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Your question comes because of the vague use of the term purity. That's ok, moreover because Morel and Voevodsky didn't explain in that paper why that result deserved the term purity. That result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) you should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. Might be good, fun or pretty, but it is not cohomology what you are doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are, Mayer-Vietoris, inverse image, ect... In general all Grothendieck 6 functors formalism is usually direct from definition. In motivic homotopy as well. Purity is always a theorem and requires hypothesis. Surprisingly for me, Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because it is preliminary to purity).

As I told you purity is not important for applications, it is a requirement to start speanking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher $K$-theory Riemann-Roch relies essentially on purity.

Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is the ring spectrum giving your cohomology. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).

Your question comes because of the vague use of the term purity. That's ok, moreover because Morel and Voevodsky didn't explain in that paper why that result deserved the term purity. That result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) one should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. It night be good, fun or pretty, but it is not cohomology what one is doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes (which is expected to describe cohomologies) then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are Mayer-Vietoris, inverse image, ect... are usually direct from definition. Purity is always a theorem and requires hypothesis. Surprisingly for me, the Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because it is preliminary to purity).

As I said purity is not important for applications, it is a requirement to start speaking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher Riemann-Roch relies essentially on purity.

Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is the ring spectrum giving your cohomology. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).

Your question comes because you don't know what purity is. That's ok, morevoer because you cannot learn it from Morel and Voevodsky because they didn't explain it in that paper. The result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) you should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. Might be good, fun or pretty, but it is not cohomology what you are doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are, Mayer-Vietoris, inverse image, ect... In general all Grothendieck 6 functors formalism is usually direct from definition. In motivic homotopy as well. Purity is always a theorem and requires hypothesis. Surprisingly for me, Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because it is preliminary to purity).

As I told you purity is not important for applications, it is a requirement to start speanking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher $K$-theory Riemann-Roch relies essentially on purity.

Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is the ring spectrum giving your cohomology. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).

Your question comes because of the vague use of the term purity. That's ok, moreover because Morel and Voevodsky didn't explain in that paper why that result deserved the term purity. That result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) you should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. Might be good, fun or pretty, but it is not cohomology what you are doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are, Mayer-Vietoris, inverse image, ect... In general all Grothendieck 6 functors formalism is usually direct from definition. In motivic homotopy as well. Purity is always a theorem and requires hypothesis. Surprisingly for me, Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because it is preliminary to purity).

As I told you purity is not important for applications, it is a requirement to start speanking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher $K$-theory Riemann-Roch relies essentially on purity.

Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is the ring spectrum giving your cohomology. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).

Your question comes because you don't know what purity is. That's ok, morevoer because you cannot learn it from Morel and Voevodsky because they didn't explain it in that paper. The result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) you should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. Might be good, fun or pretty, but it is not cohomology what you are doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are, Mayer-Vietoris, inverse image, ect... In general all Grothendieck 6 functors formalism is usually direct from definition. In motivic homotopy as well. Purity is always a theorem and requires hypothesis. Surprisingly for me, Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because is preliminary to purity).

As I told you purity is not important for applications, it is a requirement to start speanking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher $K$-theory Riemann-Roch relies essentially on purity.

Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is a ring spectrum. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).

Your question comes because you don't know what purity is. That's ok, morevoer because you cannot learn it from Morel and Voevodsky because they didn't explain it in that paper. The result that you mention is a final step on a sequence of arguments which gives purity, but it is not purity itself. Find the link between them at the end.

What is purity: The term purity comes from the purity isomorphism. As Mayer-Vietoris, the purity isomorphism is a property you expect on any cohomology. Let's be more precise: whatever context you are if $H^\bullet$ is a cohomology and $Z\hookrightarrow X$ is a "good" closed immersion (smooth and sometimes even proper-compact) you should have an isomorphism $$ \bar i_*\colon H^{*-2d,*-d}(Z) \buildrel{\sim}\over{\to} H^{*,*}_Z(X) \qquad \mbox{Purity isomorphism.} $$ In the formula $d$ is the codimension of $Z$ in $X$. Denote $U$ the open complement of $Z$. Together with the local long exact sequence and forgetting support the purity isomorphism gives the so called Gysin long exact sequence $$ \cdots\to H^{*-1,*}(U)\to H^{*-2d,*-d}(Z)\buildrel{i_*}\over{\to}H^{*,*}(X)\to H^{*,*}(U)\to\cdots $$ The term "Gysin" comes from the fact that $i_*$ is called the "Gysin morphism".

And now back to your question:

Why is purity important?: There go my two main reasons:

1.- Purity is a standard property of cohomology. Take any book of any type of cohomology, you will find it. No purity? Then no cohomology. Might be good, fun or pretty, but it is not cohomology what you are doing. It is as simple as that. If Morel and Voevodsky want to show that they found the adequate homotopy category of schemes then they have to prove purity.

2.- Purity is the "nontrival" basic property of cohomology. Whatever context you are, Mayer-Vietoris, inverse image, ect... In general all Grothendieck 6 functors formalism is usually direct from definition. In motivic homotopy as well. Purity is always a theorem and requires hypothesis. Surprisingly for me, Local long exact sequence in motivic homotopy is not trivial, and you will see a lot fuzz around it too (even more because it is preliminary to purity).

As I told you purity is not important for applications, it is a requirement to start speanking about cohomology. But if you want a concrete application, an important one, there it goes: nowadays higher $K$-theory Riemann-Roch relies essentially on purity.

Link between MV result and purity: People in Algebraic Geometry nowadays usually rely on the so called "Thom isomorphism" for establishing the purity. By construction of the Thom space you have $$ H(\mathrm{Th}(N_{Z/X}))\simeq H(Z). $$ With the isomorphism proved by Morel and Voevodsky you have: $$ H(Z)\simeq H(\mathrm{Th}(N_{Z/X}))=\mathrm{Hom}_{\mathbf{SH}(X)}(\mathrm{Th}(N_{Z/X}),E)\buildrel{\mathrm{MV}}\over{\simeq} \mathrm{Hom}_{\mathbf{SH}(X)}(X/X-U,E)= H_Z(X) $$ Where $\mathbf{SH}(X)$ is the stable homotopy category and $E$ is the ring spectrum giving your cohomology. (If you dont want to use $\mathbf{SH}$ you can put the $\mathbb{Z}\times BGL$ instead of $E$ and $\mathrm{Hom}$ in the homotopy category and obtain it for $K$-theory).