Establishing a right-exact sequence in K-theory

I am confused about problem II.6.10 b) in Hartshorne's Algebraic Geometry. The question goes:

For a scheme $X$ let $K(X)$ be the Grothendieck group of coherent sheaves on $X$, i.e. the free abelian group generated by isomorphism classes of coherent sheaves, modulo the relations $\mathcal{F}-\mathcal{F'}-\mathcal{F''}$ for each exact sequence $0\rightarrow\mathcal{F'}\rightarrow\mathcal{F}\rightarrow\mathcal{F''}\rightarrow0$.

Now, fix $X$ to be a Noetherian scheme, and $Z$ a closed subscheme. Then show that there is an exact sequence $$K(Z)\rightarrow K(X) \rightarrow K(X-Z)\rightarrow 0$$

where the first arrow is pushforward and the second is restriction.

I am having difficulty showing that an arbitrary class in $K(X)$ that goes to zero once restricted to $X-Z$ is in the image of the first map. There is a hint that basically tells you how to show that for $\mathcal{F}$ a coherent sheaf on $X$ with support contained in $Z$, $\mathcal{F}$ is in the image of $K(Z)$, but I do not know how to reduce to this case, from starting with a general class in $K(X)$ which may at best be a difference of classes of coherent sheaves.

• By $X-Z$, do you mean the open subscheme of $X$ given by the topological / set complement of $Z$? – Arthur Jun 4 '17 at 5:44
• If you're okay with doing something else, you know the pullback induces a map $K(X)/K(Z)\rightarrow K(X-Z)$. You can try to define an inverse $K(X-Z)\rightarrow K(X)/K(Z)$ to this map, which shows the first map is an injection, hence isomorphism, hence the sequence is exact. This is apparently done in this master's thesis as theorem 1.9. – Eoin Jun 4 '17 at 7:20
• As an aside, I don't know how it will help, every element can be written as the difference of classes of coherent sheaves, like $[\mathcal{F}]-[\mathcal{G}]$. – Eoin Jun 4 '17 at 7:31
• @Arthur Yes, that is what I mean. – A. S. Jun 4 '17 at 16:57
• @Eoin I will look at your reference later, thank you. – A. S. Jun 4 '17 at 16:58

The following proof is adapted from Grothendieck's original proof in [SGA6, Exp. 0, App., Prop. 2.10], although my argument ended up being much longer since I couldn't verify some of the statements he gives.

If $P$ is an object of an essentially small abelian category $\mathcal{A}$, then we denote $\beta(P)$ to be its associated isomorphism class in the free abelian group generated by all isomorphism classes of coherent sheaves on $\mathcal{A}$, and $\gamma(P)$ to be the image of $\beta(P)$ in the Grothendieck group $K(\mathcal{A})$ of $\mathcal{A}$. When $\mathcal{A}$ is the category of coherent sheaves on a noetherian scheme $X$ (which is essentially small by [ATJLPRVG, Thm. 4.2]), we denote $\beta$ and $\gamma$ by $\beta_X$ and $\gamma_X$, respectively.

We will need the following:

Lemma 1 [Weibel, Ch. II, Exer. 6.4]. Let $\mathcal{A}$ be an essentially small abelian category, and let $K(\mathcal{A})$ be its Grothendieck group. If $A_1,A_2$ are two objects of $\mathcal{A}$ such that $\gamma(A_1) = \gamma(A_2)$ in $K(\mathcal{A})$, then there are objects $C',C''$ in $\mathcal{A}$ and two extensions $$\label{eq:extns}\tag{1} 0 \longrightarrow C' \longrightarrow C_1 \longrightarrow C'' \longrightarrow 0, \qquad 0 \longrightarrow C' \longrightarrow C_2 \longrightarrow C'' \longrightarrow 0$$ in $\mathcal{A}$, such that $A_1 \oplus C_1 \simeq A_2 \oplus C_2$.

