On commutative diagrams consisting of low term exact sequences

Chang Lv State Key Laboratory of Information Security
Institute of Information Engineering
Chinese Academy of Sciences
Beijing 100093, P.R. China lvchang@amss.ac.cn

Abstract.

We establish several useful commutative diagrams consisting of low term exact sequences attached to Grothendieck spectral sequences, which extends and integrates the previous ones appeared in literature such as Alexei N. Skorobogatov [Beyond the Manin obstruction, Invent. Math. (1999)], and [On the elementary obstruction to the existence of rational points, Mathematical Notes (2007)]. Parts of the diagrams was frequently used in local-global principle to rational points.

Key words and phrases:

Ext functor, spectral sequences, derived functors

2000 Mathematics Subject Classification:

13D07, 18G40, 18G10

This work was supported by National Natural Science Foundation of China (Grant No. 11701552).

1. The commutative diagram

Suppose that $\Phi:\mathcal{A}\rightarrow\mathcal{B}$ and $\Psi_{t}:\mathcal{B}\rightarrow\mathcal{C}$ are left exact additive functors between abelian categories, $t=1,2,3$ . Assume that $\mathcal{A}$ and $\mathcal{B}$ have enough injectives and $\Psi_{t}$ takes injectives to $\Phi$ -acyclics. Then for any $A\in\operatorname{Ob}(\mathcal{A})$ , we have the Grothendieck spectral sequence

(S)_{\Phi,\Psi_{t},A}

^{t}E_{2}^{p,q}=(R^{p}\Psi)(R^{q}\Phi)A\Rightarrow\ ^{t}E^{p+q}=R^{p+q}(\Psi\Phi)A

and the low term exact sequence

(E)_{\Phi,\Psi_{t},A}

0\rightarrow\ ^{t}E_{2}^{1,0}\rightarrow\ ^{t}E^{1}\rightarrow\ ^{t}E_{2}^{0,1}\rightarrow\ ^{t}E_{2}^{2,0}\rightarrow\ ^{t}E_{1}^{2}\rightarrow\ ^{t}E_{2}^{1,1}\rightarrow\ ^{t}E_{2}^{3,0}

attached to them, where ${}^{t}E_{1}^{2}=\ker(^{t}E^{2}\rightarrow\ ^{t}E_{2}^{0,2})$ . Let $\mathbf{D}^{+}(\mathcal{A})$ , $\mathbf{D}^{+}(\mathcal{B})$ , $\mathbf{D}^{+}(\mathcal{C})$ be the corresponding derived category of complexes bounded below, $\mathbf{R}\Psi$ , $\mathbf{R}\Psi_{t}$ the corresponding derived functor and $\mathbb{R}^{\bullet}$ (or $\mathbb{H}^{\bullet}$ ) the hyppercohomology functor.

Proposition 1.1.

With the previous notation, suppose that there are morphism of functors $u:\Psi_{1}\rightarrow\Psi_{2}$ and $v:\Psi_{2}\rightarrow\Psi_{3}$ such that for any $F\in\mathbf{D}^{+}(\mathcal{B})$ ,

(1.2)

is a distinguished triangle functorial in $F$ .

(i)

We have the long exact sequence

$\dots\rightarrow\ ^{t}E_{2}^{i,0}\rightarrow\ ^{t}E_{\leq 1}^{i}\rightarrow\ ^{t}E_{2}^{i-1,1}\rightarrow\ ^{t}E_{2}^{i+1,0}\rightarrow\dots$

where ${}^{t}E_{\leq 1}^{i}=\mathbb{R}^{i}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))$ .

(ii)

We have the following commutative diagram with exact rows and columns

(1.3)

where ${}^{1}E_{\leq 1}^{3}$ fits into the exact sequence

(1.4)

0\rightarrow\ ^{1}E_{1}^{2}\rightarrow\ ^{1}E^{2}\rightarrow\ ^{1}E_{2}^{0,2}\rightarrow\ ^{1}E_{\leq 1}^{3}\rightarrow\ ^{1}E^{3},

the rows are parts of the low term exact sequences $(E)_{\Phi,\Psi_{t},A}$ attached to the spectral sequences $(S)_{\Phi,\Psi_{t},A}$ with $t$ numbered on the left upper corner of each object, and the columns are induced by taking cohomology at $1$ of (1.2) in which $F$ is substituted with $\tau_{[0]}\mathbf{R}\Phi(A)$ , $\tau_{\leq 1}\mathbf{R}\Phi(A)$ , $\tau_{[1]}\mathbf{R}\Phi(A)$ , $\tau_{[0]}\mathbf{R}\Phi(A)[1]$ , $\tau_{\leq 1}\mathbf{R}\Phi(A)[1]$ , respectively.

