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Weierstrass preparation theorem and singularities in the space of non-degenerate arcs

We study the Weierstrass preparation and division theorems over arbitrary test rings and the local structure of singularities in the space of nondegenerate arcs on algebraic varieties. As an application, we prove a strengthened version of a theorem of Grinberg-Kazhdan and Drinfeld.


Introduction
It has been long expected that there exists a deep connection between singularities of certain arc spaces and harmonic analysis over nonarchimedean fields. For instance, certain functions appearing naturally in harmonic analysis can be interpreted as the function attached to the trace of the Frobenius operators on what should be the stalks of the intersection complex of certain arc spaces, see [1]. However, a proper foundation of a theory of perverse sheaves on arc spaces is still missing even though a recent work of Bouthier and Kazhdan [2] outlines a strategy for setting it up.
As the theory of perverse sheaves is originally built for schemes of finite type, the basic difficulty in extending it to arc spaces is that those spaces are almost always infinite dimensional. The first inroad into this new territory is made by Grinberg and Kazhdan who prove that the formal completion of the arc spaces at a point representing a non-degenerate arc is isomorphic to the formal completion of a scheme of finite type, augmented by infinitely many free formal variables, under the assumption that the base field is the field of complex numbers. This result is later improved by Drinfeld who prove it over an arbitrary base field.
Let us fix the notations in order to state Grinberg-Kazhdan-Drinfeld's theorem. Let k be a field. Let X be an affine k-scheme of finite type. For every n ∈ , we consider the space of nth jets on X representing the functor R → n X (R) = X (R[t]/t n+1 ) on the categories of k-algebras. The arc space of X is the limit of n X as n → ∞: (1.1) Introduction dimension of X . If X is not smooth, the situation is much more complicated: the transition morphism n+1 X → n X is neither smooth nor surjective. Let X ′ be a smooth open subscheme of X and Z a closed subscheme of X complement of X ′ . We are mainly interested in the open subscheme of non-degenerate arcs whose k-points form the set The definition of non-degenerate arc space depends thus on the choice of a smooth open subscheme X ′ of X . Although we may of take X ′ to be the smooth locus of X , it is often more convenient to keep the possibility of choosing X ′ smaller than the smooth locus of X . For instance, the case of the affine line X = 1 and X ′ = m will be of special interest for the study of the Weierstrass preparation theorem. Some cares are in order to enunciate the functorial description of • X .
Definition 1.1. The non-degenerate arc space of X relative to the smooth subscheme X ′ is the functor on the category of k-algebras which attaches to each k-algebra R the set • X (R) consisting of maps x : Spec(R[[t]]) → X such that the projection x −1 (X ′ ) → Spec(R) is surjective.
We can now state the theorem of Grinberg-Kazhdan and Drinfeld, [3], [4]. Grinberg-Kazhdan-Drinfeld's theorem gives the hope that a reasonable theory of perverse sheaves on formal arc spaces may exist. In [1], it is proven that in a weak sense the formal finite dimensional model is independent of choices i.e ifŶ y andŶ ′ y ′ are finite type formal models ofX x ( X ) x ≃Ŷ y×D ∞ ≃Ŷ ′ y ′×D ∞ then there exists integers m, n ∈ such that there exists an isomorphism Introduction function although we don't know yet to define the intersection complex on the space of nondegenerate arcs. If x ∈ X (k) is a non-degerate arc on X , andŶ y is a finite-dimensional formal as in 1.2, then we set IC X (x) = IC Y ( y).
The existence of the isomorphism (1.2) implies that IC Y ( y) = IC Y ′ ( y ′ ) so that this number doesn't depend on the choice of the finite type formal model as argued in [1,Prop. 1.2].
In order to define perverse sheaves on the space of non-degenerate arcs, one may hope a stronger version of the Drinfeld-Grinberg-Kazhdan theorem in which formal completions are replaced by strict Henselizations for instance. In other words, instead of formal charts as in 1.2, one would like to construct Henselian charts. There are good reasons to believe that the analogue of Grinberg-Kazhdan theorem for henselizations doesn't hold.
In [2], Bouthier and Kazhdan attempt to construct certain type of coverings of the space of non-degenerate arcs which admit essentially smooth surjective map to schemes of finite type. The strategy of [2] consists in generalizing Drinfeld's construction in [4], which works over local Artinian test rings, to general test rings. There are unfortunately some gaps in the construction of [2].
Both constructions [4] and [2] rely in the Weierstrass preparation theorem and the Newton method of solving algebraic equation by approximation. We will study with some care the Weierstrass preparation and division theorem, and the Newton method over a general test rings. The study of the Weierstrass division theorem reveals that the coordinate ring of the arc space of the curve of equation x y = 0 contains non-zero functions which vanish at all points to the infinite order. We observe that these functions are annihilated in completed local rings but generally not in henselizations. We also observe that if X is a k-scheme of finite type, henselizations of the ring of coordinates X × ∞ don't contain non-zero elements which vanish at all points to the infinite order.
Nevertheless, following the method of [4] and [2], one can obtain a theorem on local structure of non-degenerate arc space. Although the description of the local structure is not very pleasant, one can derive from it the following statement. Theorem 1.3. Let X be a k-scheme locally of complete intersection. Let x ∈ X (k) be a non-degenerate arc. There exist a k-scheme Y of finite type, a point y ∈ Y (k), a morphism φ : Y → X mapping y to x such that for every field k ′ containing k, for every point y ′ ∈ Y (k ′ ) mapping to x ′ ∈ X (k ′ ), there exists an isomorphism Schemes of finite type as in the above statement should be thought of as good slices in non-degenerate arc spaces. One may hope to construct a reasonable theory of perverse sheaves on arc spaces using these slices. For instance, one may think of the hypothetical intersection complex IC X of X as the unique object whose restriction to every slice Y of finite type is the intersection complex IC Y of Y up to a shift by the dimension.

