Draft lesson.

Real-parameter complexes and cohomology continuity

Written with GPT-6.1 Sol (OpenAI). Original text: CC0.

Taking cohomology need not commute with an inverse limit. For a countable tower with surjective maps in every degree, the failure is exactly a first derived-limit term in the preceding cohomology degree. We prove this comparison with its actual maps, then use it to propagate cohomology isomorphisms along a real-parameter diagram.

Fix a unital ring \(A\), a Grothendieck universe containing \(A,\mathbb R,\mathbb N\), and the ambient axiom of choice. Complexes are cochain complexes of small left \(A\)-modules, with \(d:X^n\to X^{n+1}\) and \(d^2=0\). No bound on their terms is imposed. Write \(H^nX=\ker d^n/\operatorname{im}d^{n-1}\), with the canonical maps induced by chain maps.

Hom, tensor and the exactness of module limits, Sections 2–5, gives small limits, filtered representative tests and exact module products. Countable Mittag-Leffler towers and formal Pro derived limits, Sections 1–2 and 6, proves surjectivity of the difference map for ML towers and identifies its cokernel with the formal Pro right derived functor. Local tests and global maps in real-parameter module diagrams, Sections 1–3, supplies the two local comparisons and their full local-to-global proofs.

1. The real-parameter hypotheses

Let \(s\mapsto X_s\) be a covariant diagram of complexes on the real order. Its chain maps are \(v_{r,s}:X_r\to X_s\) for \(r\leq s\), with identity and composition as usual. Assume:

  1. Every \(v_{r,s}^n:X_r^n\to X_s^n\) is surjective, for every integer \(n\) and \(r\leq s\).
  2. The canonical right comparison of terms is an isomorphism for every \(s,n\):
\[ \rho_s^n:X_s^n\xrightarrow{\sim}\lim_{t>s}X_t^n, \qquad x\longmapsto(v_{s,t}^nx)_{t>s}. \tag{1.1} \]
  1. The canonical left comparison of cohomology is an isomorphism for every \(s,n\):
\[ a_s^n:\operatorname{colim}_{t<s}H^nX_t\xrightarrow{\sim}H^nX_s, \qquad [t,\xi]\longmapsto H^n(v_{t,s})\xi. \tag{1.2} \]

The right cohomology comparison is

\[ b_s^n:H^nX_s\longrightarrow\lim_{t>s}H^nX_t, \qquad \xi\longmapsto(H^n(v_{s,t})\xi)_{t>s}. \tag{1.3} \]

These are the local maps from the module prerequisite applied to the term diagrams and to the cohomology diagrams. They are natural in the real parameter and in morphisms of diagrams. In particular, (1.1) specifies the isomorphism used; an unspecified isomorphism between the two modules would not be this hypothesis.

2. Reconstruct a whole future from a countable tower

Fix \(s\) and put \(t_m=s+1/(m+1)\) for \(m\geq0\). Then \(t_{m+1}<t_m\), and

\[ M_m=X_{t_m},\qquad w_m=v_{t_{m+1},t_m}:M_{m+1}\to M_m \tag{2.1} \]

is an inverse tower of complexes. Its maps are termwise surjective by hypothesis 1.

Lemma 2.1. Restriction to these times gives canonical isomorphisms

\[ \lim_{t>s}X_t\simeq\lim_m M_m, \qquad \lim_{t>s}H^nX_t\simeq\lim_m H^nM_m. \tag{2.2} \]

The first is an isomorphism of complexes, with limits computed degreewise.

Proof. The construction works for any covariant module diagram \(Y_t\) on \(t>s\). A compatible full tuple restricts to a compatible tower tuple. Conversely, given \((y_m)\) in the tower limit, for every \(t>s\) choose \(m\) with \(t_m\leq t\), and set

\[ y_t=v_{t_m,t}y_m. \tag{2.3} \]

Such an \(m\) exists by the Archimedean property of the reals. If \(m'\) is another choice, take \(k\geq m,m'\). Tower compatibility and the diagram equations show that both values in (2.3) equal \(v_{t_k,t}y_k\). Thus the result is independent of choices. For \(s<t\leq u\), choose one \(m\) with \(t_m\leq t\); equation (2.3) then gives \(v_{t,u}y_t=y_u\).

