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Marsault Chabat
Marsault Chabat
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The main references for this question are

1 : V.Voevodsky's paper Triangulated categories of motives over a field

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $\mathit{Hom}$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the NisniechNisnevich site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}\mathit{Hom}_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I am interested in the case where $R=M$ with $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with it? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

The main references for this question are

1 : V.Voevodsky's paper Triangulated categories of motives over a field

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $\mathit{Hom}$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisniech site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}\mathit{Hom}_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I am interested in the case where $R=M$ with $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with it? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

The main references for this question are

1 : V.Voevodsky's paper Triangulated categories of motives over a field

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $\mathit{Hom}$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisnevich site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}\mathit{Hom}_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I am interested in the case where $R=M$ with $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with it? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

Name of paper; `\mathit`
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LSpice
LSpice
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Motivic cohomology as $Hom$$\mathit{Hom}$ in the category of geometric motives, with coefficient in a Chow motive

The main references for this question are

1 : V.Voevodsky's paper V.Voevodsky's paperTriangulated categories of motives over a field

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $Hom$$\mathit{Hom}$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisniech site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}Hom_{\text{geo. motives}}(M(X),R(i)[n]).$$$$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}\mathit{Hom}_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I interstingam interested in the case where $R=M$ whithwith $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with it? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

Motivic cohomology as $Hom$ in the category of geometric motives, with coefficient in a Chow motive

The main references for this question are

1 : V.Voevodsky's paper

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $Hom$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisniech site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}Hom_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I intersting in the case where $R=M$ whith $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

Motivic cohomology as $\mathit{Hom}$ in the category of geometric motives, with coefficient in a Chow motive

The main references for this question are

1 : V.Voevodsky's paper Triangulated categories of motives over a field

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $\mathit{Hom}$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisniech site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}\mathit{Hom}_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I am interested in the case where $R=M$ with $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with it? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?

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Marsault Chabat
Marsault Chabat
  • 1.3k
  • 5
  • 15

Motivic cohomology as $Hom$ in the category of geometric motives, with coefficient in a Chow motive

The main references for this question are

1 : V.Voevodsky's paper

2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, Vladimir Voevodsky and Charles Weibel.

Context: in both 1 and 2 we can find a possible definition of motivic cohomology using $Hom$ in the category of geometric motives over $k$. I begin to recall the main ideas of this.

This category looks to be in 1 the pseudo-abelianisation of a localisation of the bounded complexes of smooth schemes over a field $k$, and it is in 2 the same construction, but now on the category of sheaves (of $R$-algebra) with transfers for the Nisniech site (questions: why this particular site? Do we obtain the same category if $R=\mathbb{Z}$?). Then the category of effective Chow motives over $k$ embeds through a functor $M$ into effective geometric motives over $k$ (see proposition 20.1 of 2). Now, given a smooth scheme $X$ we define as in [2, definition 14.16] the motivic cohomology (for a ring $R$) to be $$H^{n,i}_{\text{mot}}(X,R) \overset{\text{def}}{=}Hom_{\text{geo. motives}}(M(X),R(i)[n]).$$

My question is: Are we able to generalize the above construction, replacing $R$ by a sheaf. In particular, I intersting in the case where $R=M$ whith $M$ a Chow motive over $X$ seen as a locally constant sheaf over $X$. If the answer is "Yes and it is exactly the same construction" then is there any reference which deals with? More generally do you have any reference for motivic cohomology of a scheme $X$ with coefficient in a Chow motive over $X$?