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Relations between slices and quotients of the algebraic cobordism spectrum. (English) Zbl 1209.14019
Let $$MGL$$ be the algebraic cobordism spectrum. V. Voevodsky [Int. Press Lect. Ser. 3, No. I, 3–34 (2002; Zbl 1047.14012)] introduced the slice filtration and formulated a conjecture known as the slice conjecture [Conjecture(5)]. This conjecture describes the slice filtration using motivic Eilenberg-Mac Lane spectrum amd the coefficients of the complex cobordism $$MU_{*}.$$ The author discusses the relation between the slice conjecture and quotients of $$MGL.$$ Let $${ L}_{*}$$ be the graded Lazard ring. In motivic homotopy theory the filtration on $$MGL$$ obtained by dividing out by ideals of $$L_{*}$$ generated by the $$x_{i},$$ or more precisely ideals of $$L_{*}$$ consisting of elements of degree greater than a given bound, is conjecturally the slice filtration. The author proves that if this holds on the level zero then it holds for all levels. The situation for rational slices and $$K$$-theory is also described.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 19E20 Relations of $$K$$-theory with cohomology theories
##### Keywords:
slice filtration; algebraic cobordism; K-theory
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