Let $k$ be an algebraically closed field and $X$ be a normal variety over $k$. I am trying to show that there is a surjective group homomorphism $G_0(X)\rightarrow \mathbb{Z}\oplus \mathrm{Cl}(X)$, by using Quillen’s spectral sequence for the $G$-theory of a Noetherian scheme of finite Krull dimension. But I am not sure about the reasoning.
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$\begingroup$ See (4.1) and the following Lemma in Karmazyn-Kuznetsov-Shinder, "Derived categories of singular surfaces". $\endgroup$– Evgeny ShinderCommented Apr 28, 2023 at 6:37
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$\begingroup$ Thank you very much for your kind help. $\endgroup$– BorisCommented Apr 28, 2023 at 13:32
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$\begingroup$ Although Sasha's map is the same as the one you get from Quillen's spectral sequence, the result you're asking about follows almost immediately from the construction of the spectral sequence (with some knowledge on how spectral sequences work). Quillen constructs his spectral sequence in his paper on ``Higher algebraic K-theory", see Theorem 5.4. The spectral sequence converges to G_0(X) along the diagonal p=-q, and it clearly degenerates at E_2^{0,0} and E_2^{1,-1} (there are no nontrivial morphisms to or from here after the second page). $\endgroup$– EoinCommented Jul 8, 2023 at 1:34
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$\begingroup$ Quillen shows that the differential on the first page can be described in terms of divisors (see the proof of Proposition 5.14) when the local rings are regular, i.e. in codimension <2 when X is normal. This gives a filtration on G_0(X) with terms F^0=G_0(X), F^1, and F^2 (from the spectral sequence construction) which has the properties F^0/F^1=Z and F^1/F^2=Cl(X). The map can be gotten formally from this. $\endgroup$– EoinCommented Jul 8, 2023 at 1:37
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$\begingroup$ @Eoin Thank you so much for your insightful guidance. This is very useful information for me. $\endgroup$– BorisCommented Jul 9, 2023 at 1:24
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1 Answer
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You can construct the homomorphism directly by $$ F \mapsto (\mathrm{rank}(F), c_1(F)). $$ Here $c_1(F)$ is defined by $$ c_1(F) = c_1(F\vert_{X_0}) \in \mathrm{Pic}(X_0) = \mathrm{Cl}(X), $$ where $X_0 \subset X$ is the smooth locus of $X$.
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$\begingroup$ I would like to ask whether the restriction of a coherent sheaf $F$ on $X$ to the smooth locus of $X$ is locally free, since you are using the first Chern class for this restriction. Thanks. Maybe you are using the fact that on a smooth variety, every coherent sheaf has a finite resolution by locally free sheaves of finite rank? $\endgroup$– BorisCommented Apr 26, 2023 at 17:43
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$\begingroup$ The restriction is not necessarily locally free, so you need to use a locally free resolution after restriction. $\endgroup$– SashaCommented Apr 26, 2023 at 18:06
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$\begingroup$ Thank you very much for your kind help. I would like to ask how to show that the definition of the first Chern class of coherent sheaves is independent of the resolution by locally free sheaves. Thanks. $\endgroup$– BorisCommented Apr 26, 2023 at 19:42
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$\begingroup$ This is a standard fact, I think you can find it, e.g., in "Intersection theory" by Fulton. $\endgroup$– SashaCommented Apr 26, 2023 at 20:03
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$\begingroup$ Could I ask why you used the first chern class here? I don’t understand why the first chern class of the restriction of a coherent sheaf to the smooth locus of X is an element of the Picard group of the smooth locus. Thank you very much. $\endgroup$– BorisCommented Apr 27, 2023 at 19:48