Restricting this tuple back to a \(t_m\) recovers \(y_m\), and reconstructing a restricted full tuple recovers each original coordinate by its compatibility. The maps are linear and commute with morphisms of diagrams, since their coordinate formulas do. This proves the two inverse equations and naturality. Apply it to the module diagrams \(X_t^n\) in every degree and to \(H^nX_t\). The chain-map equations make (2.3) commute with differentials, proving the complex assertion. \(\square\)

Hypothesis 2 consequently identifies \(X_s\) with the ordinary complex limit of (2.1). The canonical map \(\rho_s\) is a chain map by naturality of the differentials, and its inverse is a chain map because its components are invertible.

3. The exact countable cohomology comparison

For any module tower \(N_m\) with adjacent maps \(u_m\), define

\[ \Delta_N:\prod_mN_m\longrightarrow\prod_mN_m, \qquad (\Delta_Nz)_m=z_m-u_mz_{m+1}, \quad \ell^1(N)=\operatorname{coker}\Delta_N. \tag{3.1} \]

Its kernel is the actual module limit. The ML prerequisite proves that \(\ell^1(N)=0\) for an ML tower. It also identifies \(\ell^1(N)\) naturally with \(R^1\pi(\lim_m^{\rm form}cN_m)\). This identification concerns a module tower regarded as a formal Pro object in degree zero.

Theorem 3.1. For any countable tower \(M_m\) of complexes with termwise surjective transitions there is a natural short exact sequence, for every integer \(n\),

\[ 0\longrightarrow\ell^1(H^{n-1}M_m) \xrightarrow{\partial}H^n(\lim_m M_m) \xrightarrow{\kappa}\lim_m H^nM_m\longrightarrow0. \tag{3.2} \]

The map \(\kappa\) takes the class of a compatible cycle tuple to its compatible tuple of cycle classes. No boundedness assumption is needed.

Proof. Put \(C=\prod_m M_m\), with degreewise products, and let \(\Delta:C\to C\) be the degreewise difference map. It is a chain map since all tower transitions are chain maps. Each \(\Delta^q\) is onto: its degree-\(q\) tower has surjective transitions, hence is ML, so the difference-map theorem applies in that degree. The subcomplex \(K=\ker\Delta\) is the ordinary complex limit. Thus

\[ 0\to K\to C\xrightarrow{\Delta}C\to0 \tag{3.3} \]

is termwise short exact.

The canonical coordinate map is an isomorphism

\[ H^q(C)\xrightarrow{\sim}\prod_mH^qM_m. \tag{3.4} \]

It is surjective by choosing a cycle representative in each coordinate. If a cycle tuple has boundary class in every coordinate, choose a preimage under the differential for each coordinate; their tuple is a boundary in \(C\). Thus the map is injective. Its linearity and naturality follow from its coordinate projections. Under (3.4), \(H^q\Delta\) is exactly the difference operator of the cohomology tower. This also gives a direct verification of the product-cohomology interface, including arbitrary modules and unbounded complexes.

We construct the connecting map. Given a cycle \(w\in C^{n-1}\), choose \(z\in C^{n-1}\) with \(\Delta z=w\). Then \(d z\) belongs to \(K^n\), since \(\Delta d z=d w=0\), and it is a cycle. Send \([w]\) to \([d z]\). Changing the lift \(z\) changes \(d z\) by a boundary in \(K\). If \(w\) is changed by \(d a\), choose \(b\in C^{n-2}\) with \(\Delta b=a\); then \(z+d b\) lifts \(w+d a\) and has the same differential. Thus the map is independent of both choices. Sums and scalar multiples can be lifted by sums and scalar multiples of lifts, proving linearity. If \([w]\) is in the image of \(H^{n-1}\Delta\), a cycle lift gives zero connecting class. Hence it factors through the cokernel in (3.1).

The factored map is injective. If \([d z]=0\) in \(H^nK\), write \(d z=d h\) for \(h\in K^{n-1}\). Then \(z-h\) is a cycle and \(\Delta(z-h)=w\). Thus \([w]\) was already in the image of \(H^{n-1}\Delta\), so its cokernel class is zero.

