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Let $S$ be a smooth noetherian scheme, and let $Sm/S$ be the category of smooth schemes over $S$. Voevodsky constructs the homotopy category of motives (resp. the stable homotopy category of motives) $H(S)$ (resp. $SH(S)$) by localizing the category of simplicial presheaves on $Sm/S$.

Since I am a newcomer to this idea, I'm not sure how all the technical details work. I'd like to know what role smoothness plays in the construction of $H(S)$ and $SH(S)$. What prevents us for defining analogous categories for all schemes over $S$?

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    See https://mathoverflow.net/questions/289941/why-is-the-motivic-category-defined-over-the-site-of-smooth-schemes-only?rq=1 – dhy Feb 20 '19 at 03:45
  • Long story short: you can easily define analogous categories in a vast more generality. Now, proving that they have interesting properties is another matter... – Denis Nardin Feb 20 '19 at 11:18

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