MATHEMATICS

V. KUZ’MINOV

ON THE LIMIT SEQUENCE OF AN EXACT SEQUENCE OF INVERSE SPECTRA

(Presented by Academician P. S. Aleksandrov, 13 I 1965)

In this note, by methods of homological algebra, we study the question of the cases in which an exact sequence of spectral groups of homologies is exact.

Let \(I\) be an arbitrary directed set, and let inverse spectra of abelian groups \(\{A_\alpha^n,(\varphi_\alpha^\beta)^n\}_{\alpha\in I,\ n=0,1,\ldots}\), together with homomorphisms \(d_\alpha^n:A_\alpha^n\to A_\alpha^{n+1}\), form an exact sequence of inverse spectra. In other words, let
\[ (\varphi_\alpha^\beta)^{n+1}d_\beta^n=d_\alpha^n(\varphi_\alpha^\beta)^n \quad(\beta>\alpha,\ n=0,1,\ldots) \]
and let the sequences
\[ 0\to A_\alpha^0\to A_\alpha^1\to\cdots\to A_\alpha^n\to\cdots \]
be exact for every \(\alpha\in I\).

Let \(\varprojlim^{(i)}\) denote the \(i\)-th right derived functor of the functor \(\varprojlim\), which assigns to each inverse spectrum with directed set \(I\) its projective limit. The sequences
\[ 0\to \varprojlim\nolimits^{(i)} A_\alpha^0\to \varprojlim\nolimits^{(i)} A_\alpha^1\to\cdots\to \varprojlim\nolimits^{(i)} A_\alpha^n\to\cdots, \tag{\(L_i\)} \]
obtained by applying the functors \(\varprojlim^{(i)}\) to the exact sequence of spectra
\[ 0\to\{A_\alpha^0\}\to\{A_\alpha^1\}\to\cdots\to\{A_\alpha^n\}\to\cdots, \tag{M} \]
are, in general, not exact, but they are complexes in the sense that the composition of any two adjacent homomorphisms in each of these sequences is equal to zero. Consequently, the groups
\[ H^p \varprojlim\nolimits^{(i)} A_\alpha^* \]
of homologies of these complexes are defined.

Theorem 1. There exists a spectral sequence beginning with the term
\[ E_2^{p,q}=H^p \varprojlim\nolimits^{(q)} A_\alpha^* \]
and converging to zero.

This spectral sequence is obtained as the spectral sequence of the composition of the functors \({}^{(3)} Z^0\varprojlim\), where \(\varprojlim\) is regarded as a functor from the category of inverse spectra of complexes to the category of complexes, and \(Z^0\) is the functor assigning to a complex its group of zero-dimensional cycles.

Corollary 1. The sequence \(L_0\) is exact at the term \(\varprojlim A_\alpha^n\) if the sequences \(L_i\) are exact at the terms \(\varprojlim^{(i)} A_\alpha^{\,n-i-1}\) for \(i>0\). In particular, the sequence \(L_0\) is exact at the term \(\varprojlim A_\alpha^n\) if \(\varprojlim^{(i)} A_\alpha^{\,n-i-1}=0\) for \(i>0\).

If the spectra \(\{A_\alpha^n\}\) are such that \(\varprojlim^{(i)} A_\alpha^n=0\) for \(i\ge 2\), then the spectral sequence yields the natural isomorphisms
\[ H^n\varprojlim A_\alpha^*=H^{n-2}\varprojlim\nolimits^{(1)} A_\alpha^*. \]

Roos, who studied the functors \(\lim_{\leftarrow}^{(i)}\) in notes \((^{4-6})\), established that in the case when the directed set \(I\) coincides with the set \(Z_{+}\) of positive integers, \(\lim_{\leftarrow}^{(i)} A_{\alpha}=0\) for \(i\geq 2\) for any spectrum and \(\lim_{\leftarrow}^{(1)} A_{\alpha}=0\) if the projections in the spectrum \(\{A_{\alpha}\}\) are epimorphisms. Therefore, in the case \(I=Z_{+}\), the sequence \(L_{0}\) is exact at the term \(\lim_{\leftarrow} A_{\alpha}^{n}\) if the projections in the spectrum \(\{A_{\alpha}^{\,n-2}\}\) are epimorphisms. This last assertion coincides with a theorem of N. Tynjanski \((^8)\).

