Derived jet and arc spaces

Abstract: We study jet schemes and arc spaces in the context of derived algebraic geometry. Explicitly, we consider the jet and arc functors in the category of schemes and study their animations to the category of derived schemes -- what we call the derived jet and arc spaces. We show that the derived constructions agree with the classical versions when the base scheme is smooth, or more generally for local complete intersection log canonical singularities, giving a derived interpretation to a theorem of Mustaţă. For more singular spaces we get new singularity invariants in the form of higher homotopy groups. We also study cotangent complexes for derived jet and arc spaces, generalizing previous formulas for sheaves of differentials of classical jet and arc spaces. Several applications are obtained. Specifically, we revisit recent results on the local structure of arc spaces from the lens of cotangent complexes, giving more unified proofs and removing unnecessary hypotheses. In particular, we extend a version of Reguera's curve selection lemma for arc spaces to the case of non-perfect base fields.

• The paper provides a foundational theory for derived jet and arc spaces in derived algebraic geometry, demonstrating the representability of these functors and integrating them with truncation and étale localization, with precise results for derived arc spaces.
• Offers explicit Hasse-Schmidt presentation for jet equations and unveiled higher homotopical singularity invariants, using derived techniques to improve classical structural results for arc spaces.
• Shows that in characteristic zero over a field that is of finite type and lci, the scheme is log canonical if and only if jets derived are classical.

Overview

The paper develops the theory of jet schemes and arc spaces within derived algebraic geometry. Working over an arbitrary base ring $k$ and using simplicial algebras and animation (rather than differential graded algebras), Docampo, Miller, and Overton-Walker construct derived jet schemes $\mathbf{L}J_n(X)$ and a derived arc space $\mathbf{L}J_\infty(X)$ for any derived $k$-scheme $X$, prove that these represent the natural moduli functors of derived jets and arcs, compute their cotangent complexes, and apply the resulting machinery to sharpen several structural results on classical arc spaces—most notably removing perfectness hypotheses on the ground field from Reguera's curve selection lemma and related statements about embedding dimensions of local rings on arc spaces.

A recurring theme is that the derived constructions are invisible when $X$ is smooth but carry genuine higher homotopical information in the presence of singularities; the homotopy groups $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$ are proposed as new singularity invariants, computable via Koszul homology in the lci case.

Construction and representability

Two candidate definitions are compared. The first extends the functor of points to derived schemes:

$$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$

and analogously for arcs using animated power series $A_\bullet\llbracket t\rrbracket$. The second takes the animation $\mathbf{L}J_n$ of the classical jet functor $\mathbf{L}J_n(X)$0. The paper's first main theorem establishes that these coincide: $\mathbf{L}J_n(X)$1 corepresents $\mathbf{L}J_n(X)$2 for all $\mathbf{L}J_n(X)$3, and $\mathbf{L}J_n(X)$4 is naturally equivalent to both the homotopy limit $\mathbf{L}J_n(X)$5 and the limit of the functor-of-points description. The proof rests on a general result that animation preserves adjunctions under mild hypotheses ($\mathbf{L}J_n(X)$6 for compact projective generators), applied to the adjunction $\mathbf{L}J_n(X)$7 where $\mathbf{L}J_n(X)$8.

Three further foundational facts anchor the theory. First, truncation behaves as expected: $\mathbf{L}J_n(X)$9, so derived jets enrich rather than replace their classical counterparts. Second, étale localization holds: for formally étale maps $\mathbf{L}J_\infty(X)$0, one has $\mathbf{L}J_\infty(X)$1, which permits gluing along Zariski covers and yields the global objects. Third, the first derived jet scheme is the total space of the cotangent complex:

$\mathbf{L}J_\infty(X)$2

the expected animation of the classical identification $\mathbf{L}J_\infty(X)$3 with the tangent bundle.

Quasi-smoothness, lci singularities, and Mustaţă's theorem

For quasi-smooth schemes—affine-locally given by derived quotients $\mathbf{L}J_\infty(X)$4—the construction is fully explicit. The key computation is that

$\mathbf{L}J_\infty(X)$5

where $\mathbf{L}J_\infty(X)$6 denotes the $\mathbf{L}J_\infty(X)$7-th Hasse-Schmidt differential. Thus the familiar recipe for generating jet equations via Hasse-Schmidt differentials survives verbatim once quotients are replaced by derived quotients. Since derived quotients of classical rings have homotopy groups identified with Koszul homology, this gives an explicit presentation whenever $\mathbf{L}J_\infty(X)$8 is a regular sequence, i.e., whenever $\mathbf{L}J_\infty(X)$9 is lci.

