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To keep things simple, let us assume we work over a perfect field. The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying space of the multiplicative group $\mathbb G_m=\mathbb A^1-\{0\}$ has the $\mathbb A^1$-homotopy type of the infinite dimensional projective space. Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that $H^1(X,\mathbb G_m)=\text{Pic}(X)$ reads as $[X,B\mathbb G_m]=\text{Pic}(X)=CH^1(X)$, where $[?,?]$ stands for the $\text{Hom}$ in the $\mathbb A^1$-homotopy category of $k$-schemes, denoted by $H(k)$.

In general, we denote by $K(\mathbb Z(n),2n)$ the $n$-th motivic Eilenberg-MacLane space, i.e. the object of $H(k)$ which represents the $n$-th Chow group in $H(k)$: for any smooth $k$-scheme $X$, one has

$$[X,\Omega^i K(\mathbb Z(n),2n)]=H^{2n-i}(X,\mathbb Z(n))$$

(where $\Omega^i$ stands for the $i$-th loop space functor). For $i=0$, we just get the usual Chow groups:

$$H^{2n}(X, \mathbb Z (n)))\simeq CH^n(X) .$$

Then, there are several models for $K(\mathbb Z(n),2n)$, one of the smallest being constructed as follows. What I explained above is that $K(\mathbb Z(1),2)$ is the infinite projective space. $K(\mathbb Z(0),0)$ is simply the constant sheaf $\mathbb Z$. For higher $n$, here is a construction (this is Voevodsky's).

Given a $k$-scheme $X$, denote by $L(X)$ the presheaf with transfers associated to X, that is the presheaf of abellian groups whose sections over a smooth $k$-scheme $V$ are the finite correspondences from $V$ to $X$ (i.e. the finite linear combinations of cycles $\sum n_iZ_i$ in $V \times X$ such that $Z_i$ is finite and surjective over $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the $Z_i$; are finite and surjective over a smooth (hence normal) scheme $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U \to V$ with $U$ and $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf $L(X)$ is a sheaf for the Nisnevich topology. This construction is functorial in $X$ (I will need this functoriality only for closed immersions).

Let $X$ (resp. $Y$) be the cartesian product of $n$ (resp. $n-1$) copies of the projective line. The point at infinity gives a family of $n$ maps $u_i : Y \to X$. Then a model of $K(\mathbb Z(n),2n)$ in $H(k)$ is the sheaf of sets obtained as the quotient (in the category of Nisnech sheaves of abelian groups) of $L(X)$ by the subsheaf generated by the images of the maps $L(u_i):L(Y)\to L(X)$.

If you want a more conceptual definition, there is also the direction of algebraic cobordism (but for this, you need to understand the stable homotopy of $\mathbb P^1$-spectra, but maybe it is enough at first to think of $\mathbb P^1$-spectra simply as the cohomology theories allowed in homotopy theory of schemes): the idea is that there is an algebraic cobordism, which is represented by a $\mathbb P^1$-spectrum $MGL$ (the analog of of the spectrum $MU$ which represents complex cobordism in algebraic topology). The idea is that $MGL$ is the universal oriented cohomology theory (for short, this means that, if a cohomology theory $E$ satisfies the projective bundle formula, then the choice of an orientation, i.e. of a generating class in the second cohomology group of $\mathbb P^1$ with coefficients in $E(1)$, is the same as a map of ring spectra $MGL \to E$. The idea is that, as in algebraic topology, formal groups laws classify oriented cohomology theories ($MGL$ corresponding to the initial formal group law). The cohomology theory which corresponds to the multiplicative formal group law is $KGL$, the $\mathbb P^1$-spectrum which represents algebraic K-theory, while the cohomology theory corresponding to the additive formal group law is motivic cohomology (this latter characterization has been announced by F. Morel and M. Hopkins if $k$ is of characteristic zero, but is not published yet, and it is known for any field $k$ if we work with rational coefficients (this is a result of Spitzweck, Nauman, Ostvaer)).
show/hide this revision's text 3 fixed slight inaccuracy

To keep things simple, let us assume we work over a perfect field. The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 2)1). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying space of the multiplicative group $\mathbb G_m=\mathbb A^1-\{0\}$ has the $\mathbb A^1$-homotopy type of the infinite dimensional projective space. Moreover, as the Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that $H^1(X,\mathbb G_m)=\text{Pic}(X)$ reads as $[X,B\mathbb G_m]=\text{Pic}(X)=CH^2(X)$G_m]=\text{Pic}(X)=CH^1(X)$, where $[?,?]$ stands for the $\text{Hom}$ in the $\mathbb A^1$-homotopy category of $k$-schemes, denoted by $H(k)$.

