Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2013
   • (edited Jun 17th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexlast Friday, when the nForum was down, I had created very brief entries * _[[graded field]]_ and * _[[infinity-field]]_

   last Friday, when the nForum was down, I had created very brief entries

   • graded field

   and

   • infinity-field
2. • CommentRowNumber2.
   • CommentAuthorDylan Wilson
   • CommentTimeJun 18th 2013
   • PermaLink
   Author: Dylan Wilson
   Format: TextI have some trouble with what you've written in "infinity-field." First of all: I think the standard definition of a 'field' *in the category of spectra* only requires that one have a homotopy associative ring spectrum (very very weak). This sort of jives: the Morava K-theories should really be thought of as the 'unique' prime fields, but they admit many different A_infty structures, so if we required A_\infty in the definition, we would think we had lots more fields, which is a bit awkward. Second: This definition might lead one to suspect that if I wanted to check if a map of R-modules is nilpotent (over an E_\infty ring spectrum, R), then it's enough to check this after tensoring with every 'infty-field' over R. This is true when R=S, and I'm pretty sure this is a wide open problem for every other E_\infty ring spectrum. Indeed, however we define 'infty-field' it should probably have this property...

   I have some trouble with what you've written in "infinity-field."
   
   First of all: I think the standard definition of a 'field' *in the category of spectra* only requires that one have a homotopy associative ring spectrum (very very weak). This sort of jives: the Morava K-theories should really be thought of as the 'unique' prime fields, but they admit many different A_infty structures, so if we required A_\infty in the definition, we would think we had lots more fields, which is a bit awkward.
   
   Second: This definition might lead one to suspect that if I wanted to check if a map of R-modules is nilpotent (over an E_\infty ring spectrum, R), then it's enough to check this after tensoring with every 'infty-field' over R. This is true when R=S, and I'm pretty sure this is a wide open problem for every other E_\infty ring spectrum. Indeed, however we define 'infty-field' it should probably have this property...
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 18th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexre first: I have added "$A_\infty$-algebra or in fact just an H-space". re second: not sure what you want me to do here. I didn't invent the term "field" for this notion. But feel free to edit the entry as you see further need!

   re first: I have added “$A_\infty$-algebra or in fact just an H-space”.

   re second: not sure what you want me to do here. I didn’t invent the term “field” for this notion.

   But feel free to edit the entry as you see further need!