Exposé I. Global and local cohomological invariants with respect to a closed subspace

1. The functors ${\Gamma }_Z$, $\Gamma Z$

Throughout this Exposé we write $\Gamma Z$ for the sheafified section-with-support functor (underlined in the original source) and ${\Gamma }_Z$ for the global one.

Let $X$ be a topological space, and let C_X be the category of abelian sheaves on $X$. Let $\Phi$ be a family of supports in the sense of Cartan; one defines the functor ${\Gamma }_{\Phi }$ on C_X by:

Γ_Φ(F) = subgroup of Γ(F) formed by the sections f such that support f ∈ Φ.

If $Z$ is a closed part of $X$, we designate by abuse of language by ${\Gamma }_Z$ the functor ${\Gamma }_{\Phi }$, where $\Phi$ is the set of closed parts of $X$ contained in $Z$. Hence one has:

Γ_Z(F) = subgroup of Γ(F) formed by the sections f such that support f ⊂ Z.

We wish to generalize this definition to the case where $Z$ is a locally closed part of $X$, hence closed in a suitable open part $V$ of $X$. In this case we shall set:

$${\Gamma }_Z(F)={\Gamma }_Z(F|V).$$

It must be verified that ${\Gamma }_Z(F)$ "does not depend" on the open set chosen. It suffices to show that if $V^{\prime }$, with $V\supset V^{\prime }\supset Z$, is an open set, then the map ${\rho }_{V^{\prime }}^V:F(V)\to F(V^{\prime })$ maps ${\Gamma }_Z(F|V)$ isomorphically onto ${\Gamma }_Z(F|V^{\prime })$. Now

$${\Gamma }_Z(F|V)=\ker {\rho }_{V-Z}^V,$$

so if $f\in {\Gamma }_Z(F|V)$ and if ${\rho }_{V^{\prime }}^V(f)={\rho }_{V-Z}^V(f)=0$, then $f=0$, since $(V^{\prime },V-Z)$ is a covering of $V$. Likewise, if $f^{\prime }\in {\Gamma }_Z(F|V^{\prime })$, then $f^{\prime }\in F(V^{\prime })$ and $0\in F(V-Z)$ define an $f\in F(V)$ such that ${\rho }_{V^{\prime }}^V(f)=f^{\prime }$, $f\in {\Gamma }_Z(F|V)$; hence ${\rho }_{V^{\prime }}^V$ induces an isomorphism ${\Gamma }_Z(F|V)\to {\Gamma }_Z(F|V^{\prime })$.

Note that every open set $W$ of $Z$ is induced by an open set $U$ of $X$ in which $W$ is closed. It follows that $W\mapsto {\Gamma }_W(F)$ defines a presheaf on $Z$, and one verifies that this is a sheaf, which we shall denote $i^{!}(F)$, where $i:Z\to X$ is the canonical immersion. One finds:

$${\Gamma }_Z(F)=\Gamma (i^{!}(F)).$$

The sheaf $i^{!}(F)$ is a subsheaf of $i\ast (F)$; indeed, the canonical homomorphism

Γ(F|U) = Γ(U, F) ⟶ Γ(U ∩ Z, i*(F))

is injective on ${\Gamma }_{U\cap Z}(F|U)\subset \Gamma (F|U)$. Summarizing, we have the following result:

Proposition.

There exists a unique subsheaf $i^{!}(F)$ of $i\ast (F)$ such that, for every open set $U$ of $X$ such that $U\cap Z$ is closed in $U$,

Γ(F|U) = Γ(U, F) ⟶ Γ(U ∩ Z, i*(F))

induces an isomorphism ${\Gamma }_{U\cap Z}(F|U)\to \Gamma (U\cap Z,i^{!}(F))$.

Note that if $Z$ is open, one will simply have

i^!(F) = i*(F) = F|Z, Γ_Z(F) = Γ(Z, F).