Formal completion and tensor product
Although it may be possible to drop the complete intersection assumption, in this paper, I want, as much as possible, to stick with the computational approach of [4]. One should also note that in many moduli problems, slices are given to us by replacing maps from the formal disc to a given target scheme, or algebraic stack, by maps from a given curve. This idea is in fact behind the calculations in the complete intersection case as in [4]. Drinfeld explains how one can approach the general case by means of the Newton groupoid in the Geometric Langlands Seminar in Spring 2017.

Formal completion and tensor product
There is a potential danger in dealing with formal series with coefficients in a varying commutative ring R for the operation of tensor product doesn't commute with the operation of completion. The purpose of this section is to provide warnings by means of some examples where what we observe isn't quite what we would naively expect. Our examples are all related to the problem of Weierstrass division.
For a commutative ring R, an ideal I of R, and an R-module M , we define the I-completion of M to be the limitM For instance, the p-completion of the ring of integers is the ring p of p-adic integers. At this place, it may be necessaty to clear some confusion in terminology: although the field p of p-adic numbers, defined as the quotient field of p , is p-adically complete in the sense that all Cauchy sequences in p for the p-adic distance converge, the p-completion of p is zero.
For every commutative ring R, the ring of formal series R [[t]] consists in set of all series of the form x = x 0 + x 1 t + x 2 t 2 + · · · where x 0 , x 1 , . . . are arbitrary elements subjected to usual rules of addition and multiplication. It contains the ring R[t] of polynomials consisting of x = x 0 + x 1 t + x 2 t 2 + · · · with coefficients x 0 , x 1 , . . . equal to 0 except for finitely many of them. We may also define R [[t]] as the t-completion of R[t]: As unproblematic as this definition may appear, some good cares are in order. If R is a commutative ring R, I is an ideal of R, M is a R-module, for an R-algebra R ′ , the canonical morphismM As an abelian group, R is canonically isomorphic with . In fact, we may see R as the [t]module on which t acts as the multiplication by p.
, then on the one hand we haveM I ⊗ R R ′ = p , but on the other hand the t-completion of [t]/(t − p) is zero as p ∈ × . In particular, in this example the map (2.1) is surjective but not injective.
The following lemma won't be used in the sequel except for a side observation, but it may serve as yet another warning. ]. This kernel consists in formal series x = x 0 + x 1 t + · · · such that x n ∈ I for all n. If a 1 , . . . , a m is a system of generators of I, then for every n ∈ , the coefficient x n of x can be written possibly non uniquely in the form x n = a 1 y n,1 +· · ·+a m y n,m . If we choose such an expression for each n ∈ then we have x = a 1 y 1 + · · · + a m y m where y i is the formal series y i = y i,0 + y i,1 t + · · · . It follows that x belongs to the ideal of R [[t]] generated by I.
The assertion doesn't hold without the assumption that I is finitely generated. Let R = k[x 0 , x 1 , x 2 , . . .] be the ring of polynomials in infinitely many variables x 0 , x 1 , x 2 , . . . with coefficients in a field k, I the ideal of R generated by x 0 , x 1 , x 2 , . . .,R = R/I = k. Then the formal series x 0 + x 1 t + x 2 t 2 + · · · , lying in the kernel of R

Variations on the Weierstrass preparation theorem
If f is a nonzero germ of a holomorphic function at the origin in n+1 , given with coordinates t, x 1 , . . . , x n , the classical Weierstrass preparation theorem asserts the existence of a germ of holomorphic function u at the origin in n+1 non vanishing at the origin and a polynomial where q 0 , . . . , q d−1 are germs of holomorphic functions at the origin n , and vanishing at the origin, such that f = uq.
Let R n = [[x 1 , . . . , x n ]] be the ring of formal series of variables x 1 , . . . , x n . Let f ∈ R n [[t]] be a formal series of variables t, x 1 , . . . , x n . We denotef ∈ [[t]] the reduction of f modulo x 1 , . . . , x n and writef = t dū where u ∈ [[t]] × . By usual manipulations with formal series, we can prove that there exist a unique formal series u ∈ R n [[t]] × with reductionū ∈ [[t]] × and a unique polynomial q ∈ R n [t] of the form (3.1) whose coefficients q 0 , . . . , q d−1 ∈ R n are formal series vanishing constant coefficients such that f = uq. The holomorphic content of the Weierstrass preparation theorem consists in showing that if f is absolutely convergent in a neighborhood of 0 in n+1 then q 0 , . . . , q d−1 are also absolutely convergent in a neighborhood of 0 in n .
Let A n denote the ring of germs of holomorphic functions defined in a neighborhood of 0 in n . A n is a Henselian local ring whose completion with respect to the maximal ideal is can be factored uniquely in the form f = uq with u ∈ R n [[t]] × and q ∈ R n [t] a Weierstrass polynomial as above but there is no guarantee that the coefficients q 0 , . . . , q d−1 belong to A n unless f is assumed to be a convergent series. We draw the conclusion that the existence of Weierstrass factorization for is a Weierstrass polynomial doesn't hold generally for Henselian local ring A as opposed to complete local rings. The aim of this section is to study the existence of the Weierstrass factorization over a general test ring and its connection with the geometry of the arc space of the affine line.
The arc space 1 of the affine line 1 is spectrum of the ring of polynomials with countably infinite number of variables: for a R-point x ∈ 1 (R) is uniquely represented by a formal series x = x 0 + x 1 t + · · · for every k-algebra R. We consider the space • 1 of non-degenerate arcs with respect to the open subset m of 1 . On the level of k-points we have The description of set of R-points of • 1 is as follows: is the ring of Laurent formal series with coefficients in R. In other words, we have an inclusion ∩ R((t)) × is closely related to the affine Grassmannian of m defined as the the sheafification of the functor R → R((t)) × /R[[t]] × . Although our purpose is not to review the theory of affine Grassmannian, we will borrow some familiar arguments from that theory helping to clarify the relation between the properties of being invertible in R((t)) × and the existence of a strict Weierstrass factorization. (3.5) We will prove that the equality holds for all commutative rings R. If these conditions are satisfied, then the strict Weierstrass factorization of x is unique.
. It follows that t n = x( y t n ) ∈ (x).

