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Stable 𝔸1-connectivity over a base

  • Anderi Eduardovich Druzhinin EMAIL logo

Abstract

Morel’s stable connectivity theorem states the vanishing of the sheaves of the negative motivic homotopy groups π ¯ i s ( Y ) and π ¯ i + j , j s ( Y ) , i < 0 , in the stable motivic homotopy categories 𝐒𝐇 S 1 ( k ) and 𝐒𝐇 ( k ) for an arbitrary smooth scheme Y over a field k. Originally the same property was conjectured in the relative case over a base scheme S. In view of Ayoub’s conterexamples the modified version of the conjecture states the vanishing of stable motivic homotopy groups π ¯ i s ( Y ) (and π ¯ i + j , j s ( Y ) ) for i < - d , where d = dim S is the Krull dimension. The latter version of the conjecture is proven over noetherian domains of finite Krull dimension under the assumption that residue fields of the base scheme are infinite. This is the result by J. Schmidt and F. Strunk for Dedekind schemes case, and the result by N. Deshmukh, A. Hogadi, G. Kulkarni and S. Yadavand for the case of noetherian domains of an arbitrary dimension. In the article, we prove the result for any locally noetharian base scheme of finite Krull dimension without the assumption on the residue fields, in particular for 𝐒𝐇 S 1 ( ) and 𝐒𝐇 ( ) . In the appendix, we modify the arguments used for the main result to obtain the independent proof of Gabber’s Presentation Lemma over finite fields.

Award Identifier / Grant number: 19-71-30002

Funding statement: The research is supported by the Russian Science Foundation grant 19-71-30002; the author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.

A Étaleness criterion

In this section, we prove some criterion for étale morphisms. Namely, the criterion on étaleness of a morphism f : X Y at a point x if f induces the isomorphism f : x f ( x ) . Let us mention that the latter assumption is contained in the definition of Nisnevich topology coverings. The criterion is used in the previous section, in Subsection 4.1.

Lemma A.1.

Let π : X Y be a morphism of essentially smooth schemes over a regular scheme S, dim X = dim Y . Let x X be a closed point finite over S, and let E X be the first order thickening of X at x. Assume that the restriction π | x : x Y is a closed immersion. Then the morphism π is étale at x if and only if the restriction π | E : E Y is a closed immersion.

Proof.

Assume that π | E : E Y is a closed immersion. Consider the closed subscheme

F = π ( x ) × Y X X .

Then F x × Y X , since π | x is a closed immersion. Since π | E is a closed immersion, it follows that

F E = π ( x ) × Y X × X E = π ( x ) × Y E = x .

Since E is the first order thickening of X at x, it follows that F = x F ˇ . Hence π is quasi-finite over π ( x ) and unramified at x. Since π is quasi-finite, and dim X = dim Y , π is flat by [1, Corollary V.3.6]. Finally, π induces the isomorphism x π ( x ) , so the residue filed extension is separable, in particular. Thus the morphism π is étale.

Assume that π is étale at x. Then

F = x F ˇ for  F = π ( x ) × Y X X .

Hence

(A.1) π ( x ) × Y E = F × X E = F E = x .

Let us show firstly that the morphism

(A.2) π | E : E Y

is finite. Since π is étale, it follows that π is quasi-finite. Hence by Zariski’s main theorem [8, Theorem 8.12.6], π equals the composition

π : X 𝑗 X ¯ π ¯ Y

with j being an open immersion, and a finite morphism π ¯ . The morphism π | E equals the composition

(A.3) E X X ¯ Y .

Then E equals the first order thickening of j ( x ) in X ¯ , and hence the composite morphism (A.3) is finite. Note that j ( x ) is closed in X ¯ , since X is finite over S.

Hence the morphism π | E (A.2) can be decomposed as

E 𝑐 E ¯ Y ,

where E ¯ Y is a closed subscheme that is the image of E, and c is a finite surjective map of Artin schemes. Then because of (A.1) it follows that

π ( x ) × E ¯ E = π ( x ) × Y E = x .

Hence since E and E ¯ are noetherian, Nakayama’s Lemma implies that π induces the isomorphism

c : E E ¯ .

B Gabber’s Presentation Lemma over a field

In this section, we apply the results and the technique of Section 4.1 to prove Gabber’s Presentation Lemma over fields [7, Lemma 3.1], [5, Theorem 3.2.1], [10]. The proof is a small modification of the argument from Lemma 4.1. The essential difference of Gabber’s Presentation Lemma (discussed here) in comparing with Lemma B.1 is additional condition (2) below. We write these two lemmas separately to stress that the following full form of Gabber’s Presentation Lemma is not needed for the proof of Theorem 5.1, while Lemma 4.1 is enough.

Lemma B.1.

