Theorem of absolute purity

In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given a regular scheme X over some base scheme, i : Z → X {\displaystyle i:Z\to X} a closed immersion of a regular scheme of pure codimension r, an integer n that is invertible on the base scheme, F {\displaystyle {\mathcal {F}}} a locally constant étale sheaf with finite stalks and values in Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } , for each integer m ≥ 0 {\displaystyle m\geq 0} , the map H m ⁡ ( Z ét ; F ) → H Z m + 2 r ⁡ ( X ét ; F ( r ) ) {\displaystyle \operatorname {H} ^{m}(Z_{\text{ét}};{\mathcal {F}})\to \operatorname {H} _{Z}^{m+2r}(X_{\text{ét}};{\mathcal {F}}(r))} is bijective, where the map is induced by cup product with c r ( Z ) {\displaystyle c_{r}(Z)} .

Source: Wikipedia — Theorem of absolute purity (CC BY-SA 4.0)

Theorem of absolute purity

In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given a regular scheme X over some base scheme, i : Z → X {\displaystyle i:Z\to X} a closed immersion of a regular scheme of pure codimension r, an integer n that is invertible on the base scheme, F {\displaystyle {\mathcal {F}}} a locally constant étale sheaf with finite stalks and values in Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } , for each integer m ≥ 0 {\displaystyle m\geq 0} , the map H m ⁡ ( Z ét ; F ) → H Z m + 2 r ⁡ ( X ét ; F ( r ) ) {\displaystyle \operatorname {H} ^{m}(Z_{\text{ét}};{\mathcal {F}})\to \operatorname {H} _{Z}^{m+2r}(X_{\text{ét}};{\mathcal {F}}(r))} is bijective, where the map is induced by cup product with c r ( Z ) {\displaystyle c_{r}(Z)} .

Source: Wikipedia "Theorem of absolute purity" · CC BY-SA 4.0

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