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Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra".

As next step, I moved to the paper by Kapranov regarding derived categories of Grassmanians and quadrics, "On the derived categories of coherent sheaves on some homogeneous spaces". As it is written also in the introduction of this paper, the result about quadrics was also obtained by Swan independently in his article "$K$-theory of quadric hypersurfaces".

Even if I don't know much yet about $K$-theory, I thought that it is worth to try to read it, in order to get a different point of view to the problem. Neverthless, the equivalence of the results in the two papers is to me quite obscure. I wonder if anyone could enlighten me about the connection with finding generators of the derived category on the quadric.

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  • $\begingroup$ Working on the paper I found that there is at most one connection, i.e. the minimal resolution proposed by Swan in Lemma 7.3 coincides with the "Tate resolution" considered by Kapranov in (4.1). But this is not very deep, I think $\endgroup$ – Ramac Jun 3 '16 at 17:39
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The connection is via the additivity theorem in (connective) algebraic $K$-theory or, better, the fact that algebraic $K$-theory is the universal additive invariant as proven in BGT. Let me try to give an answer that blackboxes $K$-theory in terms of its universal property.

So Beilinson provided a semiorthogonal decomposition of the derived category of coherent sheaves $\mathbb{P}^n$ which reads as $D(Coh(\mathbb{P}^n)) \simeq D(Coh(k)) \oplus ... D(Coh(k))$. Just as decompositions of an abelian group comes from splitting of short exact sequences, this decomposition takes place as the splitting of "short exact sequences" in some category. One form of this category is the category of idempotent complete small stable $\infty$-categories which 1 denotes by $Cat^{perf}_{\infty}$ and a short exact sequence is really a cofiber sequence in this category which splits (morphisms are exact functors, a cofiber sequence $A \rightarrow B \rightarrow C$ roughly witnesses $C$ as a Verdier quotient of $B$ by $A$ up to idempotent completion, splitting means that there appropriate adjoints splitting the sequence above). If you are more inclined with the $dg$-language, this is also explained nicely in Tabuada's survey.

Anyway algebraic $K$-theory is a functor $Cat^{perf}_{\infty} \rightarrow Spectra$. By the additivity theorem (see also [BGT, Proposition 7.10]) this functor carries split cofiber sequences to sums. It turns out that it is actually the "universal" such functor (see 1 for precise statements). So the Beilinson decomposition above is yet another way to see that $K(\mathbb{P}^n_k) \simeq K(k) \oplus ... K(k)$, ditto for the quadrics and Grassmanian stuff

To summarize: if you find a nice set of generators for your derived category, then you can hope for a semiorthogonal decomposition. This is interpreted as a split cofiber sequence in some category and algebraic $K$-theory eats in stuff in this category and takes these cofiber sequences to sums in spectra. Furthermore this property "characterizes" the $K$-theory functor.

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