Let X be a smooth scheme over a base field k, let Z X be a closed subscheme of positive codimension, and let x Z be a point. Let V = X x be the local scheme of X at x. Then there is an étale map π : V A k 1 × Y such that

  1. Y = ( 𝔸 k n - 1 ) y for a point y 𝔸 k n - 1 ,

  2. the local scheme Z x is finite over Y , and

  3. π induces the isomorphism Z x π ( Z x ) .

Proof.

Without loss of generality, we can assume that x X is a closed point in the finite type scheme X. Otherwise, we can replace x with a closed point in the closure of x in X. Next, we can assume that X is an affine smooth irreducible scheme of finite type over k; otherwise, we can replace X with a Zariski open neighbourhood of x with such properties.

Let E = Z ( I 2 ( x ) ) E be a closed subscheme that is the first order thickening of x in X. By Lemma 4.4 there is a map

(B.1) f = ( f 1 , , f n ) : E 𝔸 k n

that is a closed immersion.

Since X is affine, there is a closed immersion X 𝔸 k N for some N . Let X ¯ k N be the closure. Let Z ¯ k N be the closure of the image of Z = Z X , and let

Z = Z ¯ k N - 1

be the complement of Z in Z ¯ . Since X is of finite type over k, and x is a closed point, x is of finite type over k, and moreover, x is a finite scheme over k. Hence x is a closed subscheme and a closed point in X ¯ .

To construct the required map π, we are going to construct a map

p = ( p 1 , , p n ) : 𝔸 k N 𝔸 k n , p i = s i / t d i 𝒪 ( 𝔸 k N ) ,

where

s i Γ ( k N , 𝒪 ( d i ) ) , d i , i = 1 , , n ,

and t 1 , , t N , t Γ ( k N , 𝒪 ( 1 ) ) denote the coordinate sections. We do this using Lemma 4.2 and constructing for large enough d 1 , , d n the global sections s i , for i = 1 , , n , such that

(B.2) ( s i / t d i ) | E = f i ,
(B.3) codim ( B i - 1 - x ) ( B i - x ) > 0 , or  B i = , i = 1 , , n ,
(B.4) codim C i - 1 Z ( C i Z ) > 0 , or  C i Z = , i = 1 , , n ,

where

C i = Z ( s 1 , , s i ) k N , B i = π i - 1 ( π i ( x ) ) Z , π i = ( s 1 / t d 1 , , s i / t d i ) : Z 𝔸 n i , i = 0 , , n .

Namely, we apply Lemma 4.2 inductively to the schemes

T = C i - 1 Z k N - 1 , B = B i - 1 Z 𝔸 k n

for all i, i = 1 , , n . Note that C i are closed subschemes of k N , and C 0 = k N ; B i are closed subschemes in Z; B 0 = Z , and B i B i - 1 are closed immersions. Then it follows by (B.2) that the constructed map p induces an étale map V 𝔸 k n . Let y = π ( x ) = f ( x ) 𝔸 k n , and y = π n - 1 ( x ) 𝔸 k n - 1 . Denote by

π : V 𝔸 k 1 × Y

the induced map, where Y = ( 𝔸 n - 1 ) y . We are going to check that π satisfies required conditions (2) and (3).

Since

dim ( C 0 Z ) = dim Z < dim Z < n ,

we have dim ( C 0 Z ) n - 2 . Then by (B.4), it follows that

Z ( s 1 , , s n - 1 ) Z = C n - 1 Z = ,

and hence the induced map Z 𝔸 k n - 1 is finite. This implies assertion (2).

Then we see that Z 𝔸 k n is finite as well. By injection (B.1) π induces the isomorphism

(B.5) x π ( x ) = f ( x ) = y .

To prove assertion (3), we need to prove that π induces isomorphism Z x π ( Z x ) . Consider the fibred product Z × 𝔸 k n y . Since π is étale, it follows that

Z × 𝔸 k n x = x F ˇ

for some F ˇ . By definition of B n we have the equality Z × 𝔸 k n y = B n . Since

dim B 0 = dim Z n - 1 ,

by (B.2), it follows that B n - x = . Hence

Z × 𝔸 k n y = x .

Then since Z is finite over 𝔸 k n , since the scheme Z and consequently the scheme π ( Z ) are noetherian, and because of isomorphism (B.5), it follows by Nakayama’s Lemma that π induces the isomorphism

Z x π ( Z x ) .

Acknowledgements

The author acknowledges Hakon Kolderup for many helpful remarks on the article. The author is grateful for the hospitality of the Mathematical Department of the University of Oslo and the Frontier research group project “Motivic Hopf Equations”.

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Received: 2020-05-30
Revised: 2022-06-05
Published Online: 2022-09-29
Published in Print: 2022-11-01

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