(iii)

Suppose we are given $\beta\in\ ^{3}E^{1}$ and $\gamma\in\ ^{2}E_{2}^{0,1}$ such that they map to the same element in ${}^{3}E_{2}^{0,1}$ . Then there exists $\alpha\in\ ^{1}E_{2}^{2,0}$ such that $\alpha,\ \beta$ map to the same element in ${}^{1}E_{1}^{2}$ and $-\alpha,\ \gamma$ map to the same element in ${}^{2}E_{2}^{2,0}$ . In other words, we have the zig-zag diagram

(iv)

The statement of (iii) is also correct if we move our focus one step right. That is, we are given $\beta\in\ ^{3}E_{2}^{0,1}$ and $\gamma\in\ ^{2}E_{2}^{2,0}$ , and so on.

Proof.

The proof widely extends [5, Lem. 3] and [7, Prop. 1.1]. For any $F\in\mathbf{D}^{+}(\mathcal{A})$ , the truncation functors determine the distinguished triangle

(1.5)

\tau_{\leq 0}F\rightarrow F\rightarrow\tau_{\geq 1}F\rightarrow(\tau_{\leq 0}F)[1].

Note that $\mathbf{R}\Psi_{t}$ , $t=1,2,3$ are triangulated. Along with the functorial distinguished triangles (1.2) in which $F$ is substituted with $\tau_{\leq 0}F$ , $F$ and $\tau_{\geq 1}F$ respectively, we obtain the following commutative diagram

(1.6)

Let $F=\tau_{\leq 1}\mathbf{R}\Phi(A)=\tau_{[0,1]}\mathbf{R}\Phi(A)$ and taking cohomology at $1$ , the diagram becomes

(1.7)

with exact rows and columns.

We now identify the objects appearing in (1.7) with the ones in (1.3). Clearly for any $t=1,2,3$ and $i,j,k\in\mathbb{Z}$ with $j,i-j+k\geq 0$ , we have

\mathbb{R}^{i}\Psi_{t}(\tau_{[j]}\mathbf{R}\Phi(A))[k]=(R^{i-j+k}\Psi_{t})(R^{j}\Phi)A=\ ^{t}E_{2}^{i-j+k,j}.

It remains to identify $\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))$ with ${}^{t}E^{1}$ and $\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))[1]$ with ${}^{t}E_{1}^{2}$ .

For any $F$ consider the distinguished triangle

\tau_{\leq 1}F\rightarrow F\rightarrow\tau_{\geq 2}F\rightarrow(\tau_{\leq 1}F)[1]

and note that $\Psi_{t}(\tau_{\geq 2}F)$ is acyclic in $0$ and $1$ . Then we have the long exact sequences

	$\displaystyle\mathbb{R}^{0}\Psi_{t}(\tau_{\geq 2}F)\rightarrow$	$\displaystyle\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}F)\rightarrow\mathbb{R}^{1}\Psi_{t}(F)\rightarrow\mathbb{R}^{1}\Psi_{t}(\tau_{\geq 2}F)$
	$\displaystyle\rightarrow$	$\displaystyle\mathbb{R}^{2}\Psi_{t}(\tau_{\leq 1}F)\rightarrow\mathbb{R}^{2}\Psi_{t}(F)\rightarrow\mathbb{R}^{2}\Psi_{t}(\tau_{\geq 2}F)$
(1.8)		$\displaystyle\rightarrow$	$\displaystyle\mathbb{R}^{3}\Psi_{t}(\tau_{\leq 1}F)\rightarrow\mathbb{R}^{3}\Psi_{t}(F)$

where $\mathbb{R}^{i}\Psi_{t}(\tau_{\geq 2}F)=0$ with $t=1,2,3$ and $i=0,1$ . Now we take $F=\mathbf{R}\Phi(A)$ . It follows that

\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))=\mathbb{R}^{1}\Psi_{t}(\mathbf{R}\Phi(A))=\ ^{t}E^{1}

where the last equality follows from the isomorphism of functors

(1.9)

(\mathbf{R}\Psi_{t})(\mathbf{R}\Phi)\cong\mathbf{R}(\Psi_{t}\Phi).