It will also follow that R[[t]]/xR[[t]] is a R-module of finite type as a quotient of the R-module of finite type R[[t]]/t n R[[t]]
.
with exact horizontal lines. Since x ∈ R((t)) × , the middle vertical map is an isomorphism.
The snake lemma provides us with an exact sequence of R-modules

It follows that R((t))/R[[t]] is flat as R-module and thus R[[t]]/xR[[t]] is also flat as R-module. Now we prove that if R[[t]]/xR[[t]]
is a projective R-module of finite rank d and if t n ∈ (x) for some n ∈ , then x has a strict Weierstrass factorization. We consider the exact sequence of R-modules where (x) and (t n ) are ideals of R [[t]] generated by x and t n respectively. We know that

R((t))/t n R[[t]] and R((t))/xR[[t]]
are flat R-modules of finite type. It follows that (x)/(t n ) is a flat R-module. Being generated by x, x t, . . . , x t n−1 , it is a R-module of finite type, and therefore (x)/(t n ) is a projective module of finite type. Let q ∈ R[t] denote the characteristic polynomial of the R-linear operator t acting on

R[[t]]/xR[[t]]. By the Cayley-Hamilton theorem, the multiplication by q in R[[t]]/(x) is
null. This implies that q ∈ (x) or in other words x divides q.
We claim that q divides t n . Indeed if q ′ is the characteristic polynomial of t acting on the projective R-module of finite type (x)/(t n ), then we have t n = qq ′ by the exact sequence (3.9). Now we consider morphisms of R-modules: is also known to be projective R-modules of rank d by assumption. Since q divides t n , t acts nilpotently on R[t]/q. It follows that the map ] × and q is a monic polynomial dividing t n thus x has a strict Weierstrass factorization.
] has a strict Weierstrass factorization x = uq with u ∈ R[[t]] × and q is a monic polynomial dividing t n for some n ∈ and we prove that Finally if x = uq is a strict Weierstrass factorization then the linear maps (3.10) are isomophisms of free R-modules. It follows that q is the characteristic polynomial of t acting on R[[t]]/(x). Thus the strict Weierstrass factorization of x is unique if it exists.
The assertion can now be reduced to the obvious equality For every prime ideal p of R, we will denote ν p : R → k p the canonical homomorphism from R to the fraction field ] is nonzero. In this case, we will denote ord p (x) the vanishing order of the formal series

Proposition 3.5. Let x ∈ R[[t]] be a non-degenerate formal series with coefficients in R.
We consider the two assertions: (1) x has a strict Weierstrass factorization; (2) the function p → ord p (x) is a locally constant function on Spec(R).
For an arbitrary k-algebra R, (1) implies (2). The converse statement that (2) implies (1) holds under the assumption that Nil(R) n = 0 for some n ∈ , Nil(R) being the nilradical of R. In particular if R is either reduced or Noetherian then (1) and (2) are equivalent.
Proof. Assume that x has a strict Weierstrass factorization i.e. x ∈ β+ 1 (R), then by Cor.

3.4, R[[t]]/(x)
is a projective R-module of finite type compatible with base change. In particular, for every prime ideal p we have ] be a formal series with coefficients in R such that the function p → ord p (x) is locally constant on Spec(R). We will prove that x has a strict Weierstrass factorization. We may assume that Spec(R) is connected and ord p (x) = d for all p ∈ Spec(R). Let us consider the coefficients x 0 , x 1 , . . . of the formal It follows that x i belongs to Nil(R) the intersection of all prime ideals of R. We recall that Nil(R) can also be characterized as the ideal of all nilpotent elements of R.
] with x d ∈R × obviously has a strict Weierstrass factorization. Under the assumption Nil(R) n = 0 for some n ∈ N , we deduce that x has a strict Weierstrass factorization in virtue of Prop. 3.6 that follows.

