Milnor excision for motivic spectra

<jats:title>Abstract</jats:title> <jats:p>We prove that the <jats:inline-formula id="j_crelle-2021-0040_ineq_9999"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">∞</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2021-0040_eq_0142.png" /> <jats:tex-math>{\infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-category of motivic spectra satisfies Milnor excision: if <jats:inline-formula id="j_crelle-2021-0040_ineq_9998"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>→</m:mo> <m:mi>B</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2021-0040_eq_0063.png" /> <jats:tex-math>{A\to B}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a morphism of commutative rings sending an ideal <jats:inline-formula id="j_crelle-2021-0040_ineq_9997"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>I</m:mi> <m:mo>⊂</m:mo> <m:mi>A</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2021-0040_eq_0085.png" /> <jats:tex-math>{I\subset A}</jats:tex-math> </jats:alternatives> </jats:inline-formula> isomorphically onto an ideal of <jats:italic>B</jats:italic>, then a motivic spectrum over <jats:italic>A</jats:italic> is equivalent to a pair of motivic spectra over <jats:italic>B</jats:italic> and <jats:inline-formula id="j_crelle-2021-0040_ineq_9996"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>/</m:mo> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2021-0040_eq_0062.png" /> <jats:tex-math>{A/I}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that are identified over <jats:inline-formula id="j_crelle-2021-0040_ineq_9995"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>B</m:mi> <m:mo>/</m:mo> <m:mi>I</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mi>B</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2021-0040_eq_0065.png" /> <jats:tex-math>{B/IB}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoub’s étale motives over schemes of finite virtual cohomological dimension.</jats:p>