Draft lesson.

Local tests and global maps in real-parameter module diagrams

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

A module diagram on the real line has two canonical comparisons at every point: from everything strictly to its left, and to everything strictly to its right. If both comparisons are injective, every transition is injective. If both are surjective, every transition is surjective. The second implication requires compatible lifts at a cut, and we construct those lifts explicitly.

Fix a unital ring \(A\), a Grothendieck universe \(\mathcal U\) containing \(A\) and \(\mathbb R\), and the ambient axiom of choice. All modules and index sets below are \(\mathcal U\)-small. Universes and small categories, Sections 1–2, supplies the size and choice conventions. The complete module limit and filtered representative/zero tests used here are Hom, tensor and the exactness of module limits, Sections 2–3. No topological structure on the modules is assumed.

1. The two comparisons, with their actual maps

View \(\mathbb R\) as a category with a single arrow \(r\to s\) when \(r\leq s\). A covariant diagram \(X\) consists of modules \(X_s\) and linear maps \(v_{r,s}:X_r\to X_s\), with

For each \(s\), put

The left interval is a nonempty directed set. Each colimit element has a representative \([t,x]\), and

This is the exact zero test proved in the prerequisite. The right limit is the submodule of the product consisting of tuples \((x_t)_{t>s}\) satisfying \(v_{t,u}x_t=x_u\) whenever \(s<t\leq u\). Its projections have the ordinary limit universal property.

The cocone \((v_{t,s})_{t<s}\) and cone \((v_{s,t})_{t>s}\) define

The first formula respects the colimit relations by (1.1); the second is compatible by the same equation. The universal properties prove linearity and uniqueness. In particular, surjectivity of \(\lambda_s\) means that every element of \(X_s\) comes from one stage \(t<s\). Surjectivity of \(\mu_s\) means that every whole compatible future tuple is the image of one element of \(X_s\).

For \(r\leq s\), inclusion of left intervals induces \(L_rX\to L_sX\), taking \([t,x]\) to the same representative. Restriction of tuples induces \(R_rX\to R_sX\). The equations

hold on representatives and on coordinates respectively. Thus \(L\) and \(R\) are real-parameter diagrams and \(\lambda:LX\to X\), \(\mu:X\to RX\) are natural transformations. A morphism \(f:X\to Y\) sends \([t,x]\) to \([t,f_tx]\) and \((x_t)\) to \((f_tx_t)\). Its commuting transition squares prove that these operations are well defined and intertwine both maps in (1.4). This includes naturality in the diagram itself.

2. Injectivity at a finite death cut

Theorem 2.1. If every \(\lambda_s\) and every \(\mu_s\) is injective, then every \(v_{r,s}\) is injective.

Proof. Identity transitions are injective. Fix \(r<s\), and suppose a nonzero \(x\in X_r\) has \(v_{r,s}x=0\). For \(t\geq r\) write \(x_t=v_{r,t}x\). The set

is nonempty because it contains \(s\). It is upward closed inside \([r,s]\), by (1.1). Its infimum \(c\) exists in \([r,s]\). There is no death time below \(c\). Every \(t>c\) is a death time, allowing \(t\) beyond \(s\) as well: for \(c<t\leq s\), the infimum property gives a \(d\in D\) with \(d<t\), and then \(x_t=v_{d,t}x_d=0\); for \(t>s\), use \(x_s=0\).

If \(x_c\ne0\), all its future images are zero, so \(\mu_c(x_c)=0\). This contradicts injectivity of \(\mu_c\). This case includes \(c=r\), since \(x_r=x\ne0\).

Otherwise \(x_c=0\), and consequently \(c>r\). The class \([r,x]\in L_cX\) is nonzero: by (1.3), its vanishing would give a \(u\) with \(r\leq u<c\) and \(x_u=0\), contradicting the definition of \(c\). But \(\lambda_c[r,x]=x_c=0\), contradicting injectivity of \(\lambda_c\).