Prop, 3.5 provides a handy way to check whether a formal series in x ∈ R[[t]] has a strict Weierstrass factorization by calculating the function
We also observe that both statement and proof of the above proposistion can be simplified if R is a k-algebra of finite type. For a k-algebras of finite type, we may replace in the statement of Prop, 3.5 Spec(R) by the space Spm(R) of maximal ideals For maximal ideals, the base change formula (3.11) holds in virtue of Lemma 2.1 so that we don't need to appeal to Prop. 3.3. This assures that (1) implies (2). The converse statement holds because for a k-algebra of finite type R, the intersection of all maximal ideals, its Jacobson radical Jac(R), is equal to the intersection of all prime ideals, its nilradical. Proposition 3.6. The morphism β+ 1 → • 1 is surjective and formally étale: for every kalgebra R, every nilpotent ideal I of R i.e. I n = 0 for some n ∈ , In particular, if R is an Artinian local k-algebra then In other words, every non-degenerate formal series x ∈ • 1 (R) has a unique strict Weierstrass factorization under the assumption that R is an Artinian local ring .
Since I n = 0, we have I((t)) n = 0. It follows that the homomorphism R((t)) →R((t)) induces a bijection between their spectra. For an element in a ring is invertible if it doesn't belong to any maximal ideal, we deduce that x ∈ R((t)) is invertible if and only its reduction x ∈R((t)) is invertible. Proof. The uniqueness of the strict Weierstrass factorization implies that We have a map Q which derives from the latter by base change is also a closed immersion.
We have thus proved that Q 0 d is a strict colimit of Artinian schemes based at the point t d of Q d . It follows that there is a canonical morphism of formal schemes Q 0 d →Q d . It follows from Prop. 3.6 that the morphism Q 0 d →Q d is an isomorphism.
To summarize, the property of having a strict Weierstrass factorization is controlled by the morphism γ : which induces isomorphism on k-points. We have seen that γ is formally étale. On the negative side γ is not schematic because otherwise β+ 1 would be schematic and it isn't according to its explicit description in Prop. 3.7.
We will now study the geometry related to the notion of non-strict Weierstrass factorization. The natural action of m on 1 induces on the level of arc spaces an action of m on 1 . For every d ∈ we have a morphism 1 The union of those open subschemes is the space • 1 of non-degenerate arcs. We will consider the affine subspace Q d of 1 defined by the equations x d = 1 and x d+1 = x d+2 = · · · = 0. The action of m on 1 induces a morphism given by β(q, u) = uq. By definition a formal series x ∈ 1 (R) has Weierstrass factorization if it lies in the image β D (Q d (R) × m (R)) for some d ∈ . We will see that as opposed to the morphism γ of (3.12), the Weierstrass morphism β d of (3.13) is not formally étale but only formally smooth in a weak sense. On the other hand, it has the advantage of being schematic. Proposition 3.8. To simplify notation, we will write X = Q d × m and Y = ≤d 1 . The morphism β d : X → Y is surjective on k-points. Moreover, for every x ∈ X (k) mapping on y ∈ Y (k), the induced mapping on formal completionsX x →Ŷ y is formally smooth.
Proof. The surjectivity on k-points is obvious. To prove that the induced mapping on formal completionsX x →Ŷ y is formally smooth is equivalent to prove the usual lifting properties for Artinian local algebras: for R an Artinian algebra, I a ideal of R,R = R/I, for everyx ∈ X (R) mapping toȳ ∈ Y (R), if y ∈ Y (R) reducing toȳ modulo I then there exists x ∈ X (R) mapping to y ∈ Y (R) and reducing tox ∈ X (R) mod I.
Let y ∈ Y (R) be a non degenerate formal series with coefficients in an Artinian local ring R. By Prop. 3.6, there exists a unique strict Weierstrass factorization y = q 0 u 0 where u 0 ∈ R[[t]] × and q 0 is a monic polynomial of some degree e, q 0 dividing t n for some n ∈ .
The pointx ∈ X (R) mapping toȳ ∈ Y (R) corresponds to a Weierstrass factorizationȳ = qū ofȳ, its reduction modulo I whereq is a monic polynomial of degree d with coefficients inR andū ∈R[[t]] × . By reducing y = u 0 q 0 modulo I we have the unique strict Weierstrass factorizationȳ =ū 0q0 ofȳ that may be different from the Weierstrass factorizationȳ =ūq given byx ∈ X (R).
where µ n and ν n are projective limits of systems of smooth surjective maps with smooth surjective transition morphisms. For every x n ∈ X n (k) mapping to y n ∈ Y n (k), there exists x ∈ X (k) over x n mapping to y ∈ Y (k) over y n . We have a commutative diagram of formal completionsX xŶŷ X n,x nŶ n, y n β d µ nνn β d,n (3.16) whereβ d ,μ n ,ν n are known to be formally smooth. It follows thatβ d,n is also formally smooth. Since X n and Y n are k-schemes of finite type, we can derive from the smoothness of β d,n from the formal smoothness of induced morphism on formal completions of points.
Here we prove that a certain morphism between finite order jet spaces, which are of finite type, is smooth by proving that the morphism on the level of arc spaces is formally smooth in a weak sense. It is also possible to prove directly the smoothness of the morphism between jet spaces by calculation on their tangent spaces as we will see later in the study of the Weierstrass division theorem, see the proof of Prop. 4.1.
Let us draw two consequences of the smoothness of β d,n .
Proof. This is the Henselian lifting property for smooth morphisms.
Corollary 3.11. The morphism β d : Q d × m → ≤d 1 is a projective limit of a system of smooth morphisms with possibly non-smooth transition morphisms.
Proof. By base change we obtain smooth morphisms that form a projective system whose limit if β d . One should pay a special attention to the fact that the transition maps in this system are not smooth. Indeed, on finite levels, we have a commutative diagram Based on the discussion on the property of the Weierstrass morphism β of (3.13), we will introduce a new type of morphisms which seems to be useful in dealing with arc spaces.