In a same manner, (1) and (1.9) yield

	$\displaystyle\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))[1]$	$\displaystyle=\mathbb{R}^{2}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))$
		$\displaystyle=\ker\left(\mathbb{R}^{2}\Psi_{t}(\mathbf{R}\Phi(A))\rightarrow\mathbb{R}^{2}\Psi_{t}(\tau_{\geq 2}\mathbf{R}\Phi(A))\right)$
		$\displaystyle=\ker\left(\mathbb{R}^{2}(\Psi_{t}\Phi)A\rightarrow\mathbb{R}^{2}\Psi_{t}(\tau_{\geq 2}\mathbf{R}\Phi(A))\right).$

Then the low term exact sequence attached to the hyppercohomology spectral sequence [4, Appendix C (g)]

E_{2}^{p,q}=R^{p}g(H^{q}(F))\Rightarrow\mathbb{R}^{p+q}g(F)

gives the isomorphism when $F=\tau_{\geq 2}\mathbf{R}\Phi(A)$ and $g=\Psi_{t}$

\mathbb{R}^{2}\Psi_{t}(\tau_{\geq 2}\mathbf{R}\Phi(A))\cong\Psi_{t}(R^{2}\Phi(A)),

since in this case $E_{2}^{p,q}=0$ for all $p$ and $q=0,1$ . Thus

\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))[1]=\ker\left(\mathbb{R}^{2}(\Psi_{t}\Phi)A\rightarrow\Psi_{t}(R^{2}\Phi(A))\right)=\ ^{t}E_{1}^{2}.

Similarly,

\mathbb{R}^{1}\Psi_{1}(\tau_{\leq 1}\mathbf{R}\Phi(A))[2]=\mathbb{R}^{3}\Psi_{1}(\tau_{\leq 1}\mathbf{R}\Phi(A))=\ ^{1}E_{\leq 1}^{3}

and the exact sequence (1.4) is also deduced from (1) and (1.9). This completes the identification of objects.

Finally, in diagram (1.7), the identification of the vertical arrows is clear. For that of the horizontal ones, it follows from a general fact for such spectral sequences. See, for example, [5, Appendix B], which shows that $\mathbb{R}^{1}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))\rightarrow\mathbb{R}^{1}\Psi_{t}(\tau_{[1]}\mathbf{R}\Phi(A))$ is exactly the edge map ${}^{t}E_{1}\rightarrow\ ^{t}E_{2}^{0,1}$ . This completes the proof of (ii) as well as (i).

Consider the subdiagram of (1.6)

(1.10)

whose rows and columns are all distinguished triangles. Since up to an isomorphism, every distinguished triangle in a derived category arises from some short exact sequence of complexes [2, Chap. IV.2 8. Prop.], we may view (1.10) as a commutative diagram consisting of three rows and three columns of short exact sequences of complexes. Then the result follows from [6, Lem. 4.3.2] by taking $i=1$ . This completes the proof of (iii).

(iv) is similar as (iii). The proof is complete. ∎

Next we describe a variant of Proposition 1.1.

Proposition 1.11.

Keeping assumptions in Proposition 1.1, suppose that there are $A\in\mathbf{D}^{+}(\mathcal{A})$ and $B\in\mathbf{D}^{+}(\mathcal{B})$ with a morphism

f:B\rightarrow\tau_{\leq 1}\mathbf{R}\Phi(A).