Suppose again that $Z$ is arbitrary. Then, for $U$ a variable open set of $X$, one sees that

U ⟼ Γ_{U∩Z}(F|U) = Γ(U ∩ Z, i^!(F))

is a sheaf on $X$, which we shall denote $\Gamma Z(F)$; more precisely, by the preceding formula (expressing that $i^{!}$ commutes with restriction to open sets) one has an isomorphism

$$\Gamma Z(F)=i_{\ast }(i^{!}(F))$$

by definition, one has, for every open set $U$ of $X$,

$$\Gamma (U,\Gamma Z(F))={\Gamma }_{U\cap Z}(F|U).$$

Let us note here a characteristic difference between the case where $Z$ is closed and the case where $Z$ is open. In the first case, formula (8) shows us that $\Gamma Z(F)$ can be regarded as a subsheaf of $F$, and one thus has a canonical immersion

$$\Gamma Z(F)\hookrightarrow F.$$

In the case where $Z$ is open, on the contrary, one sees from (6) that the right-hand side of (8) is $\Gamma (U\cap Z,F)$, so receives $\Gamma (U,F)$, hence one has a canonical homomorphism in the opposite direction from the previous one:

$$F⟶\Gamma Z(F),$$

which is moreover none other than the canonical homomorphism¹

$$F⟶i_{\ast }i\ast (F),$$

taking into account the isomorphism

$$\Gamma Z(F)\simeq i_{\ast }i\ast (F)$$

deduced from (6) and (7).

Of course, for $F$ variable, ${\Gamma }_Z(F)$, $\Gamma Z(F)$, $i^{!}(F)$ may be considered as functors in $F$, with values respectively in the category of abelian groups, of abelian sheaves on $X$, and of abelian sheaves on $Z$. It is sometimes convenient to interpret the functor

$$i^{!}:C_X⟶C_Z$$

as the right adjoint of a well-known functor

$$i_{!}:C_Z⟶C_X$$

defined by the following proposition:

Proposition.

Let $G$ be an abelian sheaf on $Z$. Then there exists a unique subsheaf of $i_{\ast }(G)$, say $i_{!}(G)$, such that, for every open set $U$ of $X$, the (identity) isomorphism

Γ(U ∩ Z, G) = Γ(U, i_*(G))

defines an isomorphism

Γ_{Φ_{U∩Z,U}}(U ∩ Z, G) = Γ(U, i_!(G)),

where ${\Phi }_{U\cap Z,U}$ denotes the set of parts of $U\cap Z$ that are closed in $U$.

The verification reduces to noting that the left-hand side is a sheaf for $U$ variable, i.e. that the property, for a section of $i_{\ast }(G)$ on $U$ considered as a section of $G$ on $U\cap Z$, of having support closed in $U$ is of local nature on $U$. The sheaf $i_{!}(G)$ just defined is also known under the name: sheaf deduced from $G$ by extension by 0 outside $Z$, cf. [Godement]. In particular, if $Z$ is closed, one has

$$i_{!}(G)=i_{\ast }(G);$$

but in the general case, the canonical injection $i_{!}(G)\to i_{\ast }(G)$ is not an isomorphism, as is already well known for $Z$ open. Evidently, $i_{!}(G)$ depends functorially on $G$ (and is even an exact functor in $G$). This said, one has:

Proposition.

There exists an isomorphism of bifunctors in $G$, $F$ ($G$ an abelian sheaf on $Z$, $F$ an abelian sheaf on $X$):

Hom(i_!(G), F) = Hom(G, i^!(F)).

To define such an isomorphism, it amounts to the same to define functorial homomorphisms

i_! i^!(F) ⟶ F, G ⟶ i^! i_!(G),

satisfying the well-known compatibility conditions (cf. for example Shih's exposé in the Cartan seminar on cohomological operations).

Recalling that $i_{!}$ is exact, hence transforms monomorphisms into monomorphisms, one concludes:

Corollary.

If $F$ is injective, $i^{!}(F)$ is injective, hence $\Gamma Z(F)=i_{\ast }{i}^{!}(F)$ is also injective.

Replacing $X$ by a variable open set $U$ of $X$, one also concludes from 1.3:

Corollary.

One has an isomorphism functorial in $F$, $G$:

ℋom(i_!(G), F) = i_*(ℋom(G, i^!(F))).

Taking for $G$ the constant sheaf on $Z$ defined by $\mathbb{Z}$, say ${\mathbb{Z}}_Z$, 1.3 and 1.5 specialize into

Corollary.

