Given a simplicial commutative ring $R$, you have the usual K-theory spectrum $K(R)$ defined as the group completion of the symmetric monoidal category of finitely-generated projective modules over $R$.

A more naive construction would be to just define a simplicial spectrum $[n] \mapsto K(R_n)$ and take a geometric realization. Is the realization of this simplicial spectrum related to $K(R)$?


1 Answer 1


When we talk about simplicial commutative rings, we are usually interested in simplicial commutative rings up to weak equivalences, i.e. not about the $1$-category $\mathrm{Fun}(\Delta^{\mathrm{op}}, \mathrm{CRing})$ but rather an $\infty$-categorical localisation of it known as "animated commutative rings" nowadays.

For the construction $|K(R_\bullet)|$ you describe to behave well, we would want for it to be invariant under such weak equivalences. But that doesn't hold, for example the discrete ring $\mathbb{F}_p$ also admits a resolution by polynomial rings $\mathbb{Z}[x_1,\ldots,x_\bullet]$ with $s$ generators in simplicial degree $s$. But $K(\mathbb{Z}[x_1,\ldots])\simeq K(\mathbb{Z})$ by $\mathbb{A}^1$-invariance of $K$-theory over regular rings, and so the simplicial diagram is constant. But of course $K(\mathbb{Z})\neq K(\mathbb{F}_p)$.

A fancy way of saying what goes wrong is that (the correctly defined) $K$-theory of animated commutative rings does not preserve sifted colimits (and the above description gives an example, by presenting the discrete animated ring $\mathbb{F}_p$ as sifted colimit of polynomial rings over $\mathbb{Z}$).

More generally, any simplicial commutative ring is weakly equivalent to one of polynomial rings, so if you universally localize $K$-theory on animated commutative rings to preserve sifted colimits, you obtain the constant functor with value $K(\mathbb{Z})$. This localisation is a nonabelian version of left deriving a functor, and you may think of this situation as analogous to what happens in ordinary homological algebra when deriving a non-right exact functor: It can destroy the original behaviour on non-projective objects, for example it might be constant on projectives.


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