In general, we denote by $K(\mathbb Z(n),2n)$ the $n$-th motivic Eilenberg-MacLane space, i.e. the object of $H(k)$ which represents the $n$-th Chow group in $H(k)$: for any smooth $k$-scheme $X$, one has

$$[s^i(X),K(\mathbb $[X,\Omega^i K(\mathbb Z(n),2n)]=H^{2n-i}(X,\mathbb Z(n)) \quad \text{(where Z(n))$$

(where $s^i$ \Omega^i$ stands for the $i$-th suspension (in the usual sense))}.$$loop space functor).

Then, there are several models for $K(\mathbb Z(n),2n)$, one of the smallest being constructed as follows. What I explained above is that $K(\mathbb Z(1),2)$ is the infinite projective space. $K(\mathbb Z(0),0)$ is simply the constant sheaf $\mathbb Z$. For higher $n$, here is a construction (this is Voevodsky's).

Given a $k$-scheme $X$, denote by $L(X)$ the presheaf with transfers associated to X, that is the presheaf of abellian groups whose sections over a smooth $k$-scheme $V$ are the finite correspondences from $V$ to $X$ (i.e. the finite linear combinations of cycles $\sum n_iZ_i$ in $V \times X$ such that $Z_i$ is finite and surjective over $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the $Z_i$; are finite and surjective over a smooth (hence normal) scheme $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U \to V$ with $U$ and $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf $L(X)$ is a sheaf for the Nisnevich topology. This construction is functorial in $X$ (I will need this functoriality only for closed immersions).

Let $X$ (resp. $Y$) be the cartesian product of $n$ (resp. $n-1$) copies of the projective line. The point at infinity gives a family of $n$ maps $u_i : Y \to X$. Then a model of $K(\mathbb Z(n),2n)$ in $H(k)$ is the sheaf of sets obtained as the quotient (in the category of Nisnech sheaves of abelian groups) of $L(X)$ by the subsheaf generated by the images of the maps $L(u_i):L(Y)\to L(X)$.

If you want a more conceptual definition, there is also the direction of algebraic cobordism (but for this, you need to understand the stable homotopy of $\mathbb P^1$-spectra, but maybe it is enough at first to think of $\mathbb P^1$-spectra simply as the cohomology theories allowed in homotopy theory of schemes): the idea is that there is an algebraic cobordism, which is represented by a $\mathbb P^1$-spectrum $MGL$ (the analog of of the spectrum $MU$ which represents complex cobordism in algebraic topology). The idea is that $MGL$ is the universal oriented cohomology theory (for short, this means that, if a cohomology theory $E$ satisfies the projective bundle formula, then the choice of an orientation, i.e. of a generating class in the second cohomology group of $\mathbb P^1$ with coefficients in $E(1)$, is the same as a map of ring spectra $MGL \to E$. The idea is that, as in algebraic topology, formal groups laws classify oriented cohomology theories ($MGL$ corresponding to the initial formal group law). The cohomology theory which corresponds to the multiplicative formal group law is $KGL$, the $\mathbb P^1$-spectrum which represents algebraic K-theory, while the cohomology theory corresponding to the additive formal group law is motivic cohomology (this latter characterization has been announced by F. Morel and M. Hopkins if $k$ is of characteristic zero, but is not published yet, and it is known for any field $k$ if we work with rational coefficients (this is a result of Spitzweck, Nauman, Ostvaer)).
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The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 2). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying space of the multiplicative group G_m=A^1-{0} $\mathbb G_m=\mathbb A^1-\{0\}$ has the A^1-homotopy $\mathbb A^1$-homotopy type of the infinite dimensionnal dimensional projective space. Moreover, as the picard Picard group is homotopy invariant for regular schemes (semi-normal is even enough), the fact that H^1(X,G_m)=Pic(X) $H^1(X,\mathbb G_m)=\text{Pic}(X)$ reads as [X,BG_m]=Pic(X)=CH^2(X), $[X,B\mathbb G_m]=\text{Pic}(X)=CH^2(X)$, where [?,?] $[?,?]$ stands for the Hom $\text{Hom}$ in the A^1-homotopy $\mathbb A^1$-homotopy category of k-schemes, $k$-schemes, denoted by H(k).$H(k)$.

