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I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme.

Let $E$ be an $\Omega$-spectrum and $\varphi \colon F\to E$ be a morphism of spectra satisfying that for any smooth $S$-scheme $X$ the induced map $$ \mathrm{Hom}_{\mathbf{SH}(S)}(X,F)\longrightarrow \mathrm{Hom}_{\mathbf{SH}(S)}(X,E) $$ is an isomorphism of groups. The question is: Is $\varphi\colon F\to E$ an isomorphism of spectra?

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    $\begingroup$ If you mean induces a $\pi_{*,*}$ isomorphism then I believe this is correct, because the collection $\{ \Sigma^{p,q}X \mid X \in Sm/S,p,q \in \mathbb{Z} \}$ is a collection of compact generators of $SH(S)$. $\endgroup$
    – Drew Heard
    Commented Aug 7, 2018 at 10:54

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Here is a slightly more fleshed out version of the comment above.

First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of compact generators, is Theorem 9.1 of https://arxiv.org/pdf/math/0310190.pdf (there may be earlier references). In particular, the smallest localizing subcategory of $SH(S)$ generated by $\mathcal{C}$ is $SH(S)$ itself.

Now let $\phi \colon F \to E$ be as in your statement, with cofiber $Z$, and assume that the induced map $Hom_{SH(S)}(X,F) \to Hom_{SH(S)}(X,E)$ is a $\pi_{*,*}$ equivalence. It follows that $[X,Z] = 0$ for all $X \in \mathcal{C}$. Since $\mathcal{C}$ generates, this implies that $Z \simeq \ast$.

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