after we observe that $f\in{\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jsc}}$. This proves the claim.

Every divisorial valuation $v$ on $X=\operatorname{Spec}R$ determines an arc $\alpha_{v}\colon\operatorname{Spec}k_{v}[[t]]\to X$ given by

[TABLE]

Here $\mathcal{O}_{v}$ is the valuation ring of $v$, $k_{v}=\mathcal{O}_{v}/\mathfrak{m}_{v}$ is the residue field, $\widehat{\mathcal{O}}_{v}$ is the $\mathfrak{m}_{v}$-adic completion, and the isomorphism $\widehat{\mathcal{O}}_{v}\simeq k_{v}[[t]]$ is given by Cohen Structure Theorem. Since for every divisorial valuation $v$ we have $\operatorname{ord}_{\alpha_{v}}=v$, Eq. 1 implies that $v({\mathfrak{a}}^{\mathrm{jsc}})=v(\mathfrak{a})$ for every divisorial valuation $v$. It follows that $\overline{{\mathfrak{a}}^{\mathrm{jsc}}}=\overline{\mathfrak{a}}$ by the valuative characterization of integral closure, and hence we have ${\mathfrak{a}}^{\mathrm{jsc}}\subset\overline{\mathfrak{a}}$. This proves (a).

Assume now that $R$ is regular. For any ideal $\mathfrak{b}\subset R$ and any $m\in{\mathbb{N}}$, denote

[TABLE]

Note that

[TABLE]

We have

[TABLE]

Since $X$ is smooth, the projections $\psi_{m}\colon X_{\infty}\to X_{m}$ are surjective, and therefore we have $(V(\mathfrak{a})_{m})_{\mathrm{red}}=(V(\overline{\mathfrak{a}})_{m})_{\mathrm{red}}$ for all $m\in{\mathbb{N}}$. By restricting to the reduced fibers over $o\in X$, we see that

[TABLE]

and therefore we have ${\mathfrak{a}}^{\mathrm{jsc}}={(\overline{\mathfrak{a}})}^{\mathrm{jsc}}$ by 4.5 and the definition of jet support closure. On the other hand, part (a) applied to $\overline{\mathfrak{a}}$ implies that ${(\overline{\mathfrak{a}})}^{\mathrm{jsc}}=\overline{\mathfrak{a}}$. Therefore we have ${\mathfrak{a}}^{\mathrm{jsc}}=\overline{\mathfrak{a}}$, which proves (b). ∎

The following example shows that in general, if $R$ is not regular, the jet support closure is a tighter operation than integral closure.

Example 4.13**.**

Let $R=k[[x,y,z]]/(x^{2}+y^{2}+z^{2})$ and $\mathfrak{a}=(x,y)\subset R$ (with slight abuse of notation, we still denote by $x,y,z$ the classes of these elements in $R$). Since $z$ is integral over $\mathfrak{a}$, we have $\overline{\mathfrak{a}}=(x,y,z)$. On the other hand, note that $R/\mathfrak{a}\cong k[[x]]/(x^{2})$. Since $(x^{2})$ is integrally closed in the ring $k[[x]]$, it is also jet support closed $k[[x]]$. It follows by applying 4.9 twice that $\mathfrak{a}$ is jet support closed in $R$.

5. The embedded local isomorphism problem revisited

Our first result of this section provides a characterization of the embedded local isomorphism problem in terms of jet closures.

Proposition 5.1**.**

Let $(R,\mathfrak{m})$ be a local $k$-algebra, and let $X=\operatorname{Spec}R$ with closed point $o\in X$. Then the germ $(X,o)$ has the embedded local isomorphism property if and only if the zero ideal of $R$ is arc closed.

Proof.

If $\phi\colon(Y,o)\hookrightarrow(X,o)$ is closed immersion of $k$-scheme germs and $I_{Y}\subset R$ is the ideal of $Y$, then for any $m\in{\mathbb{N}}\cup\{\infty\}$ the map $\phi_{m}^{\mathrm{loc}}\colon Y_{m}^{o}\to X_{m}^{o}$ is an isomorphism if and only if $I_{Y}\subset{(0)}^{m\mathchar 45\relax\mathrm{jc}}$. This implies that $(X,o)$ has the embedded local isomorphism property if and only if

[TABLE]

By 3.12, this last condition is equivalent to the condition that ${(0)}^{\mathrm{ac}}=(0)$. ∎

Remark 5.2*.*

What 5.1 says is that Problem 2.3 would not change if the condition that $\phi_{m}^{\mathrm{loc}}\colon Y_{m}^{o}\to X_{m}^{o}$ is an isomorphism for all $m\in{\mathbb{N}}\cup\{\infty\}$ is replaced with the weaker requirement that just $\phi_{\infty}^{\mathrm{loc}}\colon Y_{\infty}^{o}\to X_{\infty}^{o}$ is an isomorphism.

Remark 5.3*.*

In view of 3.12, 5.1 can equivalently be stated by saying that $(X,o)$ has the embedded local isomorphism property if and only if

[TABLE]

In the formulation of Problem 2.3, this means that it is equivalent to only assume that $\phi_{m}^{\mathrm{loc}}\colon Y_{m}^{o}\to X_{m}^{o}$ is an isomorphism for all $m\in{\mathbb{N}}$, instead of assuming it for all $m\in{\mathbb{N}}\cup\{\infty\}$.

