What is the difference of Bass and Quillen K-theory groups of a ring $R$.

More concretely, what is $K^{Bass}_{i}(R)$? does it equal to $K^{Quillen}_{i}(R)$ if $R$ is a regular ring ?

Does it make sense to say that for some rings, the Bass and the Quillen K-theory agree ?
I read that Bass K-theory is sometimes called the non-connective K-theory what does it mean ? If all the answers are negative, what happens in the case when $R$ is a field ?


I read that Bass K-theory is sometimes called the non-connective K-theory

Basically, Quillen $K$-theory can be viewed as a spectrum (in the sense of topology), instead of just as a space (this is an important enrichment, not just a generalization for sake of generalization!). A spectrum $X$ also has homotopy groups in negative degrees, but if they are all trivial groups in negative degrees we call $X$ connective. Any spectrum admits a "connective cover" which has the same homotopy groups in non-negative degrees, but trivial ones in negative degrees.

The result then is that the Quillen $K$-theory spectrum $K(R)$ of a ring $R$ is the connective cover of the Bass $K$-theory spectrum $K^B(R)$. This immediately implies that $K^B_i(R) \simeq K_i(R)$ for $i \geq 0$, but the left hand side can be non-trivial if $i<0$. If $R$ is regular, it is true that $K^B_i(R) \simeq 0$ for $i<0$, and so the natural map of spectra $K^B(R) \to K(R)$ will be a homotopy equivalence of spectra. The point of introducing Bass $K$-theory is that it gives you a localization sequence and "Mayer-Vietoris" which are critical in applications, and are false for usual $K$-theory specifically because certain maps fail to be exact on $\pi_0$, which $\pi_{-1}$ ends up correcting.

This is all in Thomason-Trobaugh, but it'll be a bit of a march given the length and abstractness of the paper. Don't let that scare you though, it's fairly complete in terms of proofs.

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