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nLab > nLab General Discussions: Adams tower as Tot-tower

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMar 3rd 2016
- (edited Mar 3rd 2016)
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Author: Urs
Format: MarkdownItexWhat would be a source for an explicit derivation -- for $E$ an $E_\infty$-ring and $X$ a spectrum -- that the classical $E$-Adams tower corresponding to the canonical $E$-Adams resolution $N(X \wedge E^{\wedge^\bullet})$ is equivalently the Tot-tower of the cosimplicial spectrum $X \wedge E^{\wedge^\bullet}$? That's not immediate, is it?

What would be a source for an explicit derivation – for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>$ an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>$ -ring and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ a spectrum – that the classical $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>$ -Adams tower corresponding to the canonical $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>$ -Adams resolution $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><msup><mi>E</mi> <mrow><msup><mo>∧</mo> <mo>•</mo></msup></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(X \wedge E^{\wedge^\bullet})</annotation></semantics></math>$ is equivalently the Tot-tower of the cosimplicial spectrum $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>∧</mo><msup><mi>E</mi> <mrow><msup><mo>∧</mo> <mo>•</mo></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X \wedge E^{\wedge^\bullet}</annotation></semantics></math>$ ? That’s not immediate, is it?

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nLab > nLab General Discussions: Adams tower as Tot-tower