In general, we denote by K(Z(n),2n) $K(\mathbb Z(n),2n)$ the nth $n$-th motivic Eilenberg-MacLane space, i.e. the object of H(k) $H(k)$ which represents the nth $n$-th Chow group in H(k): $H(k)$: for any smooth k-scheme X, $k$-scheme $X$, one has

[s^i(X),K(Z(n),2n)]=H^2n-i(X,Z(n)) (where s^i

$$[s^i(X),K(\mathbb Z(n),2n)]=H^{2n-i}(X,\mathbb Z(n)) \quad \text{(where $s^i$ stands for the ith $i$-th suspension (in the usual sense)).sense))}.$$

Then, there are several models for K(Z(n),2n)$K(\mathbb Z(n),2n)$, one of the smallest being constructed as follows. What I explained above is that K(Z(1),2) $K(\mathbb Z(1),2)$ is the infinite projective space.K(Z(0),0) $K(\mathbb Z(0),0)$ is simply the constant sheaf Z$\mathbb Z$. For higher n, $n$, here is a construction (this is Voevodsky's).

Given a k-scheme X, $k$-scheme $X$, denote by L(X) $L(X)$ the presheaf with transfers associated to X, that is the presheaf of abellian groups whose sections over a smooth k-scheme V $k$-scheme $V$ are the finite correspondances correspondences from V $V$ to X $X$ (i.e. the finite linear combinations of cyclesΣᵢ nᵢ Zᵢ $\sum n_iZ_i$ in $V x X \times X$ such that Zᵢ $Z_i$ is finite and surjective over V). $V$). This is a presheaf, where the pullbacks are defined using the pullbacks of cycles (the condition that the Zᵢ $Z_i$; are finite and surjective over a smooth (hence normal) scheme V $V$ makes that this is well defined without working up to rational equivalences, and as we consider only pullbacks along maps $U -> V \to V$ with U $U$ and V $V$ smooth (hence regular) ensures that the multiplicities which will appear from these pullbacks will always be integers). The presheaf L(X) $L(X)$ is a sheaf for the Nisnevich topology. This construction is functorial in X $X$ (I will need this functoriality only for closed immersions).

Let X $X$ (resp. Y) $Y$) be the cartesian product of n $n$ (resp. n-1) $n-1$) copies of the projective line.The point at infinity gives a family of n $n$ maps uᵢ $u_i : Y -> X\to X$. Then a model of K(Z(n),2n) $K(\mathbb Z(n),2n)$ in H(k) $H(k)$ is the sheaf of sets obtained as the quotient (in the category of Nisnech sheaves of abelian groups) of L(X) $L(X)$ by the subsheaf generated by the images of the maps L(uᵢ):L(Y)->L(X).$L(u_i):L(Y)\to L(X)$.

If you want a more conceptual definition, there is also the direction of algebraic cobordism (but for this, you need to understand the stable homotopy of P^1-spectra$\mathbb P^1$-spectra, but maybe it is enough at first to think of P^1-spectra $\mathbb P^1$-spectra simply as the cohomology theoriesallowed in homotopy theory of schemes): the idea is that there is an algebraic cobordism, which is represented by a P^1-spectrum MGL $\mathbb P^1$-spectrum $MGL$ (the analog of of the spectrum MU $MU$ which represents complex cobordism in algebraic topology). The idea is that MGL $MGL$ is the universal oriented cohomology theory (for short, this means that, if a cohomology theory E $E$ satisfies the projective bundle formula, then the choice of an orientation, i.e. of a generating class in the second cohomology group of P^1 $\mathbb P^1$ with coefficients in E(1), $E(1)$, is the same as a map of ring spectra MGL->E$MGL \to E$. The idea is that, as in algebraic topology, formal groups laws classify oriented cohomology theories (MGL $MGL$ corresponding to the initial formal group law). The cohomology theory which corresponds to the multiplicative formal group law is KGL, $KGL$, the P^1-spectrum $\mathbb P^1$-spectrum which represents algebraic K-theory, while the cohomology theory corresponding to the additive formal group law is motivic cohomology (this latter characterization has been announced by F. Morel and M. Hopkins if k $k$ is of characteristic zero, but is not published yet, and it is known for any field k $k$ if we work with rational coefficients (this is a result of Spitzweck, Nauman, Ostvaer)).
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