Its image is the kernel of \(\kappa\). The class of \(d z\) is a boundary in \(C\), so all of its coordinate classes vanish. Conversely, if a cycle \(y\in K^n\) has zero coordinate classes, (3.4) makes it a boundary \(d z\) in \(C\). Now \(\Delta z\) is a cycle because \(\Delta y=0\), and its connecting class is exactly \([y]\).

Finally \(\kappa\) is onto the compatible cohomology tuples. Given such a tuple, choose a cycle tuple \(z\in C^n\) representing it. Compatibility says that \(\Delta z\) is a boundary in \(C\); choose \(w\in C^{n-1}\) with \(d w=\Delta z\). Choose \(u\in C^{n-1}\) with \(\Delta u=w\). The cycle \(z-d u\) belongs to \(K\), since

\[ \Delta(z-d u)=\Delta z-d\Delta u=d w-d w=0. \tag{3.5} \]

It has the prescribed coordinate classes, proving surjectivity. All constructions commute with a morphism of towers: an image of a chosen lift is a lift of the image, and the proved choice-independence gives the same induced cohomology maps. This proves naturality and (3.2) with its stated maps. \(\square\)

4. Right cohomology comparisons and induction

Theorem 4.1. Under hypotheses 1–3:

  1. Every \(b_s^n\) is surjective.
  2. If \(b_s^{n-1}\) is an isomorphism for every \(s\), then \(b_s^n\) is an isomorphism for every \(s\).
  3. If there is an integer \(n_0\) such that every \(b_s^j\) is an isomorphism for \(j\leq n_0\) and every \(s\), then every cohomology transition \(H^n(v_{r,s})\) is an isomorphism, for all integers \(n\) and \(r\leq s\).

Proof. Fix \(s\) and use the future tower (2.1). Theorem 3.1 applies because its transitions are termwise onto. Lemma 2.1 and the canonical chain isomorphism \(\rho_s\) identify its comparison \(\kappa\) with \(b_s^n\): on a cycle of \(X_s\), the tuple is precisely its images at all future times. Surjectivity of \(\kappa\) proves assertion 1.

For assertion 2, apply the module local-to-global corollary to the real diagram \(H^{n-1}X_s\). Its left comparisons are \(a_s^{n-1}\), isomorphisms by hypothesis 3, and its right comparisons are \(b_s^{n-1}\), isomorphisms by the assumed preceding-degree condition. The corollary proves that every transition in this cohomology diagram is an isomorphism. Its future tower is therefore ML. The left term of (3.2) vanishes, by the ML difference-map theorem. Hence \(\kappa\), and therefore \(b_s^n\), is an isomorphism. This works at every \(s\).

For assertion 3, begin with the given isomorphisms in degrees \(j\leq n_0\). Apply assertion 2 successively at \(n_0+1,n_0+2,\ldots\). For any fixed integer \(n>n_0\), finitely many steps reach that degree. Thus every \(b_s^n\) is an isomorphism in every degree, with all lower degrees supplied by the original assumption. For each \(n\), the cohomology diagram now has both local comparisons invertible; the same module corollary makes every \(H^n(v_{r,s})\) invertible. The lower-degree assumption concerned these comparison maps, rather than a bound on the terms of all the complexes. \(\square\)

Example 4.2 (a bounded-below application). If there is a common integer \(N\) with \(X_s^j=0\) for \(j<N\) and all \(s\), then \(H^jX_s=0\) for \(j<N\). Their right comparisons are the unique isomorphisms \(0\to0\). Hypotheses 1–3 and Theorem 4.1, with \(n_0=N-1\), make every cohomology transition an isomorphism. The theorem also permits nonzero terms arbitrarily far down, when its given lower comparison condition is verified directly.

5. Four graded exercises with complete solutions

Exercise 1 (basic: a coinitial future and its coordinates). For \(t>0\), let \(Y_t=k^{\lfloor1/t\rfloor}\), with prefix restrictions as \(t\) increases. Compute its limit over all \(t>0\), identify the tower at \(t_m=1/(m+1)\), and verify both maps in Lemma 2.1 explicitly.