Definition 3.12. A morphism f : X → Y of affine k-schemes is said to be of Weierstrass type if it satisfies the following properties:
• f is a filtered limit of a projective system of smooth morphisms in which transition maps are not required to be smooth; • for every k-point x ∈ X (k) with image y ∈ Y (k), the induced morphism on formal com-pletionsX x →Ŷ y is formally smooth.
If the codomain is the point Y = Spec(k), we will say that X is a k-scheme of Weierstrass type.
Arbitrary base changes of a morphism of Weierstrass type are of Weierstrass type. Composition of morphisms of Weierstrass type is also of the same type. As examples, the Weierstrass maps α and β of (3.13) is of Weierstrass type.
We will introduce a much stronger notion: Definition 3.13.
• A morphism f : X → Y is said to be essentially smooth surjective if it is the filtered limit of a projective system of smooth surjective morphism in which transition maps are also smooth surjective.
• A k-scheme X is said to be essentially smooth if the structural morphism f : X → Spec(k) is essentially smooth surjective.

• A k-scheme X is said to be essentially of finite type if there exists an essentially smooth surjective morphism f : X → S where S is a k-scheme of finite type.
A k-scheme essentially of finite type is of course also of Weierstrass type. The converse is wrong. There are much more k-schemes of Weierstrass type than k-schemes essentially of finite type. We will prove an useful criterion to recognize a morphism of Weierstrass type.
If we denote x i and y j the images of x and y in X i and Y j respectively then we have f i, j (x i ) = y j . We consider the diagram of formal completionŝ wheref x ,μ i,x ,ν j, y are formally smooth. It follows thatf i, j,x i is also formally smooth. For X i and Y j are k-schemes of finite type, the morphism f i, j : X i → Y j is smooth at x i . Since X → X i is surjective, this implies that f i, j : X i → Y j is smooth. Now f : X → Y can be realized as the projective limit of smooth morphisms X i × Y j Y → Y as j → ∞.

Variations on the Weierstrass division theorem
In the classical setting, the Weierstrass division theorem asserts that if f is a germ of holomorphic functions at the origin of n+1 given with coordinates x 1 , . . . , x n , t, q is a polynomial of variable t of degree d of the form q = q 0 + q 1 t + · · · q d−1 t d−1 + t d where q 0 , . . . , q d−1 are germs of holomorphic functions at the origin of n equipped with coordinates x 1 , . . . , x n and vanishing at the origin, then there exist an unique expression f = qh + a where h is a germ of holomorphic function at the origin of n+1 and a is a polynomial of variable t of degree less than d whose coefficients are germs of holomorphic functions at the origin of n .
In this section we aim to study the Weierstrass division problem when we replace the ring of germs of holomorphic functions at the origin of n by an arbitrary k-algebra R, f by a formal series with coefficients in R, and q ∈ R[t] a monic polynomial of degree d.
Recall that we have denoted Q d the d-dimensional affine space representing the functor R → Q d (R) of all monic polynomials of degree d with coefficients in R. Let us now denote A d the d-dimensional affine space representing the functor R → A d (R) of all polynomials of degree less than d. The Weierstrass division problem for a varying test ring R consists in studying the morphism given by the formula: α d (q, a, v) = (q, qv + a).
In a sense, the morphism α d is the derivative of the morphism β d of (3.13). The morphism β d : X → Y of (3.13) where X = Q d × m and Y = • 1 induces a morphism of tangent bundles which is a point of Q d × m × 1 . We obtain α d from dβ d by setting the parameter u to be 1. Proof. Because β d is a morphism of Weierstrass type, so is its derivative dβ d . By base change, α d is also a morphism of Weierstrass type. The same argument applies for the property of being surjective on k-points. Over the point q = t d , β d induces an isomorphism on formal completions of points, so are dβ d and α d .
One could also reverse the argument and prove first that α d induces a smooth morphism at finite levels. Indeed, for every n ≥ d, the morphism is a linear morphism of finite rank vector bundles over A d . It is enough to prove that α d,n induces a surjective linear morphism in every fiber. It is enough to prove that for a monic polynomial q ∈ Q d (k) of degree d with coefficients in k, the k-linear map given by the formula (4.1) is surjective. We will prove that the k-linear map given by the formula (4.1) is surjective. This is equivalent to saying that the map is surjective. This map factors as