In the first of his notes Roos constructs, for a direct spectrum of modules \(\{A_{\alpha}\}\) and a module \(B\) over a fixed ring \(\Lambda\), a spectral sequence beginning with the term
\(E_{2}^{p,q}=\lim_{\leftarrow}^{p}\operatorname{Ext}_{\Lambda}^{q}(A_{\alpha},B)\) and converging to the term \(E_{\infty}^{p,q}\) associated with the module \(\operatorname{Ext}_{\Lambda}^{n}(\lim_{\rightarrow} A_{\alpha},B)\). This spectral sequence, in the case considered here of spectra of abelian groups, yields the natural exact sequence

\[ 0\to \lim_{\leftarrow}^{(0)}\operatorname{Hom}(A_{\alpha},B)\to \operatorname{Ext}(\lim_{\rightarrow} A_{\alpha},B)\to \]

\[ \to \lim_{\leftarrow}\operatorname{Ext}(A_{\alpha},B)\to \lim_{\leftarrow}^{(2)}\operatorname{Hom}(A_{\alpha},B)\to 0 \]

and the natural isomorphisms
\[ \lim_{\leftarrow}^{(i)}\operatorname{Hom}(A_{\alpha},B)\approx \lim_{\leftarrow}^{(i-2)}\operatorname{Ext}(A_{\alpha},B) \quad\text{for } i>2. \]

This result of Roos serves as the starting point in the proof of the following assertions.

Theorem 2. a) A compact spectrum, i.e. a spectrum in which the groups are compact and the projections are continuous, is acyclic. b) A spectrum of finite-dimensional vector spaces over a fixed field \(F\) is acyclic. c) If an exact functor \(\Gamma\) from the category of inverse spectra over a directed set \(I\) to the category of inverse spectra over another directed set \(I'\) takes compact spectra to compact spectra and if, moreover, the functors \(\lim_{\leftarrow I'}\Gamma\) and \(\lim_{\leftarrow I}\) are naturally equivalent, then the functors \(\lim_{\leftarrow I'}^{(i)}\Gamma\) and \(\lim_{\leftarrow I}^{(i)}\) are naturally equivalent as well. In particular, the values of the functors \(\lim_{\leftarrow}^{(i)}\) do not change when passing to a cofinal part of the spectrum and when passing to a spectrum with weakened projections.

Here the spectrum \(\{A_{\alpha}\}\) is called acyclic if
\[ \lim_{\leftarrow}^{(i)} A_{\alpha}=0 \]
for \(i\geq 1\).

Corollary 2. If in the sequence \((M)\) the spectra \(\{A_{\alpha}^{n}\}\) are compact or are spectra of finite-dimensional vector spaces over a fixed field \(F\), then the sequence \(L_{0}\) is exact.

Let us note that here, in contrast to Theorem 5.6 of the book \((^7)\), the continuity of the homomorphisms \(d_{\alpha}^{n}\) is not assumed.

Let \(\operatorname{Pext}(A,B)\) denote the group of those extensions of the group \(B\) with kernel \(A\) in which the subgroup \(A\) is pure. This group coincides with the maximal complete subgroup of the group \(\operatorname{Ext}(A,B)\) of all extensions of the group \(B\) with kernel \(A\) \((^9)\). The following is important for what follows.

Theorem 3. Let \(\{A_{\alpha}\}\) be a direct spectrum of groups with a finite number of generators and let \(B\) be an arbitrary group; then
\[ \lim_{\leftarrow}^{(i)}\operatorname{Hom}(A_{\alpha},B)=0 \]
for \(i\geq 2\) and
\[ \lim_{\leftarrow}^{(1)}\operatorname{Hom}(A_{\alpha},B)\approx \operatorname{Pext}(\lim_{\rightarrow} A_{\alpha},B),\qquad \lim_{\leftarrow}^{(i)}\operatorname{Ext}(A_{\alpha},B)=0 \]
for \(i\geq 1\).

