The Picard group of motivic $$\mathcal {A}_\mathbb {C}(1)$$

Gheorghe, Bogdan; Isaksen, Daniel C.; Ricka, Nicolas

doi:10.1007/s40062-018-0200-z

Cited by 2 publications

(6 citation statements)

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“…In fact, recent progress has determined that our results do not depend on this deep and difficult work. There are now purely topological constructions of homotopy categories that have identical computational properties to the cellular stable C-motivic homotopy category [8] [30]. In these homotopy categories, one can obtain from first principles the fundamental computations of the cohomology of a point and of the Steenrod algebra, using only well-known classical computations.…”

Section: New Ingredientsmentioning

confidence: 99%

“…Using a topological model of the cellular stable C-motivic homotopy category, one can construct a "motivic modular forms" spectrum mmf [8], whose motivic cohomology is the quotient of the C-motivic Steenrod algebra by its subalgebra generated by Sq 1 , Sq 2 , and Sq 4 . Just as tmf plays an essential role in studies of the classical Adams spectral sequence, mmf is an essential tool for motivic computations.…”

Section: New Ingredientsmentioning

confidence: 99%

“…Over C, a "motivic modular forms" spectrum mmf has recently been constructed [8]. From our computational perspective, mmf is a ring spectrum whose cohomology is A//A(2), i.e., the quotient of the C-motivic Steenrod algebra by the subalgebra generated by Sq 1 , Sq 2 , and Sq 4 .…”

Section: Motivic Modular Formsmentioning

confidence: 99%

“…(5) Deduce additional Adams differentials for the C-motivic sphere with a variety of ad hoc arguments. The most important methods are Toda bracket shuffles and comparison to the motivic modular forms spectrum mmf [8]. (6) Deduce hidden τ extensions in the C-motivic Adams spectral sequence for the sphere, using a long exact sequence in homotopy groups.…”

Section: Introductionmentioning

confidence: 99%

“…Adams d 3 differentials in the C-motivic Adams spectral sequence (s, f, w) Lemma 5.12(37,8,21) τ e 0 g c 0 (τ 2 e 0 g) (38, 4, 21) e 1 h 1 t Cτ(39,12,21) τ P d 0 e 0 P 2 c 0 d 0 h 1 (40, 4, 22) f 1 0 h 2 (46, 14, 24) i 2 τ P 2 h 1 d 2 0 tmf (47, 16, 25) τ P 2 d 0 e 0 P 3 c 0 d 0 h 1 (47, 18, 24) h 5 0 Q ′ P 4 h 0 d 0 Cτ (49, 6, 27) h 1 h 5 e 0 M h Cτ (49, 11, 26) τ 2 d 0 m P ∆h (τ 3 gm) (55, 20, 29) τ P 3 d 0 e 0 P 4 c 0 d 0 h 1(55, 7, 30) B 6 τ h 2 gn Cτ (56, 8, 31) h 5 c 0 e 0 13, 30) τ ∆h 1 d 0 e 0 P ∆h 1 c 0 d 0 d 4 (τ 2 ∆h 1 d 0 e 0 ) (57, 12, 33) τ e 0 g 2 Cτ (57, 8, 30) h 5 j M P h 2 Cτ (57, 15, 30) τ 2 P d 0 m P 2 ∆h 2 1 d 0 mmf (57, 7, 30) Q 2 τ 2 gt τ ∆h 1 g (58, 8, 33) e 1 g h 1 gt Cτ (60, 12, 35) τ g 3 10, 33) h 1 · ∆x + τ M e 0 M P c 0 Cτ (62, 10, 32) τ h 1 · ∆x 0 ?τ 2 M h 0 e 0 Cτ (63, 24, 33) τ P 4 d 0 e 0 P 5 c 0 d 0 h 1 (64, 17, 34) τ P ∆h 1 d 0 e 0 P 2 ∆h 1 c 0 d 0 d 4 (τ 2 P ∆h 1 d 0 e 0 ) (65, 13, 36) ∆h 2 2 m M P h 3 1 c 0 Cτ (65, 19, 34) τ 2 P 2 d 0 m P 3 ∆h 2 1 d 0 mmf…”

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Stable Stems

Isaksen

2019

Memoirs of the AMS

104

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We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, η, and ν, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences.We also compute the motivic stable homotopy groups of the cofiber of the motivic element τ . This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of τ are the same as the E 2 -page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known. ContentsList of Tables vii Chapter 1. Introduction 1.1. The Adams spectral sequence program 1.2. Motivic homotopy theory 1.3. The motivic Steenrod algebra 1.4. Relationship between motivic and classical calculations 1.5. Relationship to the Adams-Novikov spectral sequence 1.6. How to use this manuscript 1.7. Notation 1.8. Acknowledgements Chapter 2. The cohomology of the motivic Steenrod algebra 2.1. The motivic May spectral sequence 2.2. Massey products in the motivic May spectral sequence 2.3. The May differentials 2.4. Hidden May extensions Chapter 3. Differentials in the Adams spectral sequence 3.1. Toda brackets in the motivic Adams spectral sequence 3.2. Adams differentials 3.3. Adams differentials computations Chapter 4. Hidden extensions in the Adams spectral sequence 4.1. Hidden Adams extensions 4.2. Hidden Adams extensions computations Chapter 5. The cofiber of τ 5.1. The Adams E 2 -page for the cofiber of τ 5.2. Adams differentials for the cofiber of τ 5.3. Hidden Adams extensions for the cofiber of τ Chapter 6. Reverse engineering the Adams-Novikov spectral sequence 6.1. The motivic Adams-Novikov spectral sequence 6.2. The motivic Adams-Novikov spectral sequence for the cofiber of τ 6.3. Adams-Novikov calculations Chapter 7. Tables Index Bibliography v 45 Hidden Adams-Novikov η extensions 46 Hidden Adams-Novikov ν extensions 47 Correspondence between classical Adams and Adams-Novikov E ∞ 48 Classical Adams-Novikov boundaries 49 Classical Adams-Novikov non-permanent classes

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Section: New Ingredientsmentioning

confidence: 99%

Section: New Ingredientsmentioning

confidence: 99%

Section: Motivic Modular Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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