t]/(q) is an isomorphism by the Euclidean division theorem. The map k[t]/(q) → k[[t]]/(q) is surjective as k[[t]]/(q) = k[t]/(t e )
where e is the multiplicity of t dividing q.
We will now exhibit a strange family of finite-dimensional schemes of Weierstrass type. For every d ∈ , we denote S d the kernel of morphism α d of (4.1), in other words, we have the Cartesian diagram where 0 is the zero section q → (q, 0) ∈ A d × 1 . The scheme S d is a generalized vector bundle over Q d in the sense that its fiber over each point q ∈ Q d (k) is a k-vector space. The dimension of the fibers varies however in the opposite direction of what happens with a morphism of finite type. Indeed, for every monic polynomial q ∈ Q d (k) of degree d, the fiber S d,q (k) of S d over q is a k-vector space of dimension d − e where e is order of vanishing of q at 0. In particular, if q(0) = 0 then dim S d,q (k) = d, and if q = t d then S d,q (k) = 0.
Let us compute explicitly the ring of coordinates of S 1 . We will write Q 1 = Spec(k[q 0 ]) with the universal monic polynomial of degree one q = q 0 + t ∈ Q 1 (k[q 0 ]). We will denote A 1 = Spec(k[a 0 ]) the space of polynomial of degree less than one and 1 = k[x 0 , x 1 , . . .]. The scheme S 1 is defined as the closed subscheme of Q 1 × A 1 × 1 defined by the system of equations In other words, we have S 1 = Spec(R 1 ) where (4.7) The fiber of S 1 over a point with coordinate q 0 = 0 in Q 1 is the affine line since the coordinate a 0 will determine other coordinates by x 0 = −a −1 a 0 , x 1 = −a −1 x 0 ... On the other hand, the fiber of S 1 over the point q 0 = 0 in Q 1 is reduced to the point of coordinates a 0 = 0 and x 0 = x 1 = · · · = 0. We may look at S 1 from three different angles. First, from the equations defining S 1 we see that S 1 is a projective limit of affine planes, in other words its ring of coordinates R 1 is an inductive limit R 1 = lim −→ A n with A n = k[q 0 , x n ] with transition maps A n → A n+1 given by x n → −q 0 x n+1 . The morphism Spec(A n+1 ) → Spec(A n ) can be realized as the blow-up of the affine plane 2 = Spec(A n ) at the origin (q 0 , x n ) = (0, 0) with the strict transform of the x n -axis deleted.
Second, set theoretically speaking, S 1 looks like a "sawed plane" i.e. the plane of coordinates (q, x) with the x-axis (0, x) deleted and the origin (0, 0) added back in. This constructible subset of the plane can be thus realized as the set of points of a scheme which is of course not of finite type.
Third, the sawed plane S 1 may be seen in some sense as the glueing of the plane (q, x) with the x-axis deleted and with the q-axis (q, 0). Indeed, if A is a k-algebra of finite type and φ : R 1 → A be a A-point of S 1 , mapping some prime ideal p A of A to the prime ideal of R 1 generated by p 0 = (q 0 , a 0 , x 0 , . . .) corresponding to the origin (0, 0) of S 1 , then the morphism Spec(A) → S 1 must factor through the q-axis. Indeed, under this assumption we have φ(q 0 ) ∈ p A . Since the variables a 0 , x 0 , . . . are divisible by every power of q 0 , we have φ(a 0 ), φ(x 0 ), . . . ∈ p n A for every n ∈ . Since A is a k-algebra of finite type, it follows that φ(a 0 ) = φ(x 0 ) = · · · = 0.
Let us identify the q-axis with the subscheme of S 1 defined by the equations a 0 = x 0 = x 1 = · · · = 0. We can show that that the completion of S 1 at origin is isomorphic to the completion of the q-axis at the origin defined by the maximal ideal q of R 1 . Indeed, for every n, we have a 0 , x 0 , x 1 , . . . ∈ q n and therefore R 1 /p n 0 = k[q 0 ]/q n 0 .

The completion of S d at the point (t d , 0) is isomorphic to the completion of the zero section at the same point.
Proof. The proof proceeds in the same way as the calculations in the case d = 1 that we have just outlined.
We will now consider another family of strange schemes springing out of Weierstrass division problem. For every d ∈ , we consider the homomorphism defined by (q, x) → (q, q x), and define Z d to be the kernel of µ d . For every k-algebra R, the set Z d (R) consists of pairs (q, x) where q ∈ R[t] is a monic polynomial of degree d and x ∈ R[[t]] a formal series with coefficients in R satisfying q x = 0. We may also define Z d as the closed subscheme of S d consisting of triples (q, a, x) ∈ Q d × A d × 1 with a = 0. Set theoretically, there is no difference between Z d and the zero section Q d × {0} as for every q ∈ Q d (k), the multiplication by q is injective in k [[t]]. These spaces are however different as schemes, in other words Z d is a infinitesimal thickening of Let us consider again the case d = 1 and describe explicitly the coordinate ring of Z 1 . Using the equations (4.7) of S 1 , we have Z 1 = Spec(V 1 ) where (4.9) The algebra V 1 can be realized as the inductive limit of V 1,n = k[q 0 , x n ]/(q n+1 0 x n ) with respect to transition maps V 1,n → V 1,n+1 given by x n → −q 0 x n+1 . We note that the transition maps V 1,n → V 1,n+1 are all injective so that x n = 0 in V 1 . The scheme Z 1 is therefore the projective limit of schemes Z 1,n = Spec(V 1,n ) which topologically is a union of the q 0 -axis and the x naxis as n → ∞. We see that in the limit, the x n -axis all contract topologically to the origin although they don't algebraically in the sense that x n = 0. The ideal of V 1 generated by x 0 , x 1 , . . . defined the closed subscheme Q 1 × {0} of Z 1 .  which consists of all nilpotent elements of R. In particular, if R is reduced then both nil(R) and nil ∞ (R) = 0. Second, as we can determine whether an element x ∈ R belongs to the nilradical by evaluating it via the homomorphisms ν p : R → R/p for all prime ideal p ∈ Spec(R), we can also determine whether an element x ∈ R belongs to the infinite nilradical by evaluating it via homomorphisms ν p : R → R/p n for all p ∈ Spec(R) and n ∈ . In particular, nil ∞ (R) = 0 if and only if for every non-zero element x ∈ R, there exists a ring homomorphism ν : R → A from R to an Artinian local ring such that ν(x) = 0. Proof. Let {p 1 , . . . , p r } denote the set of associated primes of R. By the primary decomposition theorem [5, Thm. 6.8], there exist primary ideals I 1 , . . . , I r with I i = p i such that I 1 ∩ · · · ∩ I r = 0. As I i = p i , for every i, there exists m i such that p m i i ⊂ I i . It follows that nil ∞ (R) ⊂ I 1 ∩ · · · ∩ I r and therefore nil ∞ (R) = 0.
Here is an example of commutative ring R with nil ∞ (R) = 0 deriving from our study of Weierstrass division theorem. By Prop. 4.3, nil ∞ (V d ) is a nonzero ideal generated by x 0 , x 1 , . . .. Here is another example familiar in the p-adic Hodge theory. Let denote the ring of integers of p the p-adic completion of the algebraic closure of the field p of p-adic numbers. Then is a local ring with maximal ideal m satisfying m = m 2 . In particular m ⊂ nil ∞ ( ) and the latter is nonzero.
For instance, for every commutative ring, the closed embedding Spec(R/nil ∞ (R)) → Spec(R) is a nil ∞ -quasi-isomorphism. In particular, the morphism Q d → Z d in Prop. 4.3 is a nil ∞ -quasi-isomorphism. Here is a related and very instructive example of nil ∞ -quasiisomorphism which is not an isomorphism.
Let X = Spec(k[x, y]/(x y)) denote the union of the x-axis and y-axis in the plane of coordinates (x, y).
) denote the complement of the origin in X . We consider the space • X of arcs on X which are non-degenerate with respect to the open subset X ′ . The inclusion of the x-axis in X induces a morphism • 1 → • X and so does the y-asis. We have thus a morphism which induces a bijection on k-points. In fact in induces a bijection on R-points for all kalgebras R with nil ∞ (R) = 0 by 4.7 but not in general by 4.3.

