Definition 7.3.4. We define the spectrum \(\S /p^{\infty }\) as the cofiber of the map \(\S \to \S [\frac {1}{p}]\): \[ \S /p^{\infty } \quad := \quad \cofib (\S \to \S [\frac {1}{p}] = \colim (\S \xrightarrow {\cdot p} \S \xrightarrow {\cdot p} \S \xrightarrow {\cdot p} \dots )). \] By writing \(\S \) as the colimit over the identity maps \(\S \xrightarrow {\id } \S \), we may alternatively write \(\S /p^{\infty }\) as the colimit over the bottom row of the following commutative diagram, where all vertical sequences are cofiber sequences:
(This also justifies the notation \(\S /p^{\infty }\).)
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