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Motivic purity and lower-weight vanishing at a smooth closed point

PDF p. 7: Lemma2.9: smooth-point supported purity

Statement

Assume AC and the completed size conventions. Let $k$ be a perfect field, let $X$ be a qcqs $k$ -scheme of pure finite dimension $d$ , and let $x \in X$ be a closed point in its smooth locus. Write $L = κ (x)$ , a finite separable extension of $k$ . For every $i \in Z$ and $j \geq d$ there is the purity isomorphism $H_{m o t}^{i, j} (X / (X ∖ {x}), Z) ≅ H_{m o t}^{i - 2 d, j - d} (L, Z) .$ For every $i$ and $0 \leq j < d$ , the supported group on the left is zero. If the notation $C_{j} = 0$ and $H^{i, j} = 0$ is explicitly adopted for $j < 0$ , the displayed formula holds for every integer $j$ , and the left groups vanish for all $j < d$ . This extends the nonnegative EM weight system by zero; it does not assert an unstated negative-weight EM construction.

The quotient notation denotes the actual coherent restriction fiber of the completed support definition. The statement neither requires $x$ to be $k$ -rational nor $X$ to be smooth away from an affine neighbourhood of $x$ . No separability, perfectness, finite-type-neighbourhood or excision premise is suppressed. For $d = 0$ the shift is zero and the argument includes the identity neighbourhood $Spec L$ .

Facts & Assumptions

Given: The stated field, scheme, point, indices and choice/size conventions. $H^{i, j}$ is cohomology in degree $i$ of the unshifted coefficient/support complex.

Goal: Establish the actual support purity isomorphism and the full lower-weight vanishing range with the correct $(2 d, d)$ shift.

[F1]

A smooth point has the exact finite-type excision pair gives the exact smooth finite-type pair and the actual EM excision equivalence.

[F2]

The actual smooth restriction arrow gives the point-supported cycle fiber compares the actual smooth open restriction with the classical and cycle arrows for every nonnegative weight. Point-supported cycles give all-weight chain purity and low-weight zero computes its fixed-point-supported chain complex, including multiplicities and low-weight zero.

[F3]

Perfect-field cancellation removes exactly d coefficient twists supplies coefficient cancellation over the perfect field for $j \geq d$ . The identified support fiber is the classical relative Hom before Gysin identifies the already compared fiber with relative classical Hom and applies the completed Gysin input under its exact hypotheses.

[F4]

Motivic and filtered K theory with support fixes the current supported object and cohomological indexing. The negative-weight extension is the explicit zero convention in the Statement, agreeing with classical MVW Definition3.1 as recorded by External classical coefficient, derived Yoneda and effective-localization interface.

Proof

1.1F1F4given

Apply [F1]. It supplies a smooth affine separated finite-type pure- $d$ neighbourhood $V$ and smooth closed point $Spec L ↪ V$ of codimension $d$ , and identifies the actual point-support complex on $X$ with that on $V$ . Thus all subsequent operations apply to the actual restriction arrow and preserve supported cohomology.

2.1F1F2F3step 1.1

For $j \geq d$ , [F2] identifies that arrow with its classical counterpart before taking fibers. Apply [F3] to obtain relative Hom out of the cofiber of $M (V ∖ x) \to M (V)$ . The smooth-pair Gysin theorem identifies this cofiber with $M (Spec L) (d) [2 d]$ ; perfect-field cancellation then identifies its coefficient Hom group with weight $j - d$ and degree $i - 2 d$ on $L$ . This proves the displayed formula. Independently, [F2]'s chain calculation identifies the support complex with $C_{j - d} (L) [- 2 d]$ , confirming the same shift directly in the actual support model; no comparison is inferred just from equality of groups.

3.1F2F4step 1.1∎

For $0 \leq j < d$ , [F2] proves that the fixed-support cycle complex has no generators in any degree. Its actual support fiber is therefore zero, so every degree- $i$ group vanishes. This argument does not cancel a positive Tate twist into an undefined negative effective coefficient. If $j < 0$ , [F4]'s explicit zero extension makes both sides zero; if $0 \leq j < d$ , it also gives zero to the negative-weight residue-field side. This proves the all-integer notation version exactly under that convention, and completes the original lower-weight assertion.

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Printed p7, Lemma2.9 proof first display and preceding smooth-point excision paragraph