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(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in). I'm going to try with a very naive answer, although I'm not sure I understan …

Let me assume that $S$ is a regular Noetherian scheme (for example a field). Then algebraic K-theory is a motivic spectrum, and in fact it is represented by the $\mathbb{P}^1$-spectrum that is $BGL_\i …

NO such a map does not exist. Thanks to Eric Peterson for making me realize that the argument carries through even if the map is not a map of algebras. By rigidity,you can only consider the case whe …

The key here is that $SH^{eff}(S)$ is closed under suspensions, so there's an inclusion $j_{n+1}:\Sigma^{n+1}_T SH^{eff}(S)\subseteq \Sigma^n_T SH^{eff}(S)$. Hence you can write $i_{n+1}=i_n \circ j_{ …

$S^1$-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 categ …

For the equation you are asking about, you don't want $MA$ to be dualizable (which is lucky, because it's not), you want $MA\wedge X_+$ to be dualizable as an $MA$-module. This is true in the situatio …

A simple way of seeing it is to explicitly spell out what we mean when we say that a spectrum is "presented" by a prespectrum. To say that that a spectrum $E$ is presented by $$(E_0,E_1,...)$$ means t …

Let me first write what happens for classical cobordism. You are basically asking whether the map $\operatorname{MU}\to H\mathbb{Z}$ factors through the projection $\operatorname{ku}\to H\mathbb{Z}$. …

The stable motivic category is a presentable symmetric monoidal ∞-category so smashing with any motivic spectrum preserves all homotopy colimits. On the other hand smashing with a spectrum need not to …