In view of 5.1, it is natural to ask if there are general conditions that guarantee that ideals are arc closed. Note that, by 3.2, it suffices to look at the case of zero ideals of local $k$-algebras. We have already observed that ideals are not typically $m$-jet closed if $m\in{\mathbb{N}}$, even in a Noetherian setting; see 3.10. Our first result in this direction shows that, in general, the arc closure is a nontrivial operation.

Proposition 5.4**.**

There exists a local $k$-algebra $R$ whose zero ideal is not arc closed.

Proof.

Let $R=k[[x_{1},x_{2},\dots]]$ be the power series ring in infinite countably many variables, and let $\mathfrak{m}$ be its maximal ideal. Then the ideal

[TABLE]

is not $m$-jet closed for any $m\in{\mathbb{N}}\cup\{\infty\}$. In fact, the element $x_{1}\in R$, which is not in $\mathfrak{a}$, belong to ${\mathfrak{a}}^{m\mathchar 45\relax\mathrm{jc}}$ for all $m\in{\mathbb{N}}\cup\{\infty\}$. Indeed, for $m\in{\mathbb{N}}$ we have

[TABLE]

where the inclusion follows by 3.9, and hence

[TABLE]

where the equality follows by 3.12. This shows that $\mathfrak{a}\neq{\mathfrak{a}}^{\mathrm{ac}}$. We conclude by 3.2 that the zero ideal of $R/\mathfrak{a}$ is not arc closed. ∎

Corollary 5.5**.**

There exist germs $(X,o)$ of $k$-schemes that do not have the embedded local isomorphism property.

Proof.

An example is given by $X=\operatorname{Spec}R/\mathfrak{a}$ where $R$ and $\mathfrak{a}$ are as in the proof of 5.4. The fact that $(X,o)$ does not have the embedded local isomorphism property follows by 5.1. More explicitly, setting $Y=\operatorname{Spec}R/(\mathfrak{a}+(x_{1}))$, the inclusion of germs $\phi\colon(Y,o)\to(X,o)$ gives a counterexample to the embedded local isomorphism property. ∎

The example given in the proof of 5.4 is not Noetherian. We do not know whether in the Noetherian setting every ideal is arc closed.

Problem 5.6**.**

If $R$ is a Noetherian local $k$-algebra, is every ideal $\mathfrak{a}$ of $R$ arc closed?

In the positive direction, we have the following results.

Definition 5.7**.**

We say that local $k$-algebra $(R,\mathfrak{m})$ is graded if $R=\bigoplus_{n\geq 0}R_{n}$ is a graded algebra with maximal ideal $\mathfrak{m}=\bigoplus_{n\geq 1}R_{n}$. We say that a germ $(X,o)$ of a $k$-scheme is homogeneous if $\mathcal{O}_{X,o}$ is a graded local $k$-algebra.

Theorem 5.8**.**

The zero ideal of $(R,\mathfrak{m})$ is arc closed in the following cases:

1. (a)

$R$* is a graded local $k$-algebra;* 2. (b)

$R$* is a reduced Noetherian local algebra essentially of finite type over $k$;* 3. (c)

$R=S/(f)$* where $S$ is a regular ring essentially of finite type over $k$ and $f\in S$.*

Proof.

We first prove (a). By 3.3, we reduce to check that the homomorphism $\lambda_{\infty}^{*}\colon R\to(R_{\infty}/\mathfrak{m}R_{\infty})[[t]]$ is injective. Consider the $R$-valued arc $\alpha$ defined by the homomorphism

[TABLE]

given on homogeneous elements $g_{n}\in R_{n}$ by $\alpha^{*}(g_{n})=g_{n}t^{n}$. Since $\alpha^{*}(\mathfrak{m})\subset(t)$, $\alpha^{*}$ factors through $\lambda_{\infty}^{*}$ by the universal property. Since $\alpha^{*}$ is injective, it follows that $\lambda_{\infty}^{*}$ is injective too.

Regarding (b) and (c), we observe that in both cases we can write $R\simeq S/\mathfrak{b}$ where $S$ is a regular ring essentially of finite type over $k$ and $\mathfrak{b}$ is an integrally closed ideal of $S$. In this case we know that $\mathfrak{b}$ is jet support closed by 4.12, and hence arc closed by 4.11. Then we conclude by 3.2 that the zero ideal of $R$ is arc closed. ∎

Corollary 5.9**.**

The following germs of $k$-schemes have the embedded local isomorphism property:

1. (a)

homogeneous germs; 2. (b)

germs of reduced schemes of finite type over $k$; 3. (c)

germs of hypersurfaces in smooth $k$-varieties.

Proof.

By 5.1 and 5.8. ∎

In the geometric setting, Problem 5.6 leads to the following question.

Problem 5.10**.**

Do germs of Noetherian $k$-schemes have the embedded local isomorphism property?

This is one of those situations in which either a positive or negative solution would be a positive outcome. If the problem is affirmatively solved, then it would imply that, in the Noetherian setting, the local jet-schemes determine completely the germ of the singularity, which is an interesting property of jet schemes. On the other hand, if the problem is negatively solved, then it would mean that even in the setting of ideals of regular Noetherian rings, there exists a geometrically meaningful non-trivial closure operation that is tighter than integral closure.