Corollary 3. If \(\{A_{\alpha}\}\) is a direct spectrum of groups with a finite number of generators and \(B\) is an arbitrary group, then the mapping
\[ \operatorname{Ext}(\lim_{\rightarrow} A_{\alpha},B)\to \lim_{\leftarrow}\operatorname{Ext}(A_{\alpha},B) \]
is an epimorphism, and the kernel of this mapping is the maximal complete subgroup of the group
\[ \operatorname{Ext}(\lim_{\rightarrow} A_{\alpha},B). \]

Theorem 4. Let \(\{A_\alpha\}\) be an inverse spectrum of groups with a finite number of generators, and let \(B\) be an arbitrary group. Then \(\lim\limits_{\leftarrow}^{(i)}(A_\alpha\otimes B)=0\) for \(i\geq 2\),

\[ \lim_{\leftarrow}^{(1)}(A_\alpha\otimes B)\approx \operatorname{Ext}\bigl(\lim_{\to}\operatorname{Hom}(A_\alpha,Z),B\bigr), \]

\(\lim\limits_{\leftarrow}^{(i)}(A_\alpha * B)=0\) for \(i\geq 2\),

\[ \lim_{\leftarrow}^{(1)}(A_\alpha * B)\approx \operatorname{Pext}\bigl(\lim_{\to} {}_t\overline{A}_\alpha,B\bigr), \]

where \({}_t\overline{A}_\alpha\) denotes the group of characters of the maximal periodic subgroup of the group \(A_\alpha\).

Definition. An inverse spectrum of \(q\)-dimensional homology groups with coefficients in the group \(G\) of the nerves of finite open coverings of a bicompact pair \((X,Y)\), whose limit, by definition, is the group \(H_q(X,Y;G)\) of spectral homology of the pair \((X,Y)\), is called a homological spectrum.

Let \(A_\alpha\) be a homological spectrum. Then \(\lim\limits_{\leftarrow}^{(i)}A_\alpha=0\) for \(i\geq 2\),

\[ \lim_{\leftarrow}^{(1)} A_\alpha=\operatorname{Pext}(H^q(X,Y),G). \]

In particular, if all spectra of the sequence \((M)\) are homological spectra, then

\[ H^p\lim_{\to} A_\alpha^* \approx H^{p-2}\operatorname{Pext}(H^*(X^*Y^*),G). \]

From this remark there follows the following main

Theorem 5. For an arbitrary finite-dimensional bicompact pair \((X,Y)\) and coefficient group \(G\), the sequences

\[ \cdots \longrightarrow H_p(Y;G)\longrightarrow H_p(X;G)\longrightarrow H_p(X,Y;G)\longrightarrow H_{p-1}(Y;G)\longrightarrow \cdots \]

\[ \cdots \longrightarrow \operatorname{Pext}(H^{p+1}(X),G)\longrightarrow \operatorname{Pext}(H^{p+1}(X,Y);G)\longrightarrow \]

\[ \longrightarrow \operatorname{Pext}(H^p(Y);G)\longrightarrow \operatorname{Pext}(H^p(X);G)\longrightarrow \cdots \tag{1} \]

have identical homology groups in the corresponding places.

Similarly, for every exact sequence of groups \(0\longrightarrow G'\longrightarrow G\longrightarrow G''\longrightarrow 0\), the sequences

\[ \cdots \longrightarrow H_p(X,Y;G')\longrightarrow H_p(X,Y;G)\longrightarrow H_p(X,Y;G'')\longrightarrow \]

\[ \longrightarrow H_{p-1}(H,Y;G')\longrightarrow \cdots \tag{2} \]

\[ \cdots \longrightarrow \operatorname{Pext}(H^{p+1}(X,Y),G)\longrightarrow \operatorname{Pext}(H^{p+1}(X,Y),G'')\longrightarrow \]

\[ \longrightarrow \operatorname{Pext}(H^p(X,Y),G)\longrightarrow \operatorname{Pext}(H^p(X,Y),G)\longrightarrow \cdots \]

have identical homology groups in the corresponding places.

The group \(\operatorname{Pext}(A,B)=0\) if the group \(B\) is compact or if \(A\) is a direct sum of cyclic groups. If \(B'\) is a pure subgroup of the group \(B\), then the sequence \(\operatorname{Pext}(A,B')\to \operatorname{Pext}(A,B)\to \operatorname{Pext}(A,B/B')\to 0\) is exact; moreover, if \(G'\) is pure in \(G\), then the homomorphism \(\delta\) in the sequences (1) and (2) is trivial. In view of this, the following is valid.

Corollary 4. If the group \(H^{p+1}(X,Y)\), \(\bigl(H^{p+1}(X),H^p(Y)\bigr)\) of a finite-dimensional bicompact pair is isomorphic to a direct sum of cyclic groups, then the sequence (1) is exact at the term \(H_p(X;G)\) \(\bigl(H_p(Y;G), H_p(X,Y;G)\bigr)\), and the sequence (2) is exact at the terms \(H_{p+1}(X,Y;G'')\), \(H_p(X,Y;G')\), and \(H_p(X,Y;G)\).

