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Denis Nardin
Denis Nardin
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Since the question remains unanswered, let me copy Tom Goodwillie's comment:

If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely generated projective modules instead of free, it changes only π0. I believe that this is discussed somewhere in Waldhausen's big foundational paper. And in the EKMM book the corresponding issue for connective ring spectra is discussed.

In more detail, for $X$ a connected space the (∞-)category of perfect $\mathbb{S}[\Omega X]$-modules is the Spanier-Whitehead category of the category of finitely dominated retractive CW-complexes over $X$, and therefore it has the same algebraic K-theory. Restricting to finite retractive CW-complexes over $X$ corresponds to taking the stable subcategory of perfect $\mathbb{S}[\Omega X]$-modules generated by the free ones under colimit, and so by Waldhausen's cofinality theorem it just replaces the $\pi_0$ with $\mathbb{Z}$.

This is worked out in detail in Lecture 21 of Jacob Lurie's course Algebraic K-theory and manifold topology. Note in particular Warning~9 there, where Lurie remarks that his definition of A-theory differs from the "traditional" one only on $\pi_0$.

Since the question remains unanswered, let me copy Tom Goodwillie's comment:

If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely generated projective modules instead of free, it changes only π0. I believe that this is discussed somewhere in Waldhausen's big foundational paper. And in the EKMM book the corresponding issue for connective ring spectra is discussed.

In more detail, for $X$ a connected space the (∞-)category of perfect $\mathbb{S}[\Omega X]$-modules is the Spanier-Whitehead category of the category of finitely dominated retractive CW-complexes over $X$, and therefore it has the same algebraic K-theory. Restricting to finite retractive CW-complexes over $X$ corresponds to taking the subcategory of perfect $\mathbb{S}[\Omega X]$-modules generated by the free ones under colimit, and so by Waldhausen's cofinality theorem it just replaces the $\pi_0$ with $\mathbb{Z}$.

This is worked out in detail in Lecture 21 of Jacob Lurie's course Algebraic K-theory and manifold topology. Note in particular Warning~9 there, where Lurie remarks that his definition of A-theory differs from the "traditional" one only on $\pi_0$.

Since the question remains unanswered, let me copy Tom Goodwillie's comment:

If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely generated projective modules instead of free, it changes only π0. I believe that this is discussed somewhere in Waldhausen's big foundational paper. And in the EKMM book the corresponding issue for connective ring spectra is discussed.

In more detail, for $X$ a connected space the (∞-)category of perfect $\mathbb{S}[\Omega X]$-modules is the Spanier-Whitehead category of the category of finitely dominated retractive CW-complexes over $X$, and therefore it has the same algebraic K-theory. Restricting to finite retractive CW-complexes over $X$ corresponds to taking the stable subcategory of perfect $\mathbb{S}[\Omega X]$-modules generated by the free ones under colimit, and so by Waldhausen's cofinality theorem it just replaces the $\pi_0$ with $\mathbb{Z}$.

This is worked out in detail in Lecture 21 of Jacob Lurie's course Algebraic K-theory and manifold topology. Note in particular Warning~9 there, where Lurie remarks that his definition of A-theory differs from the "traditional" one only on $\pi_0$.

Source Link
Denis Nardin
Denis Nardin
  • 16.5k
  • 2
  • 69
  • 103

Since the question remains unanswered, let me copy Tom Goodwillie's comment:

If you allow finitely dominated instead of finite, it changes only π0. Analogously, in defining K(R) if you use finitely generated projective modules instead of free, it changes only π0. I believe that this is discussed somewhere in Waldhausen's big foundational paper. And in the EKMM book the corresponding issue for connective ring spectra is discussed.

In more detail, for $X$ a connected space the (∞-)category of perfect $\mathbb{S}[\Omega X]$-modules is the Spanier-Whitehead category of the category of finitely dominated retractive CW-complexes over $X$, and therefore it has the same algebraic K-theory. Restricting to finite retractive CW-complexes over $X$ corresponds to taking the subcategory of perfect $\mathbb{S}[\Omega X]$-modules generated by the free ones under colimit, and so by Waldhausen's cofinality theorem it just replaces the $\pi_0$ with $\mathbb{Z}$.

This is worked out in detail in Lecture 21 of Jacob Lurie's course Algebraic K-theory and manifold topology. Note in particular Warning~9 there, where Lurie remarks that his definition of A-theory differs from the "traditional" one only on $\pi_0$.