Lemma 4.7. Let R be a commutative ring with nil ∞ (R) = 0, x ∈ R[[t]] a non-degenerate formal series. Then the multiplication by x in R[[t]] is injective.
Proof. First we assume that R is an Artinian local ring of maximal ideal m and residue field k R . In this case, the non-degeneracy condition meansx ∈ k R [[t]] is non zero in other words

Because the map R[[t]] → R((t)) is injective, and the multiplication by x is a bijection on R((t)), the multiplication by x is injective on R[[t]].
Let R be a ring with zero infinite radical and x ∈ R[[t]] a non-degenerate formal series. Let y ∈ R[[t]] be a formal series such that x y = 0. Then for every ring homomorphism ν : R → A from R to Artinian local ring, we have ν( y) = 0. If nil ∞ (R) this implies y = 0.
We will use the Newton method to analyze the arc space of certain complete intersection following Drinfeld's paper [4]. The difference with [4] is that we will have to work over all kalgebras R while Drinfeld restricts himself to Artinian rings. We will son realize that Drinfeld's arguments generalize well only under the assumption nil ∞ (R) = 0 for some familiar linear algebra facts only hold over rings with zero infinite radical.
Proof. By the Cramer formula we have φ ′ φ = φφ ′ = ξid n where id n is the identity matrix.

Variations on the Newton method
As in [4], we will assume that X is defined as the central fiber of a morphism p : m → n with m ≥ n. The coordinates of f = ( f 1 , . . . , f n ) are regular functions on m . We will use the coordinates x = (x 1 , . . . , x m−n , y 1 , . . . , y n ) for x ∈ n . We will consider the n × n-matrix  Here B(x; v) is a function linear with respect to the variable v given by the matrix B(x), and more generally B n (x; v) is a homogenous polynomial of degree n of variables v whose coefficients are polynomials of x.
We consider the Cartesian diagram where the addition is defined as in (5.4). For α is a surjective morphism of Weierstrass type, so is α 0 . For every k-algebra R, and ξ(x) is a non-degenerate formal series. We will replace N 0 by another infinite dimensional scheme defined by congruence equations. Let N 1 be the affine k-scheme that whose R-points are As the equationsB(x, f (x)) = tξ(x) 2 v 1 defining N 1 are essentially equivalent to the congruence equationB (x; f (x)) ≡ 0 mod tξ(x) 2 (5.8) one may say that N 1 is defined in the arc space of the affine space of m by finitely many congruence equations.
Proof. We consider the Taylor expansion (5. 3) of f (x + tξ(x)v 0 ) to rewrite the equation (5.6) in the form By applying the linear formB(x), we get the equation whereB is the adjugate matrix of B. We then derive the equation We will now prove that the morphism N 0 → N 1 constructed above induces a bijection N 0 (R) → N 1 (R) for every k-algebra with zero infinite radical. For an arbitrary k-algebra R, and an arbitrary vector v 1 ∈ R[[t]] n , there exists a unique v 0 ∈ R[[t]] n such that v 1 = v 0 + tB(x; H(x; v 0 )) by the following lemma 5.2. Since v 1 ∈ N 1 (R) the relation (5.10) holds. We derives (5.9) from Prop. 4.8 under the assumption nil ∞ (R) = 0.
is contraction map with contraction ratio ≤ 1/2. It therefore admits a unique fixed point by the Banach fixed point theorem.

Local structure of arc spaces
We will simplify notations in setting c(x) = ξ 2 (x) and a(x) =B(x, f (x)) in the definition of N 1 : By definition c is a polynomial function on m and a is a polynomial vector a : m → n .
In the goal of this section is to prove that after base change by a morphism of Weierstrass type N → N 1 , there exists an essentially smooth morphism N → Y where Y is a scheme of finite type over the n-fold Cartesian product of S d over Q d where S d is the "strange" vector bundle over Q d defined in (4.6). Recall that as d varies S d form family of finite dimensional schemes, which are not of finite type, in which the first member S 1 is the "sawed plane".
where T d is a scheme of finite type over S n+1 d+1 .