One has isomorphisms functorial in $F$:

$${\Gamma }_Z(F)=\mathrm{Hom}({\mathbb{Z}}_{Z,X},F),\Gamma Z(F)=\mathcal{H}om({\mathbb{Z}}_{Z,X},F),$$

where ${\mathbb{Z}}_{Z,X}=i_{!}({\mathbb{Z}}_Z)$ is the abelian sheaf on $X$ deduced from the constant sheaf on $Z$ defined by $\mathbb{Z}$, by extension by 0 outside $Z$.

Remark.

Suppose that $X$ is a ringed space, and equip $Z$ with the sheaf of rings $O_Z=i^{-1}(O_X)$; finally, denote by C_X and C_Z the category of Modules on $X$, resp. $Z$. Then the preceding considerations extend word for word, taking $F$ to be a Module on $X$ and $G$ a Module on $Z$, and interpreting accordingly statements 1.3 to 1.6.

To finish these generalities, let us examine what happens when one changes the locally closed part $Z$. Let $Z^{\prime }\subset Z$ be another locally closed part, and let

j: Z′ ⟶ Z,   i′: Z′ ⟶ X,   i′ = i j

be the canonical inclusions. Then one has functorial isomorphisms:

(i j)^! = j^! i^!,   (i j)_! = i_! j_!.

The first isomorphism (13) defines a functorial isomorphism

Γ_{Z′}(F) = Γ(Z′, (i j)^!(F)) ≃ Γ(Z′, j^!(i^!(F))) = Γ_{Z′}(i^!(F)).

Suppose now that $Z^{\prime }$ is closed in $Z$, and let

$$Z^{\prime \prime }=Z-Z^{\prime }$$

be its complement in $Z$, which is open in $Z$, hence locally closed in $X$. The canonical inclusion (8′) applied to $i^{!}(F)$ on $Z$ equipped with $Z^{\prime }$ defines, thanks to (14), an injective functorial canonical homomorphism

$${\Gamma }_{Z^{\prime }}(F)⟶{\Gamma }_Z(F).$$

If in (14) one replaces $Z$ by $Z^{\prime \prime }$ and uses (8″), one finds a functorial canonical homomorphism:

$${\Gamma }_Z(F)⟶{\Gamma }_{Z^{\prime \prime }}(F).$$

Proposition.

Under the preceding conditions, the sequence of functorial homomorphisms

$$0⟶{\Gamma }_{Z^{\prime }}(F)⟶{\Gamma }_Z(F)⟶{\Gamma }_{Z^{\prime \prime }}(F)$$

is exact. If $F$ is flasque, the sequence remains exact when one puts a zero on the right.

Proof. Replacing $X$ by an open set $V$ in which $Z$ is closed, one reduces to the case where $Z$ is closed, hence $Z^{\prime }$ is closed. Then $Z^{\prime \prime }$ is closed in the open set $X-Z^{\prime }$, and one has a canonical inclusion

Γ_{Z″}(F) ⟶ Γ(X − Z′, F),

and the exactness of (16) simply means that the sections of $F$ with support in $Z^{\prime }$ are those whose restriction to $X-Z^{\prime }$ is zero.

When $F$ is flasque, every element of ${\Gamma }_{Z^{\prime \prime }}(F)$, considered as a section of $F$ on $X-Z^{\prime }$, can be extended to a section of $F$ on $X$, and the latter will evidently have its support in $Z$, which proves that the last homomorphism in (16) is then surjective.

Corollary.

One has a functorial exact sequence

$$0⟶\Gamma Z^{\prime }(F)⟶\Gamma Z(F)⟶\Gamma Z^{\prime \prime }(F),$$

and if $F$ is flasque, this sequence remains exact when one puts a 0 on the right.

One may interpret (1.8) in terms of results on the functors Hom and $\mathcal{H}om$ via 1.6, in the following way. Let us first note that if $G$ is an abelian sheaf on $Z$, inducing the sheaves $j\ast (G)$ and $k\ast (G)$ on $Z^{\prime }$ resp. $Z^{\prime \prime }$ (where $j:Z^{\prime }\to Z$ and $k:Z^{\prime \prime }\to Z$ are the canonical injections), one has a canonical exact sequence of sheaves on $X$:

$$0⟶k\ast (G{)}_X⟶G_X⟶j\ast (G{)}_X⟶0,$$

where, to simplify the notation, the subscript $X$ designates the sheaf on $X$ obtained by extending by 0 in the complement of the space of definition of the sheaf in question. The exact sequence (17) generalizes a well-known exact sequence in the case $Z=X$ (cf. [Godement]), and is moreover deduced from the latter by writing the exact sequence in question on $Z$, and applying the functor $i_{!}$. Taking $G={\mathbb{Z}}_Z$, one concludes in particular:

Proposition.