The contradiction covers a cut where the element dies at the endpoint and a cut where it survives there but dies at every later point. Thus the kernel of \(v_{r,s}\) is zero. \(\square\)

3. Surjectivity through a coherent family

Theorem 3.1. If every \(\lambda_s\) and every \(\mu_s\) is surjective, then every \(v_{r,s}\) is surjective.

Proof. Fix \(r<s\) and \(y\in X_s\). A partial lift family consists of a nonempty terminal interval \(J\subseteq[r,s]\) containing \(s\), together with elements \(x_t\in X_t\) for \(t\in J\), such that

Here terminal means that \(p\in J\) and \(p\leq q\leq s\) imply \(q\in J\). Order these families by extension of their domains and values. They form a set: each is encoded by a subset of the small tagged union \(\coprod_{t\in[r,s]}X_t\); the domain and values can be recovered from that subset. The initial family on \(\{s\}\) exists.

A chain has an upper bound given by the union of its families. Values agree wherever domains overlap. Any two of its domain points and their assigned values occur together in one family, since the chain is totally ordered by extension; hence (3.1) holds for the union. The union is a terminal interval and contains \(s\). The empty chain also has an upper bound, namely the initial family. Zorn's lemma therefore gives a maximal coherent family.

Let \(c=\inf J\in[r,s]\). Every \(t\) with \(c<t\leq s\) belongs to \(J\): choose a domain point below \(t\), using the infimum property, and then use terminality. Consequently \(J\) is either \([c,s]\) or \((c,s]\), with the latter case requiring \(c<s\).

First suppose \(c\notin J\). Extend the already chosen future values beyond \(s\) by setting

This is a compatible family at every \(t>c\). The only new compatibility case crosses \(s\): for \(c<p\leq s<q\), equation (3.1) gives \(v_{p,s}x_p=y\), so \(v_{p,q}x_p=v_{s,q}y\). The cases on either side follow from (3.1) and (1.1). Surjectivity of \(\mu_c\) supplies an \(x_c\in X_c\) whose future images are this entire tuple. Adjoining this value gives a strictly larger coherent family on \([c,s]\), contrary to maximality. Thus \(c\in J\).

If \(c>r\), surjectivity of \(\lambda_c\) and the one-stage representative property give \(u<c\) and \(z\in X_u\) with \(v_{u,c}z=x_c\). Set \(u_0=\max(r,u)<c\), and replace \(z\) by \(z_0=v_{u,u_0}z\) if needed. Then

For \(u_0\leq t<c\), set \(x'_t=v_{u_0,t}z_0\), retaining the existing values for \(t\geq c\). The new values commute among themselves by (1.1). Across the old boundary, for \(u_0\leq p<c\leq q\), their image is

Thus they form a strictly larger coherent family on \([u_0,s]\), again contradicting maximality. We conclude \(c=r\). Since its boundary is included, \(J=[r,s]\), and (3.1) gives \(v_{r,s}x_r=y\).

This constructs a preimage of the prescribed element. It does not require a linear choice of preimages. Surjectivity follows for every \(r<s\); the identity case is immediate. \(\square\)

The use of \(\mu_c\) fills a missing endpoint using a single coherent tuple. Choosing separate preimages for its coordinates would not establish (3.1). The use of \(\lambda_c\) then moves a closed endpoint strictly left. Both steps are needed to rule out a maximal family stopping before \(r\).

Corollary 3.2. If all the comparisons \(\lambda_s\) and \(\mu_s\) are isomorphisms, all transitions are isomorphisms.

Proof. Theorems 2.1 and 3.1 make every transition injective and surjective. Its inverse as a function is linear: applying the transition to the sums and scalar multiples of the two proposed inverse values verifies the linearity equations by injectivity. Thus it is an isomorphism of modules. \(\square\)

4. Examples and the scope of the tests

Example 4.1 (constant diagrams). If \(X_s=M\) and every transition is identity, the left colimit is \(M\), with \([t,x]\mapsto x\). Equality of any two representatives can be checked at one later point still below \(s\). The right limit is \(M\), identified with its constant tuples. The maps \(\lambda_s\) and \(\mu_s\) are identity under these identifications. These identifications also give their actual universal maps, not only the dimensions of the objects.