Local structure of arc spaces
Proof. For every d ∈ , we consider the Cartesian diagram where γ is essentially the graph of c and (β, α) is the morphism of Weierstrass type satisfying the equations: a(x + tq x ′ ) = tuqv (6.6) The idea is now to expand a(x + tq x ′ ) and c(x + tq x ′ ) in the form are deduced from a and c by the general procedure ??, a ′ (q,x, x ′ ) ∈ n and c ′ (q,x, x ′ ) ∈ 1 are respectively a nvector of formal series and a single formal series whose coefficients are polynomial functions of coefficients of q,x, x ′ .
We will now construct the morphismsā : and the expansions (6.8) and (6.9).
Let X be an affine k-scheme of finite type. The functor X Q d on the category of k-algebras is representable by a flat affine Q d -scheme X Q d . The fiber of X Q d over separable polynomial q ∈ Q d (k) is isomorphic to X d . The fiber of X Q d over the polynomial q = t d is isomorphic to the jet space d−1 X of X . If X = 1 is the affine line, we have which is a vector bundle of rank r over Q d .
where u 0 is the constant coefficient of u and u = u 0 + tu ′ where u ′ is a formal series. We "solve" this system of equations by the following diagram in which all square are Cartesian: We will describe the arrows in this diagram from right to left, and then move bottom up in each column • the morphism 0 : Q d+1 → Q d+1 × n+1 is the zero section q d+1 → (q d+1 , 0); • α d+1 is given by the formula (4.1); • the upper right square is Cartesian by the very definition of S d+1,n+1 ; • the morphism ψ on the third line is given by ψ(u 0 , q,x) = (tq, (ā(q,x),c(q,x) − u 0 q)); • the two down arrows from the second line to the third are obvious projections, the lower square between those lines is obviously Cartesian; • T d is defined so that the upper middle square is Cartesian; • the morphism g M : M → m × Q d × A m d+1 × n+1 maps a point (q, u,x, x ′ , v) ∈ M with u = u 0 + tu ′ on the point g M (q, u,x, x ′ , v) = (u 0 , q,x, (a ′ (q,x, x ′ ) − uv, c ′ (q,x, x ′ ) − u ′ )).
• as by definition N = M × Q d+1 × n+1 Q d+1 , there exists a unique morphism g N : N → S making the upper left square Cartesian.
By construction, T d is a scheme of finite type over S n+1 d+1 as ψ is a morphism between k-schemes of finite type. Since the upper left corner is Cartesian, in order to prove that there exists an isomorphism N 1,d = T d × ∞ , it is enough to prove that there exists an isomorphism compatible with the projection to m × Q d × A m d+1 × n+1 . Given a point we need to solve the equation g M (q, u,x , x ′ , v) = (u 0 , q, (tq,x ), w).
Let us write the vector w ∈ n+1 (R) as (w n , w 1 ) with w n ∈ R[[t]] n and w 1 ∈ R[[t]] and u = u 0 + u ′ t where u 0 ∈ R × and u ′ ∈ R [[t]]. We need to determine the set of The set of solutions is uniquely parametrized by x ′ ∈ R[[t]] m because the above system of equations is equivalent to These equations yield the desired isomorphism (6.18).
As consequence, we obtain the following diagram describing local structures of arc spaces where ν 0 is a nil ∞ -quasi-isomorphism, α 0 , α 1 are morphisms of Weierstrass type, and γ is a morphism of finite type. We note that the positions of ν 0 and γ 1 may be exchanged so that N 1,d → X is the composition of a nil ∞ -quasi-isomorphism with a morphism of Weierstrass type. On the lower line, T d is of of finite type over a the finite-dimensional scheme S n+1 d+1 which is of Weierstrass type but not of finite type. It doesn't seem possible to invert orders i.e. T d is likely not being of Weierstrass type over a scheme of finite type. Although, the local structure of arc spaces as described by this diagram is far from being pleasant, we can derive the existence of good slices.

Construction of good slices
We keep the notations in force in the previous section. In particular, we have a isomorphism N 1,d ≃ T d × ∞ where T d is a S n+1 d+1 -scheme of finite type. Recall that S n+1 d+1 , finite-dimensional but not of finite type, is a strange vector bundle over Q d+1 . We consider the subscheme Y d of T d constructed by the Cartesian diagram: where the right vertical map is the zero section. By base change, the morphism Y d → Q d+1 is of finite type and therefore Y d is a k-scheme affine of finite type. Let Y d = Spec(R d ) where R d is a k-algebra of finite type.
For every R d -point r ∈ ∞ (R d ), we consider the composed morphism defines a R d -point of N 1 . Since the morphism N 0 → N 1 is a nil ∞ -quasi-isomorphism after Prop. 5.1, the induced map on R d -points N 0 (R d ) → N 1 (R d ) is a bijection. We can thus lift f 1 (r) : Y d → N 1 to a morphism φ 0 (r) : Y d → N 0 and thus a morphism φ(r) : Y d → X .
Proof. A morphism f : M → N will be said to be a formal equivalence on formal completions if for every m ∈ M (k) mapping to n ∈ N (k), there exists an isomorphism It is clear that the composition of two morphisms which are formal equivalences on formal completions morphisms, is also a formal equivalence on formal completions. It is also clear that a morphism of Weierstrass type is a formal equivalence on formal completions. We claim that the morphism Y d → T d is a formal equivalence on formal completions. It follows that φ 1 (r) : Y d → N 1 is a formal equivalence on formal completions for the morphism N 1,d → N 1 is of Weierstrass type.
The morphism N 0 → N 1 being a nil ∞ -quasi-isomorphism, induces isomorphism on formal completions of points, it follows that φ 0 (r) : Y d → N 0 is a formal equivalence on formal completions. Finally the morphism N 0 → X is of Weierstrass type, and therefore a formal equivalence on formal completions. We derive that the morphism φ(r) : Y d → X is a formal equivalence on formal completions. Proof. This follows from the fact the morphism Q d+1 → S n+1 d+1 is a formal equivalence on formal completions. This property is preserved by base change by a morphism of finite type.