Consequences follow immediately. Derived jet schemes of quasi-smooth schemes are quasi-smooth; if $k$0 is smooth then all $k$1 are classical; and for lci $k$2, the scheme $k$3 is classical exactly when $k$4 is lci of the expected dimension $k$5. The paper notes that the dimension hypothesis is genuinely needed in positive characteristic: for $k$6 with $k$7, the Hasse-Schmidt derivatives vanish and $k$8 is non-classical even though $k$9 is lci—a phenomenon impossible in characteristic zero.

Combining these observations with Mustaţă's theorem and its Mather–Jacobian refinement of de Fernex–Docampo, the paper obtains a derived interpretation of log canonicity: for reduced lci schemes of finite type over a field of characteristic zero, $X$0 is log MJ-canonical if and only if $X$1 is classical for every finite $X$2. The authors flag that the lci hypothesis is essential here and leave open what classicality of derived jets implies beyond it. They also record that an earlier unpublished attempt by Bouaziz contains claims contradicting their results (reduced complete intersections would be "weakly smooth" yet can have non-classical derived jets).

The cotangent complex formula

The central technical result is a derived upgrade of the main theorem of de Fernex–Docampo on differentials on arc spaces. For an animated algebra $X$3, define animated bimodules

$X$4

where $X$5 and $X$6 are the Hasse-Schmidt pre-duals. Then, globally,

$X$7

Taking $X$8 recovers the classical formula $X$9. The proof is conceptual: animated derivations on $X$0 are computed via Lurie's description of adjunctions on overcategories, reducing them to derivations on $X$1 valued in $X$2, followed by the derived tensor–Hom adjunction and the $X$3-Yoneda lemma. The relative version for a morphism $X$4 follows by comparing fundamental triangles, using naturality of the isomorphism.

Crucially, the formula cannot be de-derived. The paper proves that for a classical algebra $X$5, the equivalence $X$6 holds if and only if the derived jet space $X$7 is classical. Since many classical schemes (e.g., cones over plane curves of degree at least 4) have non-classical derived jets, no formula for the cotangent complex of the classical jet scheme is available in those cases—the obstruction being the homology of $X$8, which the authors do not analyze. This justifies the derived framework as a necessity rather than a convenience.

Fibers, cohomological support ideals, and higher Jacobian ideals

To control fibers of cotangent complexes—André–Quillen homology groups—the paper adapts Green–Lazarsfeld's cohomological support loci into a theory of cohomological support ideals $X$9 for pseudo-coherent complexes, defined via minors of the matrices in a minimal resolution. For level $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$0 these recover Fitting ideals. Applying this to $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$1 yields higher Jacobian ideals $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$2, defined for schemes essentially of finite type over $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$3.

Using Smith normal form over the PID $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$4, the pullback of a pseudo-coherent complex along an arc decomposes into free and torsion parts governed by Betti numbers $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$5 and invariant factors $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$6, and orders of contact with support ideals are expressed through sums of invariant factors. For a non-degenerate arc $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$7 (one whose generic point lies in the smooth locus), this yields precise dimension formulas: for instance,

$\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$8

with analogous formulas at each jet level and explicit kernel/cokernel dimensions for the maps induced by truncation. These computations feed directly into the applications below.

Cotangent maps and removal of perfectness hypotheses

The paper revisits results of Chiu–de Fernex–Docampo on the local structure of arc spaces, replacing arguments based on sheaves of differentials with arguments based on cotangent complexes. The key new device is an exact $\pi_i(\mathcal{O}_{\mathbf{L}J_n(X)})$9 grid of triangles relating $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$0, $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$1, $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$2 and their $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$3-analogues, whose ninth corner $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$4 plays the role of a relative cotangent complex. Taking homotopy produces an infinite grid in which each group appears multiple times ("entanglement"), so vanishing established in one location propagates elsewhere. The usual cotangent map $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$5 appears at level $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$6 of the third column.