Under the preceding conditions, one has an exact sequence of abelian sheaves on $X$:

$$0⟶{\mathbb{Z}}_{Z^{\prime \prime },X}⟶{\mathbb{Z}}_{Z,X}⟶{\mathbb{Z}}_{Z^{\prime },X}⟶0.$$

This being so, the two exact sequences 1.8 and 1.9 are nothing other than the exact sequences deduced from (18) by application of the functor $\mathrm{Hom}(-,F)$ resp. $\mathcal{H}om(-,F)$.

This gives an evident proof of the fact that the sequences (16) and (16 bis) remain exact when one puts a zero on the right, provided that $F$ is injective.

2. The functors $H_Z^{\ast }(X,F)$ and ${\mathcal{H}}_Z^{\ast }(F)$

Definition.

One denotes by $H_Z^{\ast }(X,F)$ and ${\mathcal{H}}_Z^{\ast }(F)$ the derived functors in $F$ of the functors ${\Gamma }_Z(F)$ resp. $\Gamma Z(F)$.

These are cohomological functors, with values in the category of abelian groups resp. in the category of abelian sheaves on $X$. When $Z$ is closed, $H_Z^{\ast }(X,F)$ is, by definition, nothing other than $H_{\Phi }^{\ast }(X,F)$ where $\Phi$ denotes the family of closed parts of $X$ contained in $Z$. When $Z$ is open, we shall see that $H_Z^{\ast }(X,F)$ is nothing other than $H^{\ast }(Z,F)=H^{\ast }(Z,F|Z)$, thanks to the following proposition.

Proposition (Excision Theorem).

Let $V$ be an open part of $X$ containing $Z$. Then one has an isomorphism of cohomological functors in $F$:

H^*_Z(X, F) ⟶ H^*_Z(V, F|V).

Indeed, one has a functorial isomorphism ${\Gamma }_Z^X\simeq {\Gamma }_Z^V{j}^{!}$, where $j:V\to X$ is the inclusion, and where $j^{!}$ is thus the restriction functor (cf. (14)). This last is exact, and transforms injectives into injectives by 1.4, whence the isomorphism (19) at once.

When $Z$ is open, one may take $V=Z$ and one finds:

Corollary.

Suppose $Z$ open; then one has an isomorphism of cohomological functors:

H^*_Z(X, F) = H^*(Z, F).

One concludes from isomorphisms 1.6 and from the definitions (cf. [Tôhoku]):

Proposition.²

One has isomorphisms of cohomological functors:

H^*_Z(X, F) ≃ Ext^*(X; ℤ_{Z,X}, F),

$${\mathcal{H}}_Z^{\ast }(F)\simeq \mathcal{E}x{t}^{\ast }({\mathbb{Z}}_{Z,X},F).$$

One may therefore apply the results of [Tôhoku] on the Ext of Modules. Let us first point out the following interpretation of the sheaves ${\mathcal{H}}_Z^{\ast }(F)$ in terms of the global groups $H_Z^{\ast }(X,F)$:

Corollary.

${\mathcal{H}}_Z^{\ast }(F)$ is canonically isomorphic to the sheaf associated to the presheaf

U ⟼ H^*_{Z ∩ U}(U, F|U).

In particular, using corollary 2.3, one finds:

Corollary.

Suppose $Z$ open; then one has an isomorphism of cohomological functors:

ℋ^*_Z(F) = R^* i_* i*(F)

(where $i:Z\to X$ is the inclusion).

The spectral sequence of Ext gives the important spectral sequence:

Theorem.

One has a spectral sequence functorial in $F$, abutting to $H_Z^{\ast }(X,F)$ and with initial term

$$E_2^{p,q}(F)=H^p(X,{\mathcal{H}}_Z^q(F)).$$

Remarks.

It follows at once from 2.4 that the sheaves ${\mathcal{H}}_Z^q(F)$ are zero in $X-Z$, and also zero in the interior of $Z$ for $q\ne 0$ (so for such a $q$, ${\mathcal{H}}_Z^q(F)$ is even supported on the boundary of $Z$).

