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تسجيل مستخدم جديدGeneralized cohomology theories for algebraic stacks
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We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendiecks six operations. Objects in this category represent generalized cohomology theories for stacks like algebraic K-theory, as well as new examples like genuine motivic cohomology and algebraic cobordism. These cohomology theories admit Gysin maps and satisfy homotopy invariance, localization, and Mayer-Vietoris. We also prove a fixed point localization formula for torus actions. Finally, the construction is contrasted with a limit-extended stable motivic homotopy category: we show for example that limit-extended motivic cohomology of quotient stacks is computed by the equivariant higher Chow groups of Edidin-Graham, and we also get a good new theory of Borel-equivariant algebraic cobordism.
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We prove that the $infty$-category of $mathrm{MGL}$-modules over any scheme is equivalent to the $infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $mathbb{P}^1$-loop spaces, we deduce tha
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