With no assumption on the ground field, the paper proves:

Hypothesis on $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$7 at $$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$8	Conclusion
Unramified	$$\mathrm{dJet}_n^X(Z) = \operatorname{Maps}_{\mathrm{dSch}_k}(Z \times_k^\mathbf{L} \operatorname{Spec} k[t]/(t^{n+1}), X),$$9 surjective; $A_\bullet\llbracket t\rrbracket$0 finite separable
Smooth	$A_\bullet\llbracket t\rrbracket$1
Generically étale	$A_\bullet\llbracket t\rrbracket$2
Generically étale, $A_\bullet\llbracket t\rrbracket$3 smooth	Equality in the preceding bound
General linear projection $A_\bullet\llbracket t\rrbracket$4	$A_\bullet\llbracket t\rrbracket$5 an isomorphism; embedding dimensions equal

The last statement is proved via an explicit analysis of Jacobian matrices of linear projections, showing that a suitable open set of projections (defined over $A_\bullet\llbracket t\rrbracket$6, even when $A_\bullet\llbracket t\rrbracket$7 is not algebraically closed) makes $A_\bullet\llbracket t\rrbracket$8 unramified at $A_\bullet\llbracket t\rrbracket$9 while preserving the relevant order of contact. For generically étale $\mathbf{L}J_n$0—for example a resolution of singularities—the embedding-dimension bounds hold for all arcs, since arcs meeting the ramification locus are thin and hence have infinite embedding dimension. These results extend earlier versions that required perfect base fields, and the birational case yields a numerical analogue of the transformation rule of motivic integration.

Stable arcs and the curve selection lemma

The final application concerns Reguera's curve selection lemma, a finiteness statement on completions of local rings of arc spaces that underlies essentially all known approaches to the Nash problem. The paper works with a definition of stable arc suited to arbitrary fields: $\mathbf{L}J_n$1 is stable if it is non-degenerate and the extensions $\mathbf{L}J_n$2 are purely transcendental of degree $\mathbf{L}J_n$3 for $\mathbf{L}J_n$4; this agrees with the generic point characterization of irreducible constructible subsets not contained in the singular locus.

The main theorem states that for $\mathbf{L}J_n$5 essentially of finite type and generically smooth over any field $\mathbf{L}J_n$6, and any arc $\mathbf{L}J_n$7:

$\mathbf{L}J_n$8

and consequently stability, finiteness of jet codimension, finiteness of embedding dimension, and Noetherianity of the completed local ring $\mathbf{L}J_n$9 are all equivalent. Thin or degenerate arcs always have infinite embedding dimension. The proof combines the embedding-dimension bounds from the cotangent map section with a comparison of jet codimensions under general linear projections, together with the Denef–Loeser lemma (whose validity does not require characteristic zero). Corollaries include a bijection between stable arcs under proper birational maps, and the identification of maximal divisorial arcs as stable with

$\mathbf{L}J_n(X)$00

linking embedding dimensions to Mather discrepancies, again without perfectness assumptions.

Limitations and open questions

Several restrictions are acknowledged explicitly. The lci hypothesis in the characterization of log MJ-canonicity via classicality of derived jets is not removed, and the behavior of $\mathbf{L}J_n(X)$01 for singularities beyond lci is unexplored. The cotangent complex of the classical jet scheme remains inaccessible precisely when the derived jet scheme is non-classical; understanding the obstruction $\mathbf{L}J_n(X)$02 is posed as an open problem. On the global side, the universal arc has source a formal scheme rather than a scheme, and the module $\mathbf{L}J_n(X)$03 is a non-complete sheaf on a formal scheme; the authors work around this with certain non-coherent sheaves and note that a theory of derived formal schemes accommodating such modules is lacking. Finally, the higher homotopy groups of derived jet and arc spaces are introduced as singularity invariants but only computed via Koszul homology in the lci case; a systematic study of these invariants in general is deferred.

Conclusion

The paper supplies the foundational layer for jet and arc spaces in derived algebraic geometry over arbitrary base rings: representability of derived jet/arc functors, compatibility with truncation and étale localization, an explicit Hasse-Schmidt presentation for quasi-smooth inputs, and a cotangent complex formula that strictly subsumes the classical differential formulas and provably fails without derived techniques. Its most concrete payoff is the elimination of perfectness hypotheses from a body of results on arc spaces, culminating in Reguera's curve selection lemma over arbitrary fields. The framework also suggests, though does not yet deliver, a systematic theory of higher-homotopical singularity invariants beyond the lci case.