Let $\Delta=\Delta(\Phi,A,B,f)$ be the cone of $-f[1]$ . Denote ${}^{t}F^{p}=\mathbb{R}^{p}\Psi_{t}B$ and ${}^{t}G^{p}=\mathbb{R}^{p}\Psi_{t}\Delta$ . Then we have the long exact sequence

\dots\rightarrow F^{i}\rightarrow\ ^{t}E_{\leq 1}^{i}\rightarrow G^{i-1}\rightarrow F^{i+1}\rightarrow\dots

where ${}^{t}E_{\leq 1}^{i}=\mathbb{R}^{i}\Psi_{t}(\tau_{\leq 1}\mathbf{R}\Phi(A))$ and the following commutative diagram with exact rows and columns

(1.12)

where ${}^{1}E_{\leq 1}^{3}$ fits into the exact sequence

0\rightarrow\ ^{1}E_{1}^{2}\rightarrow\ ^{1}E^{2}\rightarrow\ ^{1}E_{2}^{0,2}\rightarrow\ ^{1}E_{\leq 1}^{3}\rightarrow\ ^{1}E^{3}.

Moreover, similar statements as (iii), (iv) in Proposition 1.1 hold. That is, if we are given $\beta\in\ ^{3}E^{1}$ $($ resp. ${}^{3}G^{0})$ and $\gamma\in\ ^{2}G^{0}$ $($ resp. ${}^{2}F^{2})$ , then the corresponding zig-zag diagram as in Proposition 1.1 is correct.

Proof.

The same as Proposition 1.1 except that in the diagram (1.7) we replace the distinguished triangle

(1.13)

\tau_{[0]}\mathbf{R}\Phi(A)\rightarrow\tau_{\leq 1}\mathbf{R}\Phi(A)\rightarrow\tau_{[1]}\mathbf{R}\Phi(A)\rightarrow\tau_{[0]}\mathbf{R}\Phi(A)[1]

B\rightarrow\tau_{\leq 1}\mathbf{R}\Phi(A)\rightarrow\Delta[-1]\rightarrow B[1].

∎

Remark 1.14.

(a)

Obviously, Proposition 1.1 is the special case of Proposition 1.11 where $f$ is the canonical map $\tau_{[0]}\mathbf{R}\Phi(A)\rightarrow\tau_{\leq 1}\mathbf{R}\Phi(A)$ .

(b)

If we replace the distinguished triangle (1.13) by

\tau_{[0]}\mathbf{R}\Phi(A)\rightarrow\mathbf{R}\Phi(A)\rightarrow\tau_{\geq 1}\mathbf{R}\Phi(A)\rightarrow\tau_{[0]}\mathbf{R}\Phi(A)[1]

we also have the long exact sequence

\dots\rightarrow\ ^{t}E_{2}^{i,0}\rightarrow\ ^{t}E^{i}\rightarrow\ ^{t}E_{\geq 1}^{i}\rightarrow\ ^{t}E_{2}^{i+1,0}\rightarrow\dots

where ${}^{t}E_{\geq 1}^{i}=\mathbb{R}^{i}\Psi_{t}(\tau_{\geq 1}\mathbf{R}\Phi(A))$ .

2. Applications

Suppose that $f_{*}:\mathcal{A}\rightarrow\mathcal{B}$ is a left exact additive functor between two abelian categories which has a left adjoint $f^{*}$ . Assume that $\mathcal{A}$ and $\mathcal{B}$ has enough injectives and $f^{*}$ is exact. For example, $f:X\rightarrow Y$ is a morphism of topoi, and $\mathcal{A}={Mod}(X,\Lambda)$ , $\mathcal{B}={Mod}(Y,\Lambda)$ . Let $M\in\operatorname{Ob}(\mathcal{B})$ and $N\in\operatorname{Ob}(\mathcal{A})$ . Then we have the Grothendieck spectral sequences

(2.1)

^{M}E_{2}^{p,q}=\operatorname{Ext}_{\mathcal{A}}^{p}(M,R^{q}f_{*}N)\Rightarrow\operatorname{Ext}_{\mathcal{B}}^{p+q}(f^{*}M,N).

For simplicity, we omit the category letter in $\operatorname{Ext}$ ’s if it does not cause a confusion.

Corollary 2.2.

Let $C,A\in\operatorname{Ob}(\mathcal{B})$ and $u\in\operatorname{Ext}^{1}(C,A)$ be the element representing the extension

(2.3)

0\rightarrow A\xrightarrow{i}B\xrightarrow{j}C\rightarrow 0.

