The definition of K-theory of a scheme $X$ is defined as $G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$. But usually the schemes are required to be (at least locally) Noetherian, and usually it is said that if it is not then the $G_i$'s are pretty bad.

But for what reasons that we really need the condition of being noetherian? (If it is not, then $\mathrm{Coh}(X)$ is not abelian, but we only need it to be exact, which might be satisfied.)

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    $\begingroup$ You don't need the scheme to be Noetherian. Usually quasi compact quasi separated is enough to get a well-behaved theory. But you need to use different definitions to work in full generality (e.g. your scheme might not have enough vector bundles for the definition you are using to be meaningful). Have you tried to have a look at Thomason-Trobaugh? $\endgroup$ – Denis Nardin Nov 18 '19 at 22:03
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    $\begingroup$ Not wanting to open a little question about notation, I'll ask here. I've seen people using the following notations A) $K_0(R):=K_0(Vect(Spec(R)))$, $K^0(X):=K_0(Vect(X))$ (upper index cause it's contravariant in $X$, I imagine), and $K_0(X):=K_0(Coh(X))$ (lower index cause it's covariant in $X$, I imagine). And B) $K^0(X):=K_0(Vect(X))$, and $G_0:=K_0(Coh(X))$. Which of A), B) is prevalent? Or the notation of the OP? $\endgroup$ – Qfwfq Nov 18 '19 at 23:04
  • $\begingroup$ @Qfwfq For algebraic-geometors, B) is good. When $X$ is not regular, $K_i$ is not $K^i$. Your B) is actually my notations, where we found ways to distinguish $K_i(\mathrm{Coh}(X))$ and $K_i(\mathrm{Vect}(X))$. But, both can be either covariant and contravariant. I guess the reason we want $K_i(\mathrm{Vect}(X))$ to be cohomological was $K_0(\mathrm{Vect}(X))$ has a natrual action on $K_0(\mathrm{Coh}(X))$, which is an analogy of cohomology rings acting on homology groups. $\endgroup$ – Li Guanyu Nov 18 '19 at 23:24
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    $\begingroup$ Yes, $K_0(Coh)$ is a module over the ring $K_0(Vect)$. Since we're here I'll ask: do these groups (together with the corresponding pieces with nonzero index) have anything to do with the homology and cohomology groups associated to the same homotopical object (such as ring spectrum...)? $\endgroup$ – Qfwfq Nov 18 '19 at 23:32
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    $\begingroup$ @Qfwfq Regarding your second question, this answer by Denis-Charles Cisinski explains the situation well. $\endgroup$ – Denis Nardin Nov 19 '19 at 6:37

You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least the following properties

  • K-theory of an affine scheme $\mathrm{Spec}\,R$ is given by the algebraic K-theory of projective $R$-modules in the sense of Quillen

  • K-theory satisfies Zariski descent (as a spectrum)

Since every scheme has a Zariski cover by affine schemes it is clear that there's at most one definition that satisfies the above two properties. At least when $X$ is quasi compact quasi separated this can be expressed more geometrically as the algebraic K-theory of perfect complexes. An introduction that does not require much background can be found in the classical paper by Thomason and Trobaugh, Higher Algebraic K-theory of schemes and of Derived categories (although if you want to invest some time learning some modern homotopy theory can only be beneficial -- many of the proofs in Thomason-Trobaugh can be simplified if you have more modern technology at your disposal).

Similarly for G-theory the "correct" analogue for qcqs schemes is the algebraic K-theory spectrum of bounded pseudocoherent complexes (also known as "almost perfect" complexes in parts of the literature).

Both notions of K-theory and G-theory recover Quillen's definition when working on Noetherian schemes (and for K-theory in a much greater generality -- whenever the scheme has an ample family of line bundles).

  • $\begingroup$ Denis, can you comment this sentence : “although if you want to invest some time learning some modern homotopy theory can only be beneficial -- many of the proofs in Thomason-Trobaugh can be simplified if you have more modern technology at your disposal” ? Could you give hints or references ? Thanks. $\endgroup$ – ACL Nov 19 '19 at 9:36
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    $\begingroup$ @ACL Essentially,algebraic K-theory is much better studied using (∞,1)-categories.Probably the standard references here are Blumberg-Gepner-Tabuada'sA universal characterization of higher algebraic K-theory and Barwick's On the K-theory of higher categories,but there's plenty of material (cfr. the recent paper by Land and Tamme on excision, or the work by Clausen,Mathew,Naumann,Noel for some stricking applications which are probably more accessible),and this is without mentioning motivic homotopy theory $\endgroup$ – Denis Nardin Nov 19 '19 at 9:40
  • $\begingroup$ T&T is available as a pdf from Rainicki's archive. And the title is Higher algebraic K-theory of schemes .... $\endgroup$ – theHigherGeometer Nov 21 '19 at 22:30
  • $\begingroup$ @DavidRoberts Thanks for the link. And I can't believe I forgot "higher" in the title... $\endgroup$ – Denis Nardin Nov 21 '19 at 22:36

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