Example 4.2 (a cut with one failed injectivity test). Let \(M\ne0\), fix \(c\), and take \(X_t=M\) for \(t\leq c\), \(X_t=0\) for \(t>c\). Transitions inside the nonzero region are identity and transitions into the zero region are zero. Every \(\lambda_s\) is an isomorphism. At \(c\), however, \(\mu_c:M\to0\) kills all of \(M\). Its noninjectivity detects precisely the transitions from the closed surviving region into the zero region. The two possible endpoint behaviors are computed in Exercise 2.

5. Four graded exercises with complete solutions

Exercise 1 (basic: changing coordinates at every real point). Let \(M\) be a module and choose an automorphism \(g_s\) of \(M\) for every \(s\in\mathbb R\). Put \(X_s=M\), \(v_{r,s}=g_sg_r^{-1}\). Verify that this is a diagram, compute both local comparisons, and identify the actual inverse of each transition.

Solution. We have \(v_{s,s}=1\) and \(v_{s,t}v_{r,s}=g_tg_s^{-1}g_sg_r^{-1}=g_tg_r^{-1}\), so (1.1) holds. The maps \(g_s^{-1}:X_s\to M\) identify this diagram with the constant one. Concretely, its left colimit identification is

This respects the transitions because \(g_u^{-1}v_{t,u}=g_t^{-1}\). It is onto by representing \(m\) by \(g_tm\) at any \(t<s\). If two representatives give the same \(m\), their images at a common later \(u<s\) both equal \(g_um\), proving injectivity. Under (5.1), \(\lambda_s\) is \(m\mapsto g_sm\).

A compatible right tuple has \(g_t^{-1}x_t\) independent of \(t>s\), by the transition equation; conversely any \(m\in M\) gives the tuple \((g_tm)_{t>s}\). Thus the right limit is \(M\), and under this identification \(\mu_s\) is \(x\mapsto g_s^{-1}x\). Both comparisons are isomorphisms. The actual inverse of \(v_{r,s}\) is \(g_rg_s^{-1}\), as verified by the two composite identities. No continuity of \(s\mapsto g_s\) was needed.

Exercise 2 (intermediate: four endpoint conventions). For a nonzero module \(M\) and a real cut \(c\), consider the following diagrams, with identity maps inside each nonzero region and zero maps when a transition touches a zero module:

Compute \(\lambda_c\) and \(\mu_c\), prove that all other local comparisons are isomorphisms, and explain which single missing local hypothesis prevents an injectivity or surjectivity conclusion.

Solution. At the cut the comparisons are

For a left interval, any terminal stretch wholly in one region is cofinal: every earlier index has an arrow into that stretch. Its colimit is therefore the constant value on that stretch. This can also be checked from representatives, moving each one into the stretch and using its identity or zero transitions. For a right interval, any initial stretch wholly in one region determines the whole compatible family. A tuple's values on the stretch are constant, and all subsequent coordinates are their transition images. Conversely that constant value uniquely supplies the whole tuple. This proves the asserted right-limit identification. When \(s\ne c\), such stretches exist next to \(s\), proving that the two comparisons are the identity of \(X_s\).

The open-death failure is noninjectivity of \(\lambda_c\); every \(\mu_s\) is an isomorphism, but transitions \(M\to0\) have nonzero kernel. The closed-death failure is noninjectivity of \(\mu_c\); every \(\lambda_s\) is an isomorphism, and again \(M\to0\) is not injective. The open-birth failure is nonsurjectivity of \(\mu_c\); every \(\lambda_s\) is an isomorphism, but transitions \(0\to M\) are not onto. The closed-birth failure is nonsurjectivity of \(\lambda_c\); every \(\mu_s\) is an isomorphism, and the same global surjectivity failure occurs. Thus neither one-sided family alone yields the corresponding theorem.