Define

	$\displaystyle\operatorname{Ext}_{1}^{2}(f^{}M,N)=\ker\left(\operatorname{Ext}^{2}(f^{}M,N)\rightarrow\operatorname{Hom}(M,R^{2}f_{*}N)\right),$
	$\displaystyle\operatorname{Ext}_{\leq 1}^{3}(f^{}M,N)=\operatorname{Ext}^{3}(M,\tau_{\leq 1}\mathbf{R}f_{}N).$

(i)

We have the following commutative diagram with exact rows and columns

where the rows are parts of the low term exact sequences attached to ${}^{A}E_{2}^{p,q}$ and ${}^{C}E_{2}^{p,q}$ defined in (2.1).

(ii)

The statement of Proposition 1.1 (iii) $($ resp. (iv) $)$ is also correct if we put $\beta,\gamma$ in the corresponding positions. That is, we are given $\beta\in\operatorname{Ext}^{1}(f^{*}A,N)$ $($ resp. $\operatorname{Hom}(A,R^{1}f_{*}N))$ and $\gamma\in\operatorname{Hom}(A,R^{1}f_{*}N)$ $($ resp. $\operatorname{Ext}^{2}(B,f_{*}N))$ , and so on.

Proof.

We shall use Proposition 1.1. Let

A\rightarrow B\rightarrow C\rightarrow A[1]

be the distinguished triangle in $\mathbf{D}^{+}(\mathcal{B})$ determined by (2.3). Take $\mathcal{C}={Ab}$ , $\Psi_{1}=\operatorname{Hom}(C,-)$ , $\Psi_{2}=\operatorname{Hom}(B,-)$ , $\Psi_{3}=\operatorname{Hom}(A,-)$ and $\Phi=f_{*}$ , which clearly satisfy the assumptions in Proposition 1.1 (c.f. [8, Thm. 10.7.4]). Then the result follows. ∎

Let $k$ be a field with characteristic $0$ and $\Gamma=\operatorname{Gal}(\overline{k}/k)$ where $\overline{k}$ is a fixed algebraic closure of $k$ . Let $p:X\rightarrow\operatorname{Spec}k$ be a $k$ -variety and $\overline{X}=X\times_{k}\overline{k}$ . In Corollary 2.2 take $\mathcal{B}$ be the category of discrete $\Gamma$ -modules, $\mathcal{A}$ the category of étale sheaves on $X$ . We write $\operatorname{Ext}_{k}$ for $\operatorname{Ext}_{\mathcal{B}}$ , $\operatorname{Ext}_{X}$ for $\operatorname{Ext}_{\mathcal{A}}$ and

\operatorname{Ext}_{1}^{2}(p^{*}T,\mathbf{G}_{m})=\ker\left(\operatorname{Ext}_{X}^{2}(p^{*}T,\mathbf{G}_{m})\rightarrow\operatorname{Hom}_{k}(T,\operatorname{Br}\overline{X})\right)\quad\text{ for any $\Gamma$-module $T$. }

Note that $\operatorname{Ext}_{k}^{1}(\mathbb{Z},-)=H^{1}(k,-)$ .

Corollary 2.4.

With previous notation, let $u\in H^{1}(k,M)$ be the element representing the extension

(2.5)

0\rightarrow M\xrightarrow{i}S\xrightarrow{j}\mathbb{Z}\rightarrow 0.

Define $H_{\leq 1}^{3}(X,\mathbf{G}_{m})=H^{3}(X,\tau_{\leq 1}\mathbf{R}p_{*}\mathbf{G}_{m})$ .

(i)

We have the following commutative diagram with exact rows and columns

(2.6)

where the top, middle and bottom row is a part of the low term exact sequences attached to the $E_{2}$ spectral sequences

\operatorname{Ext}_{k}^{p}(M,R^{q}p_{*}\mathbf{G}_{m})\Rightarrow\operatorname{Ext}_{X}^{p+q}(p^{*}M,\mathbf{G}_{m}),

\operatorname{Ext}_{k}^{p}(S,R^{q}p_{*}\mathbf{G}_{m})\Rightarrow\operatorname{Ext}_{X}^{p+q}(p^{*}S,\mathbf{G}_{m})

and

H^{p}(k,H^{q}(\overline{X},\mathbf{G}_{m}))\Rightarrow H^{p+q}(X,\mathbf{G}_{m})

respectively.