If in the sequence \(0\to G'\to G\to G''\to 0\) the subgroup \(G'\) is pure in \(G\), then the sequence (2) is exact at the terms \(H_p(X,Y;G')\) and \(H_p(X,Y;G)\).

One more application of the concepts introduced here concerns the study of the homomorphism of Sitnikov homology groups into spectral ones \((^1)\). Borel and Moore \((^2)\) defined homology groups \(\mathscr H_*(X;G)\) of a bicompact space \(X\), coinciding in the metric case with the Sitnikov groups. There exists a natural transformation

\[ \alpha:\mathscr H_*(X;G)\to H_*(X;G), \]

which, together with the formulas for the universal coefficients, forms the following commutative diagram:

\[ \begin{gathered} 0 \to \operatorname{Ext}\bigl(H^{n+1}(X),G\bigr) \longrightarrow \mathscr{H}_n(X;G) \to \operatorname{Hom}\bigl(H^n(X),G\bigr) \to 0\\ \downarrow \beta \qquad\qquad\qquad\qquad \downarrow \alpha \qquad\qquad\qquad\quad \downarrow 1\\ 0 \to \varprojlim \operatorname{Ext}\bigl(H^{n+1}(X_\alpha),G\bigr) \to H_n(X;G) \to \operatorname{Hom}\bigl(H^n(X),G\bigr) \to 0 . \end{gathered} \]

In this diagram the rows are exact. Obviously, \(\operatorname{Ker}\alpha=\operatorname{Ker}\beta\) and \(\operatorname{Coker}\alpha=\operatorname{Coker}\beta\). But \(\operatorname{Coker}\beta=0\), while \(\operatorname{Ker}\beta\) coincides with the maximal complete subgroup of the group \(\operatorname{Ext}\bigl(H^{n+1}(X),G\bigr)\). Thus \(\alpha\) is an epimorphism, and its kernel is a complete group.

Apparently, there is a possibility of extending Theorem 3 to the case of spectra of modules*. In this connection the following theorem is of interest:

Theorem 6. Let \(\Lambda\) be a Noetherian ring. There exists a sequence of functors \(T^i(A,B)\) from the category of pairs of \(\Lambda\)-modules to the category of modules such that, for every direct spectrum \(\{A_\alpha\}\) of modules with a finite number of generators and any \(\Lambda\)-module \(B\), the modules \(T^i(\varinjlim A_\alpha,B)\) and

\[ \varprojlim{}^{(i)} \operatorname{Hom}_\Lambda(A_\alpha,B) \]

are naturally isomorphic. These isomorphisms uniquely determine the functors \(T^i\).

Problem. Will the direct sum of an arbitrary number of copies of a given acyclic spectrum be acyclic? This problem is equivalent to the well-known problem of Whitehead, which asks whether a group \(A\) is free if \(\operatorname{Ext}(A,Z)=0\). Since the product of acyclic spectra is acyclic, this problem is also equivalent to the following question: is the mapping

\[ \varprojlim_{\leftarrow\,\beta}\prod \{A_\alpha\}_\beta \to \varprojlim_{\leftarrow\,\beta} \left[ \prod_{\beta}\{A_\alpha\}_\beta \big/ \sum_{\beta}\{A_\alpha\}_\beta \right] \]

an epimorphism if each spectrum \(\{A_\alpha\}_\beta\) is isomorphic to a fixed acyclic spectrum?

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
20 XII 1964

CITED LITERATURE

1. P. S. Aleksandrov, Topological duality theorems, Part II, Nonclosed sets. Moscow, 1959.
2. A. Borel, J. Moore, Michigan Math. J., 7, No. 2 (1960).
3. A. Grothendieck, On some questions of homological algebra, IL, 1961.
4. J. E. Roos, C. R., 252, 3702 (1961).
5. J. E. Roos, C. R., 254, 1556 (1962).
6. J. E. Roos, C. R., 254, 1720 (1962).
7. N. Steenrod, S. Eilenberg, Foundations of Algebraic Topology, Moscow, 1958.
8. N. Tynianskii, DAN, 141, No. 1 (1961).
9. L. Fuchs, Abelian Groups, Budapest, 1958.

* Note added in proof. After the article had been submitted for typesetting, the author succeeded in obtaining such an extension of Theorem 3.