Expression of morphisms in motivic homotopy categories in terms of Nisnevich cohomology?

Question

For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules over the motivic cobordism and motivic cohomology spectra (the latter are essentially Voevodsky motives); these categories also contain motivic spectra for all smooth varieties over $k$ that I will denote by $\Sigma^{\infty}(-_+)$. For an object $N$ of the heart of the corresponding homotopy $t$-structure, a smooth $X/k$, and $n\ge 0$ I would like to compute the morphism group $Hom(\Sigma^{\infty}(X_+),N[n])$ (in each of the aforementioned categories).

Now, to the object $N$ of the heart one can associate the presheaf $S_N$ sending a smooth $Y/k$ into $Hom(\Sigma^{\infty}(X_+),N)$; actually, $S_N$ is a Nisnevich sheaf.

I believe that $Hom(\Sigma^{\infty}(X_+),N[n])$ equals $H^n_{Nis}(X,S_N)$. Is there any reference for this fact (for any of the aforementioned motivic categories)? A natural way to prove it would be to start with the stable homotopy category $SH_{Nis}^{S^1}(k)$ of simplicial Nisnevich sheaves (of sets on smooth varieties over $k$) and to study adjoint functors connecting this category with the "motivic categories"; yet I don't have fine enough references for this argument.

Answer 1

$S^1$-spectra

Let me first show it when the category is $SH^{S^1}(k)$, that is the (∞-)category of $\mathbb{A}^1$-invariant sheaves of spectra. Then the heart of the t-structure is precisely the category of strictly $\mathbb{A}^1$-invariant sheaves of abelian groups. The embedding is given by the Eilenberg-MacLane functor, sending a sheaf of abelian groups $A$ to the Nisnevich sheafification $HA$ of $U\mapsto H(A(U))$ (as we will see strictly $\mathbb{A}^1$-invariant means precisely that $HA$ is $\mathbb{A}^1$-invariant). I will show the following more general result

Theorem: Let $A$ be a sheaf of abelian groups on a site. Then for all $X$ there is a natural isomorphism for all $i\in\mathbb{Z}$ $$\pi_{-i}\Gamma(X,HA)\cong H^i(X;A)\,.$$ Proof: First let us show the statement when $A$ is an injective sheaf of abelian groups. Then I claim that $HA=H\circ A$, that is that $U\mapsto H(A(U))$ is already a sheaf. But this is the well-known statement that the Čech cohomology of injective sheaves is trivial. Now to show the result for all sheaves it suffices to notice that both sides are $\delta$-functors that are annihilated by injective sheaves. So if we show that they coincide when $i=0$ we are done. But this is easily done by noticing that the functor $H$ sends short exact sequences of abelian sheaves into fiber sequences of sheaves of spectra, and using an injective resolution for $A$. $\square$

Since $\mathrm{Hom}(\Sigma^∞_+X,HA[i])=\pi_i\Gamma(X,HA)$ this proves the result for the case of $S^1$-spectra.

Motivic spectra

Now let us extend this result to the category $SH(k)$ of motivic spectra. Recall that a motivic spectrum is a sequence $E=\{E_n\}_{n\ge0}$ of $S^1$-spectra together with fiber sequences $$E_n(X)\to E_{n+1}(X\times\mathbb{G}_m)\to E_{n+1}(X)$$ The t-structure on motivic spectra is then given by saying that $E$ is connective iff all $E_n$'s are connective. There is a right adjoint t-exact forgetful functor to $S^1$-spectra sending $E$ to $E_0$. They fit in a diagram of adjunctions $$H(k)\rightleftarrows SH^{S^1}(k)\rightleftarrows SH(k) $$ where in the first adjunction the right adjoint map is just postcomposition with $\Omega^∞$.

The heart of this category is the category of homotopy modules, that is sequences $A={A_i}_{i\ge0}$ of strictly $\mathbb{A}^1$-invariant Nisnevich sheaves of abelian groups together with short exact sequences $$ 0\to A_i(X)\to A_{i+1}(X\times\mathbb{G_m})\to A_{i+1}(X)\to 0$$ If $A$ is a homotopy module I will denote the corresponding motivic spectrum $\{HA_i\}_{i\ge 0}$ with $HA$. Since the functor $SH(k)\to SH^{S^1}(k)$ is t-exact, there is a clear forgetful functor from homotopy modules to strictly $\mathbb{A}^1$-invariant sheaves of abelian groups sending $A$ to $A_0$. I claim the following $$\textrm{Map}(\Sigma^∞_TX_+,HA)=\Gamma(X,HA_0)$$ If this is true, clearly the thesis follows from the previous result. But this follows immediately from the diagram of adjunctions I sketched earlier, since the right hand side is just $\mathrm{Map}(\Sigma^∞_{S^1}X_+,HA_0)$.