(ii)

Let $\beta\in\operatorname{Hom}_{k}(M,\operatorname{Pic}\overline{X})$ be such that $u\cup\beta\in\operatorname{im}r$ . Then there exists $\alpha\in\operatorname{Br}_{1}X$ and $\gamma\in\operatorname{Ext}_{k}^{2}(S,\overline{k}[X]^{\times})$ such that $r(\alpha)=u\cup\beta$ , $i^{*}(\gamma)=\partial(\beta)$ and $p^{*}(j^{*})(-\alpha)=p^{*}(\gamma)$ . In other words, we have the diagram

Proof.

Use Corollary 2.2. Take $f_{*}=p_{*}$ , $N=\mathbf{G}_{m}$ and (2.3) to be (2.5). Then (i) follows from the facts that for $p,q\geq 0$ , $R^{q}p_{*}\mathbf{G}_{m}=H^{q}(\overline{X},\mathbf{G}_{m})$ and $H^{p}(X,\mathbf{G}_{m})=\operatorname{Ext}_{X}^{p}(p^{*}\mathbb{Z},\mathbf{G}_{m})$ , (see [6, p. 23] and [1, Prop. 1.4.1], respectively).

The existence of $\gamma$ follows from an easy diagram chase. Since $u\cup\beta\in\operatorname{im}\operatorname{Br}_{1}X$ ,

u\cup\partial(\beta)=d(u\cup\beta)=0.

Then there exists $\gamma\in\operatorname{Ext}_{k}^{2}(S,\overline{k}[X]^{\times})$ such that $i^{*}(\gamma)=\partial(\beta)$ . Then (ii) follows. ∎

Remark 2.7.

We have some remarks on Corollary 2.4.

(1)

One may use the variant Proposition 1.11 to replace $\operatorname{Pic}\overline{X}$ (resp. $\overline{k}[X]^{\times}$ ) appearing in the diagrams by $\operatorname{KD}^{\prime}(X)$ (resp. $\overline{k}^{\times}$ ). To be precise, take $f$ in Proposition 1.11 to be $\mathbf{G}_{m,k}\rightarrow\tau_{\leq 1}\mathbf{R}p_{*}\mathbf{G}_{m,X}$ . Then $\Delta=\operatorname{KD}^{\prime}(X)$ (see [3]).

(2)

If moreover $M$ is finitely generated (hence so is $S$ ), then its Catier dual $\hat{M}$ is a group of multiplicative type. It can be shown that

	$\displaystyle H^{p}(X,\hat{T})=\operatorname{Ext}_{X}^{p}(p^{*}T,\mathbf{G}_{m}),$
	$\displaystyle H^{p}(k,\hat{T})=\operatorname{Ext}_{k}^{p}(T,\mathbf{G}_{m})$

for $p\geq 0$ and $T=M$ or $S$ (c.f. [1, Prop. 1.4.1]). Write $H_{1}^{2}(X,T)=\ker(H^{2}(X,T)\rightarrow\operatorname{Hom}_{k}(T,\operatorname{Br}\overline{X})$ . Now (2.6) becomes

where the second and third rows are fundamental exact sequences for open varieties and the maps $\lambda$ are so-called extended type. See [3].

References

[1] Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492. MR 899402
[2] S. I. Gelfand and Y. I. Manin, Methods of homological algebra, Springer Science & Business Media, 2013.
[3] David Harari and Alexei N. Skorobogatov, Descent theory for open varieties, Torsors, étale homotopy and applications to rational points, London Math. Soc. Lecture Note Ser., vol. 405, Cambridge Univ. Press, Cambridge, 2013, pp. 250–279. MR 3077172
[4] J. S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton Univ. Press, 1980.
[5] Alexei N. Skorobogatov, Beyond the Manin obstruction, Invent. Math. 135 (1999), no. 2, 399–424. MR 1666779
[6] by same author, Torsors and rational points, vol. 144, Cambridge University Press, 2001.
[7] by same author, On the elementary obstruction to the existence of rational points, Mathematical Notes 81 (2007), no. 1-2, 97–107.
[8] C. A. Weibel, An introduction to homological algebra, Cambridge University Press, 1995.