Solution. The tower is \(Y_{t_m}=k^{m+1}\), with the usual projection from \(k^{m+2}\) to \(k^{m+1}\). Its compatible tuples are exactly prefixes of one sequence \((a_j)_{j\geq1}\in\prod_{j\geq1}k\). Such a sequence also gives a full-time tuple: at \(t\) take its prefix of length \(\lfloor1/t\rfloor\), including the empty tuple if that length is zero.

Given a tower tuple, choose \(m\) with \(t_m\leq t\). Its truncation from length \(m+1\) to length \(\lfloor1/t\rfloor\) is the value in (2.3); increasing \(m\) retains the same coordinates. Conversely a full-time tuple restricts to its successive prefixes at the \(t_m\). The two operations recover every prefix and every time coordinate, respectively. Both limits therefore identify with the product through their actual coordinate maps. This also explains the variance: the maps in the countable tower go from \(m+1\) to \(m\).

Exercise 2 (intermediate: the preceding-degree obstruction). For \(m\geq0\), define a complex \(C_m\) in degrees zero and one by

\[ C_m^0=\mathbb Z^{m+1},\quad C_m^1=\mathbb Z^m, \quad d_m(a_0,\ldots,a_m)=(a_r-2a_{r+1})_{0\leq r<m}. \tag{5.1} \]

The zero-coordinate target at \(m=0\) is zero. Take prefix projections \(C_{m+1}\to C_m\). Compute all stage cohomology and its transitions, the limit complex, and the kernel of its right degree-one cohomology comparison.

Solution. Prefix restriction commutes with (5.1), and both term maps are onto. Each \(d_m\) is onto: set \(a_m=0\) and solve backwards \(a_r=b_r+2a_{r+1}\). Its kernel is

\[ t\longmapsto(2^mt,2^{m-1}t,\ldots,2t,t). \tag{5.2} \]

Thus \(H^0C_m\simeq\mathbb Z\), \(H^1C_m=0\), and all other cohomology is zero. Under the last-coordinate identification, restricting (5.2) from \(m+1\) to \(m\) makes the new last coordinate \(2t\). Hence the degree-zero cohomology tower has transitions multiplication by two.

The limit complex is

\[ C_\infty=\left(\prod_{r\geq0}\mathbb Z \xrightarrow{\Delta_2}\prod_{r\geq0}\mathbb Z\right), \qquad (\Delta_2a)_r=a_r-2a_{r+1}, \tag{5.3} \]

in degrees zero and one. Its zeroth cohomology is zero: a compatible integer at any fixed coordinate is divisible by every power of two. Its first cohomology is \(\Gamma=\operatorname{coker}\Delta_2\), and its right comparison is \(\Gamma\to\lim_mH^1C_m=0\). Its whole kernel is therefore \(\Gamma=\ell^1(H^0C_m)\), exactly (3.2).

This kernel is nonzero. Take \(b_{2r}=0\), \(b_{2r+1}=1\). If \(a_r-2a_{r+1}=b_r\), substituting through coordinate \(2k+1\) gives

\[ 3a_0+2=4^{k+1}(2+3a_{2k+2})\quad(k\geq0). \tag{5.4} \]

The fixed integer on the left would be divisible by arbitrarily large powers of four, hence zero. That would require the impossible integer equation \(3a_0+2=0\). Thus \([b]\ne0\) in \(\Gamma\). Termwise surjectivity alone guarantees the right comparison is onto, but does not make its kernel vanish.

Exercise 3 (intermediate: an actual constant-cohomology application). Fix an integer \(p>1\) and the complex \(E=(\mathbb Z\xrightarrow p\mathbb Z)\) in degrees zero and one. For arbitrary signs \(\varepsilon_s\in\{1,-1\}\), put \(X_s=E\) and let \(v_{r,s}\) multiply both terms by \(\varepsilon_s\varepsilon_r\). Verify hypotheses 1–3 and the lower comparison condition. Compute every induced cohomology transition.