Proof. By definition of $K(\mathcal{A})$, there exist short exact sequences $$\label{eq:relns}\tag{2} 0 \longrightarrow D_i' \longrightarrow D_i \longrightarrow D_i'' \longrightarrow 0, \qquad 0 \longrightarrow D_j' \longrightarrow D_j \longrightarrow D_j'' \longrightarrow 0,$$ such that $$\beta(A_1) - \beta(A_2) = \sum_i \bigl(\beta(D_i) - \beta(D_i') - \beta(D_i'')\bigr) - \sum_j \bigl(\beta(D_j) - \beta(D_j') - \beta(D_j'')\bigr),$$ and so rearranging terms, we have $$\beta(A_1) + \sum_i \bigl(\beta(D_i') + \beta(D_i'')\bigr) + \sum_j \beta(D_j) = \beta(A_2) + \sum_j \bigl(\beta(D_j') + \beta(D_j'')\bigr) + \sum_i \beta(D_i).$$ In $\mathcal{A}$, we therefore have $$A_1 \oplus \underbrace{\bigoplus_i (D_i' \oplus D_i'') \oplus \bigoplus_j D_j}_{C_1} \simeq A_2 \oplus \underbrace{\bigoplus_j (D_j' \oplus D_j'') \oplus \bigoplus_i D_i}_{C_2}$$ and we define $C_1,C_2$ as indicated. To show the existence of the extensions in \eqref{eq:extns}, we have the exact sequences

where the bottom row of the first diagram and the top row of the second diagram are formed by taking direct sums of the sequences in \eqref{eq:relns}. $\blacksquare$

We now move to the setting of the question. Let $U = X \smallsetminus Z$.

Lemma 2. Let $\mathscr{F}$ and $\mathscr{G}$ be two coherent sheaves on $X$ such that $\mathscr{F}\rvert_U \simeq \mathscr{G}\rvert_U$. Then, we have $$\gamma_X(\mathscr{F}) - \gamma_X(\mathscr{G}) = \gamma_X(\mathscr{R}) - \gamma_X(\mathscr{S})$$ for some coherent sheaves $\mathscr{R}$ and $\mathscr{S}$ on $X$ with support in $Z$.

Proof. We consider the graph $\Gamma$ of an isomorphism $\varphi\colon\mathscr{F}\rvert_U \overset{\sim}{\to} \mathscr{G}\rvert_U$, defined as the image of $(\mathrm{id},\varphi)$ in $\mathscr{F}\rvert_U \oplus \mathscr{G}\rvert_U$. The graph fits into the commutative diagram

where the morphisms from $\mathscr{F}\rvert_U \oplus \mathscr{G}\rvert_U$ are the two projections. Since $\Gamma$ is coherent, by Chapter II, Exercise 5.15(d), there exists a coherent subsheaf $\widetilde{\Gamma}$ in $\mathscr{F} \oplus \mathscr{G}$ such that $\widetilde{\Gamma}\rvert_U = \Gamma$, and such that we have the commutative diagram

where the morphisms $f$ and $g$ are defined by composition with the projections from $\mathscr{F} \oplus \mathscr{G}$. Since $f$ and $g$ become isomorphisms once restricted to $U$, we have the exact sequences \begin{gather} 0 \longrightarrow \mathscr{K}_1 \longrightarrow \widetilde{\Gamma} \overset{f}{\longrightarrow} \mathscr{F} \longrightarrow \mathscr{Q}_1 \longrightarrow 0\\ 0 \longrightarrow \mathscr{K}_2 \longrightarrow \widetilde{\Gamma} \overset{g}{\longrightarrow} \mathscr{G} \longrightarrow \mathscr{Q}_2 \longrightarrow 0 \end{gather} where $\mathscr{K}_i$ and $\mathscr{Q}_i$ are supported in $Z$ for $i = 1,2$. We therefore obtain \begin{align} \gamma_X(\mathscr{F}) - \gamma_X(\mathscr{G}) &= \bigl(\gamma_X(\widetilde{\Gamma}) + \gamma_X(\mathscr{Q}_1) - \gamma_X(\mathscr{K}_1)\bigr) - \bigl(\gamma_X(\widetilde{\Gamma}) + \gamma_X(\mathscr{Q}_2) - \gamma_X(\mathscr{K}_2)\bigr)\\ &= \gamma_X(\mathscr{Q}_1 \oplus \mathscr{K}_2) - \gamma_X(\mathscr{K}_1 \oplus \mathscr{Q}_2) \end{align} and setting $\mathscr{R} = \mathscr{Q}_1 \oplus \mathscr{K}_2$ and $\mathscr{S} = \mathscr{K}_1 \oplus \mathscr{Q}_2$, we are done. $\blacksquare$

We can finally move on to the main proof.