Consequently, the right-hand side of (23) may be interpreted as a cohomology group on $Z$. We shall use 2.6 in the case where $Z$ is closed in $X$, in which case the right-hand side of (23)³ may be interpreted as a cohomology group computed on $Z$:

$$E_2^{p,q}(F)=H^p(Z,{\mathcal{H}}_Z^q(F)).$$

Let us also note that when $Z$ is open, the spectral sequence 2.6 is nothing other than the Leray spectral sequence for the continuous map $i:Z\to X$, taking into account the interpretation 2.5 in the calculation of the initial term of the Leray spectral sequence.

Let us return to the exact sequence (18);⁴ it gives rise to an exact sequence of Ext (cf. [Tôhoku]):

Theorem.

Let $Z$ be a locally closed part of $X$, $Z^{\prime }$ a closed part of $Z$, and $Z^{\prime \prime }=Z-Z^{\prime }$. Then one has an exact sequence functorial in $F$:

0 ⟶ H⁰_{Z′}(X, F) ⟶ H⁰_Z(X, F) ⟶ H⁰_{Z″}(X, F) ─∂─→ H¹_{Z′}(X, F) ⟶ H¹_Z(X, F) ...

... H^i_{Z′}(X, F) ⟶ H^i_Z(X, F) ⟶ H^i_{Z″}(X, F) ─∂─→ H^{i+1}_{Z′}(X, F) ...

Let us recall how this exact sequence can be obtained. Let $C(F)$ be an injective resolution of $F$; then the exact sequence (18)⁵ gives rise to the exact sequence

$$0⟶{\Gamma }_{Z^{\prime }}(C(F))⟶\Gamma (C(F))⟶{\Gamma }_{Z^{\prime \prime }}(C(F))⟶0,$$

(which is nothing other than the one defined in 1.8). One concludes an exact sequence of cohomology, which is nothing other than (24).

The most important case for us is the one where $Z$ is closed (and one can moreover always reduce to it by replacing $X$ by an open set $V$ in which $Z$ is closed). Then $Z^{\prime }$ is closed, $Z^{\prime \prime }$ is closed in the open set $X-Z^{\prime }$, and one may write

H^i_{Z″}(X, F) = H^i_{Z″}(X − Z′, F|_{X−Z′}),

which allows us to write the exact sequence (24) in terms of cohomologies with support in a given closed set. The most frequent case is the one where $Z=X$. Setting then, for simplification, $Z^{\prime }=A$, one finds:

Corollary.

Let $A$ be a closed part of $X$. Then one has an exact sequence functorial in $F$:

0 ⟶ H⁰_A(X, F) ⟶ H⁰(X, F) ⟶ H⁰(X − A, F) ─∂─→ H¹_A(X, F) ...

... H^i_A(X, F) ⟶ H^i(X, F) ⟶ H^i(X − A, F) ─∂─→ H^{i+1}_A(X, F) ...

This exact sequence shows that the cohomology group $H_A^i(X,F)$ plays the role of a relative cohomology group of $X$ mod $X-A$, with coefficients in $F$. It is on this account that it was introduced naturally in applications. By "sheafifying" (24) and (27), or by proceeding directly, one finds, taking into account that the sheaf associated to $U\mapsto H^i(U,F)$ is zero if $i>0$:

Corollary.

Under the conditions of 2.8, one has an exact sequence functorial in $F$:

... ℋ^i_{Z′}(F) ⟶ ℋ^i_Z(F) ⟶ ℋ^i_{Z″}(F) ─∂─→ ℋ^{i+1}_{Z′}(F) ...

Corollary.

Let $A$ be a closed part of $X$; then one has an exact sequence functorial in $F$:

0 ⟶ ℋ⁰_A(F) ⟶ F ⟶ f_*(F|_{X−A}) ─∂─→ ℋ¹_A(F) ⟶ 0,

and canonical isomorphisms, for $i⩾2$:

ℋ^i_A(F) = ℋ^{i−1}_{X−A}(F) = R^{i−1} f_*(F|_{X−A}),

where $f:(X-A)\to X$ is the inclusion.

This therefore defines ${\mathcal{H}}_A^0(F)$ and ${\mathcal{H}}_A^1(F)$ respectively as the kernel and cokernel of the canonical homomorphism

F ⟶ f_* f*(F) = f_*(F|_{X−A}),

and the ${\mathcal{H}}_A^i(F)$ ($i⩾2$) in terms of the derived functors of $f_{\ast }$.