Solution. Each sign map commutes with multiplication by \(p\). The transition composite is multiplication by \(\varepsilon_t\varepsilon_s\varepsilon_s\varepsilon_r=\varepsilon_t\varepsilon_r\), and identity transitions multiply by one. Thus this is a diagram of chain isomorphisms, in particular termwise surjections. Multiplication by \(\varepsilon_s\) identifies it with the constant diagram \(E\). The right compatible tuple at any cut is therefore determined by one element in each term, and the canonical term map sends that element to its sign-related future values. It is an isomorphism, proving hypothesis 2.

Since \(p\) has zero kernel in \(\mathbb Z\), we have \(H^0E=0\), \(H^1E=\mathbb Z/p\), and zero cohomology otherwise. The same sign identifications make each cohomology diagram constant up to its specified isomorphisms. Its left comparison is invertible by the one-stage constant-colimit argument, proving hypothesis 3. Every right comparison below degree one is \(0\to0\), so the lower comparison condition holds with \(n_0=0\).

The actual degree-one transition is multiplication by \(\varepsilon_s\varepsilon_r\) on \(\mathbb Z/p\), whose inverse is the same sign. In every other degree it is the unique isomorphism of zero modules. This computes the isomorphisms asserted by Theorem 4.1 even when the signs vary discontinuously with the real parameter.

Exercise 4 (advanced: why left cohomology continuity is used). Use the complexes \(C_m,C_\infty\) of Exercise 2. Define a real-parameter complex diagram by

\[ X_t=C_\infty\ (t\leq0),\qquad X_t=C_{n(t)}\ (t>0),\qquad n(t)=\lceil1/t\rceil-1. \tag{5.5} \]

Use identity transitions before zero and the corresponding prefix restrictions afterward, including from \(C_\infty\) to a finite prefix. Prove that hypotheses 1–2 hold, that all \(b_s^0\) are isomorphisms, but that \(b_0^1\) is not an isomorphism. Locate the failure of hypothesis 3.

Solution. The nonnegative integer \(n(t)\) is nonincreasing on \(t>0\). Prefix restrictions commute with the differentials in (5.1)–(5.3) and compose as prefixes, including when a transition starts at \(C_\infty\). Their maps in every degree are onto; in all other degrees the modules are zero. Thus the diagram laws and hypothesis 1 hold.

For every positive \(s\), the integer function \(n(t)\) is constant on some interval \([s,s+\eta)\): at \(s=1/k\), its value is \(k-1\), the same as immediately to the right; away from these jumps it is locally constant. A compatible tuple strictly to the right of \(s\) is determined by its value on that constant initial interval, with all later values supplied by prefix restrictions. This identifies its degreewise future limit with \(X_s\) through the canonical map (1.1). For \(s<0\), the constant interval immediately to its right similarly determines the tuple. At \(s=0\), the finite-prefix limits in both degrees are the products in (5.3), with the same difference differential. Thus (1.1) is an isomorphism at zero too. Hypothesis 2 is proved at every point.

For \(s>0\), the same initial interval argument applies to cohomology, so \(b_s^0\) is an isomorphism. For \(s<0\), its immediate future has zeroth cohomology zero, and this forces all subsequent compatible coordinates to be zero, so \(b_s^0\) is \(0\to0\). At zero the future zeroth-cohomology system is the \(\mathbb Z\)-tower with maps multiplication by two; its limit is zero by Exercise 2. Since \(H^0C_\infty=0\), \(b_0^0\) is also an isomorphism.

However \(H^1X_0=\Gamma\ne0\), while every positive-time first cohomology is zero. Thus \(b_0^1:\Gamma\to0\) is not injective. At a positive jump \(s=1/k\), the immediate left complex is \(C_k\) and the value at the cut is \(C_{k-1}\). The left zeroth-cohomology comparison is consequently

\[ a_{1/k}^0:\mathbb Z\xrightarrow{2}\mathbb Z, \tag{5.6} \]

by the transition calculation in (5.2). The immediate left interval is cofinal in the left indexing interval, so it computes this actual colimit comparison. Map (5.6) is not surjective. Hypothesis 3 fails, and its use in the preceding-degree local-to-global step cannot be omitted from that argument.

References