Exercise 3 (advanced: a projection at every strict transition). Let \(P:M\to M\) satisfy \(P^2=P\). Put \(X_s=M\), take identity at \(r=s\), and take \(v_{r,s}=P\) for \(r<s\). Show that \(L_sX\) and \(R_sX\) both identify with \(\operatorname{im}P\), compute \(\lambda_s\) and \(\mu_s\), and specialize to \(P(x,y)=(x,0)\) on \(k^2\).

Solution. Composites of two strict transitions are \(P^2=P\); identities cause no problem. The diagram laws hold. The maps \(X_t\to\operatorname{im}P\), \(x\mapsto Px\), form a left cocone, hence give a map from the colimit. An inverse sends \(z\in\operatorname{im}P\) to \([t,z]\) at any \(t<s\). These classes are independent of \(t\): at a common later index, both images are \(Pz=z\). The other composite is identity as well, since for any \(t<s\) there is a \(u\) with \(t<u<s\), and \([t,x]=[u,Px]\). Under this colimit identification, \(\lambda_s\) is the inclusion \(\operatorname{im}P\hookrightarrow M\).

For a right-limit tuple and any \(t>s\), choose \(u\in(s,t)\). Its coordinate \(x_t=Px_u\) lies in \(\operatorname{im}P\). Therefore all coordinates lie there; the transition equation makes any two equal, because \(P\) acts as identity on its image. Conversely a constant tuple with value in \(\operatorname{im}P\) is compatible. This identifies the limit with that image, and \(\mu_s\) becomes the surjection \(P:M\to\operatorname{im}P\).

For \(M=k^2\) with the displayed projection, the left map is injective but not surjective, and the right map is surjective but not injective. Every strict transition has kernel \(0\oplus k\) and image \(k\oplus0\). Thus the combination of only left injectivity and right surjectivity does not force either global property. The complete hypotheses in Theorems 2.1 and 3.1 detect these two different defects.

Exercise 4 (advanced: global surjectivity without right surjectivity). Let \(k\) be a field and \(D=\bigoplus_{j\geq1}k\). Define \(X_t=D\) for \(t\leq0\), and \(X_t=k^{n(t)}\) for \(t>0\), where \(n(t)=\lfloor1/t\rfloor\). Take identity transitions in the nonpositive region, truncations from \(D\) into finite prefixes, and prefix restrictions between positive times. Prove that this is a diagram with every transition surjective. Compute \(\lambda_0\) and \(\mu_0\), and show that the local surjectivity hypotheses are sufficient but not necessary for global surjectivity.

Solution. The function \(n(t)\) is nonincreasing for \(t>0\), with values in \(\mathbb N\), including zero for \(t>1\). Thus every specified positive transition is a well-defined prefix projection. Prefix projections compose to the corresponding shorter projection, and truncations from \(D\) compose in the same way. The other composites involve an identity in the nonpositive region. This verifies all diagram equations. Every prefix projection is onto, including a projection onto the zero module; every truncation from \(D\) is onto by inserting a finite tuple with all remaining coordinates zero. Thus all transitions are onto.

The negative-time diagram is constant \(D\), so \(L_0X=D\) and \(\lambda_0=1_D\). A compatible positive-time tuple is determined by a sequence of scalars \((a_j)_{j\geq1}\): its value at \(t\) is the first \(n(t)\) scalars. Conversely, from any such tuple recover scalar \(a_j\) at a time with \(n(t)\geq j\), for example \(t=1/j\). Two such times give the same scalar by comparing the shorter prefix. Hence

is the usual finite-support inclusion, with these actual coordinate maps. It is not onto: the sequence with every coordinate equal to \(1\) has infinite support. Thus all global transitions can be surjective even though one \(\mu_s\) is not. Theorem 3.1 asserted sufficiency of the two local hypotheses, and this example proves that their necessity would be false.

References

• The Stacks Project, Filtered colimits, Section 4.19, for the filtered representative convention. The full module representative and zero proofs used here are in the linked prerequisite, Sections 2–3.
• The Stacks Project, Lemma 10.8.8, for comparison with filtered module homology. The real-cut and coherent-lift arguments used for the two local-to-global theorems are given in Sections 2–3 above.