## On the $$K$$-theory of pullbacks.(English)Zbl 1427.19002

The Milnor-Bass-Murty (MBM) theorem associates a long exact sequence of $$K$$-groups to a Milnor square of rings. This long exact sequence starts at $$K_1$$ and goes into negative $$K$$-theory. The natural idea is to extend this sequence to all Quillen $$K$$-groups. In the most straightforward sense, this is not actually possible as shown by R. G. Swan [J. Pure Appl. Algebra 1, 221–252 (1971; Zbl 0262.16025)].
Land and Tamme, in the title under review, have discovered how to extend this sequence in a less obvious fashion using spectra (in the sense of a sequence of topological spaces). To a Milnor square of rings $\begin{matrix} A & \rightarrow & B\\ \downarrow & & \downarrow \\ A' & \rightarrow & B' \end{matrix}$ they associate a spectra $$X = A'\odot_A^{B'} B$$ such that the classic MBM sequence extends as $\cdots\to K_i(A)\to K_i(A')\oplus K_i(B)\to K_i(X)\to K_{i-1}(A)\to\cdots$ valid in all degrees.
The authors actually formulate the result more generally. A Milnor square is a special case of a pullback of $$\mathbb{E}_1$$-spectra and $$K$$-theory is a localizing invariant, so we can formulate the result with these more general objects. This comes in useful as there are many localizing invariants that are also important in the realm of $$K$$-theory (some of which can be used to compute $$K$$-theory itself). Therefore, the context of the result is appropriate.
Their theorem of course puts many classic theorems such as Suslin’s excision theorem, in a more general context. One particular I found rather interesting: if $$A$$ is an associative ring and $$I$$ a two-sided nilpotent ideal of $$A$$ such that $$NI = 0$$ for some $$N > 0$$, then all the relative $$K$$-groups $$K_i(A,A/I)$$ are annihilated by some power of $$N$$. This result was previously only available in the case where $$A$$ is a $$\mathbb{Z}/N$$ algebra, which is due to Geisser and Hesselholt. The authors also give a general cdh-descent type theorem: that any truncating invariant satisfies cdh-descent on quasicompact, quasiseparated schemes.
The paper itself is a tour de force of modern derived algebra and infinity category theory, and is an outstanding example of how some of the ideas of higher category theory have come to clarify the scene of algebraic $$K$$-theory.

### MSC:

 19D55 $$K$$-theory and homology; cyclic homology and cohomology 19D50 Computations of higher $$K$$-theory of rings 19E08 $$K$$-theory of schemes

Zbl 0262.16025
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### References:

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