Main Proof. We would like to show that every element in the kernel of $K(X) \to K(U)$ is in the image of $K(Z) \to K(X)$. By Eoin's comment, we may assume that such an element is of the form $$\gamma_X(\mathscr{A}_1) - \gamma_X(\mathscr{A}_2).$$ By assumption, we have $$\gamma_U(\mathscr{A}_1\rvert_U) - \gamma_U(\mathscr{A}_2\rvert_U) = 0$$ in $K(U)$, hence by Lemma 1, we have an isomorphism $$\mathscr{A}_1\rvert_U \oplus \mathscr{C}_1 \longrightarrow \mathscr{A}_2\rvert_U \oplus \mathscr{C}_2$$ for some coherent sheaves $\mathscr{C}_1$ and $\mathscr{C}_2$ on $U$ which fit into short exact sequences as in \eqref{eq:extns}. Now for each $i = 1,2$, let $\widetilde{\mathscr{C}}_i$ be coherent sheaves on $X$ such that $\widetilde{\mathscr{C}}_i\rvert_U = \mathscr{C}_i$; these exist by Chapter II, Exercise 5.15. We then have that $\mathscr{A}_1 \oplus \widetilde{\mathscr{C}}_1$ and $\mathscr{A}_2 \oplus \widetilde{\mathscr{C}}_2$ are coherent sheaves on $X$ that have isomorphic restrictions on $U$, hence by Lemma 2, $$\label{eq:almost}\tag{3} \gamma_X(\mathscr{A}_1) - \gamma_X(\mathscr{A}_2) = \gamma_X(\widetilde{\mathscr{C}}_2) - \gamma_X(\widetilde{\mathscr{C}}_1) + \gamma_X(\mathscr{R}) - \gamma_X(\mathscr{S})$$ for some coherent sheaves $\mathscr{R}$ and $\mathscr{S}$ with support in $Z$. By applying Chapter II, Exercise 5.15(d) to $\widetilde{\mathscr{C}}_i$ for $i = 1,2$ and the inclusions in the short exact sequences in \eqref{eq:extns}, we have two short exact sequences $$0 \longrightarrow \widetilde{\mathscr{C}}'_1 \longrightarrow \widetilde{\mathscr{C}}_1 \longrightarrow \widetilde{\mathscr{C}}''_1 \longrightarrow 0, \qquad 0 \longrightarrow \widetilde{\mathscr{C}}'_2 \longrightarrow \widetilde{\mathscr{C}}_2 \longrightarrow \widetilde{\mathscr{C}}''_2 \longrightarrow 0$$ such that $\widetilde{\mathscr{C}}'_1$ and $\widetilde{\mathscr{C}}'_2$ are equal to $\mathscr{C}'$ when restricted to $U$, and similarly, $\widetilde{\mathscr{C}}''_1$ and $\widetilde{\mathscr{C}}''_2$ are isomorphic to $\mathscr{C}''$ when restricted to $U$. By Lemma 2, we therefore have \begin{align} \gamma_X(\widetilde{\mathscr{C}}_2) - \gamma_X(\widetilde{\mathscr{C}}_1) &= \bigl(\gamma_X(\widetilde{\mathscr{C}}'_2) + \gamma_X(\widetilde{\mathscr{C}}''_2)\bigr) - \bigl(\gamma_X(\widetilde{\mathscr{C}}'_1) + \gamma_X(\widetilde{\mathscr{C}}''_1)\bigr)\\ &= \gamma_X(\mathscr{R}') - \gamma_X(\mathscr{S}') + \gamma_X(\mathscr{R}'') - \gamma_X(\mathscr{S}''). \end{align} Combining this equation with \eqref{eq:almost}, we obtain $$\gamma_X(\mathscr{A}_1) - \gamma_X(\mathscr{A}_2) = \gamma_X(\mathscr{R}) - \gamma_X(\mathscr{S}) + \gamma_X(\mathscr{R}') - \gamma_X(\mathscr{S}') + \gamma_X(\mathscr{R}'') - \gamma_X(\mathscr{S}'').$$ Now by the hint Hartshorne gives, each of these classes on the right are in the image of $K(Z)$ since their respective representatives are coherent sheaves supported on $Z$. $\blacksquare$

• Now that is some serious typing. – user2055 Sep 3 '17 at 8:30