Corollary.

Let $F$ be an abelian sheaf on $X$. If $F$ is flasque, then for every locally closed part $Z$ of $X$ and every integer $i\ne 0$, one has $H_Z^i(X,F)=0$, ${\mathcal{H}}_Z^i(F)=0$. Conversely, if for every closed part $Z$ of $X$ one has $H_Z^1(X,F)=0$, then $F$ is flasque.

Suppose that $F$ is flasque; then $F$ induces a flasque sheaf on every open set, so to prove $H_Z^i(X,F)=0$ for $i>0$, one may suppose $Z$ closed, and then the assertion follows from the exact sequence (27).⁶ One concludes, for every locally closed $Z$, by "sheafifying", i.e. applying 2.4, that ${\mathcal{H}}_Z^i(F)=0$ for $i>0$. Conversely, suppose $H_Z^1(X,F)=0$ for every closed $Z$; then the exact sequence (27)⁷ shows that for every such $Z$, $H^0(X,F)\to H^0(X-Z,F)$ is surjective, which means that $F$ is flasque.

Combining 2.6 and 2.8, we shall deduce from them:

Proposition.

Let $F$ be an abelian sheaf on $X$, $Z$ a closed part of $X$, $U=X-Z$, $N$ an integer. The following conditions are equivalent:

(i) ${\mathcal{H}}_Z^i(F)=0$ for $i⩽N$.

(ii) For every open set $V$ of $X$, considering the canonical homomorphism

H^i(V, F) ⟶ H^i(V ∩ U, F),

this homomorphism is:

a) bijective for $i<N$,

b) injective for $i=N$.

(When $N>0$, one may in (ii) restrict to requiring a)).

To prove (i) ⇒ (ii), one is reduced, thanks to the local nature of the ${\mathcal{H}}_Z^i(F)$, to proving the

Corollary.

If condition 2.13 (i) is satisfied, then

H^i(X, F) ⟶ H^i(U, F)

is bijective for $i<N$, injective for $i=N$.

Indeed, by virtue of the exact sequence (27), this also means $H_Z^i(X,F)=0$ for $i⩽N$, and this relation is an immediate consequence of the spectral sequence (23 bis).⁸

Conversely, hypothesis 2.13 (ii) means that for every open set $V$ of $X$, one has

H^i_{Z ∩ V}(V, F|V) = 0 for i ⩽ N,

which implies 2.13 (i) thanks to 2.4. If moreover $N>0$, hypothesis b) is superfluous. Indeed, if $N=1$, hypothesis a) and (28) ensure the vanishing of ${\mathcal{H}}_Z^i(F)=0$ for $i⩽N$. If $N>1$, hypothesis a) for $i=N-1>0$ and (29) ensure the vanishing of ${\mathcal{H}}_Z^i(F)$ for $i⩽N$.

Taking 2.11 into account, this further proves 2.13 (i)...

Remark.

Let $Y\to X$ be a closed immersion, and suppose that locally it is of the form $0\times Y\subset {\mathbb{R}}^n\times Y$. Suppose that $F$ is a locally constant sheaf on $X$; then one finds

ℋ^i_Y(F) ≃ { 0            if i ≠ n,
           { F ⊗ T_{Y,X}   if i = n,    where  T_{Y,X} ≃ ℋ^n_Y(ℤ_X)

is a sheaf, extension to $X$ of a sheaf on $Y$ locally isomorphic to ${\mathbb{Z}}_Y$, called the "normal orientation sheaf of $Y$ in $X$".

Using the spectral sequence (23 bis),⁹ one finds in this case:

H^i_Y(X, F) ≃ H^{i−n}(Y, F ⊗ T_{Y,X}),

and one recovers the "Gysin homomorphism":

H^j(Y, F ⊗ T_{Y,X}) ⟶ H^{j+n}(X, F).

Bibliography

[Godement] R. Godement — Topologie algébrique et théorie des faisceaux, Act. Scient. et Ind., vol. 1252, Hermann, Paris.

[Tôhoku] A. Grothendieck — "Sur quelques points d'algèbre homologique", Tôhoku Math. J. 9 (1957), pp. 119–221.

Footnotes