Dagger completions and bornological torsion-freeness
2. Basic notions
In this section, we recall some basic notions on bornological modules and bounded homomorphisms. See [11] for more details. We also study the inheritance properties of separatedness and completeness for submodules, quotients and extensions.
Abornology on a setX is a collectionBX of subsets of X, calledbounded sets, such that all finite subsets are bounded and subsets and finite unions of bounded subsets are bounded. LetV be a complete discrete valuation ring. Abornological V-module is aV-moduleM with a bornology such that every bounded subset is
2. BASIC NOTIONS 23
contained in a boundedV-submodule. In particular, theV-submodule generated by a bounded subset is again bounded. We always writeBM for the bornology onM. Let M′ ⊆ M be a V-submodule. The subspace bornology on M′ consists of all subsets of M′ that are bounded in M. The quotient bornology on M/M′ consists of all subsets of the form q(S) with S ∈ BM, where q∶M → M/M′ is the canonical projection. We always equip submodules and quotients with these canonical bornologies.
Let M andN be two bornological V-modules. A V-module map f∶M →N is bounded if f(S) ∈ BN for all S ∈ BM. Bornological V-modules and bounded V-module maps form an additive category. The isomorphisms in this category are called bornological isomorphisms. A bounded V-module map f∶M →N is a bornological embeddingif the induced mapM →f(M)is a bornological isomorphism, wheref(M) ⊆N carries the subspace bornology. It is a bornological quotient map if the induced mapM/kerf →N is a bornological isomorphism. Equivalently, for eachT∈ BN there is S∈ BM withf(S) =T.
Anextension of bornologicalV-modules is a diagram of V-modules M′Ð→f M Ð→g M′′
that is algebraically exact and such that f is a bornological embedding and g a bornological quotient map. Equivalently,g is a cokernel off andf a kernel ofg in the additive category of bornologicalV-modules. Asplit extension is an extension with a boundedV-linear maps∶M′′→M such thatg○s=idM′′.
Let M be a bornological V-module. A sequence (xn)n∈N in M converges towardsx∈M if there are S∈ BM and a sequence (δn)n∈N inV with lim∣δn∣ =0 andxn−x∈δn⋅S for alln∈N. It is aCauchysequence if there areS∈ BM and a sequence(δn)n∈NinV with lim∣δn∣ =0 andxn−xm∈δj⋅S for all n, m, j∈Nwith n, m ≥j. Since any bounded subset is contained in a bounded V-submodule, a sequence inM converges or is Cauchy if and only if it converges or is Cauchy in the π-adic topology on some boundedV-submodule of M.
We call a subsetS of M closed if x∈S for any sequence inS that converges inM tox∈M. These are the closed subsets of a topology onM. Bounded maps preserve convergent sequences and Cauchy sequences. Thus pre-images of closed subsets under bounded maps remain closed. That is, bounded maps are continuous for these canonical topologies.
2.1. Separated bornological modules. We call M separated if limits of convergent sequences inM are unique. If M is not separated, then the constant sequence 0 has a non-zero limit. Therefore,M is separated if and only if{0} ⊆M is closed. AndM is separated if and only if anyS∈ BM is contained in aπ-adically separated boundedV-submodule.
Lemma 2.1. Let M′Ð→f MÐ→g M′′ be an extension of bornologicalV-modules.
(1) If M is separated, so is M′.
(2) The quotientM′′ is separated if and only if f(M′)is closed inM. (3) If M′and M′′ are separated andM′′ is torsion-free, thenM is separated.
Proof. Assertion (1) is trivial.
IfM′′ is separated, then{0} ⊆M′′ is closed. Henceg−1({0}) =f(M′)is closed inM. If M′′ is not separated, then the constant sequence 0 inM′′ converges to some non-zerox′′∈M′′. That is, there are a bounded subsetS′′⊆M′′ and a null
sequence(δn)n∈NinV withx′′−0∈δn⋅S′′ for alln∈N. Sinceg is a bornological quotient map, there arex∈M andS∈ BM withg(x) =x′′ andg(S) =S′′. We may choose yn′′∈S′′ withx′′=δn⋅yn′′ and yn ∈S withg(yn) =yn′′. So g(x−δnyn) =0.
Thus the sequence(x−δnyn)lies inf(M′). It converges tox, which does not belong tof(M′)becausex′′≠0. Sof(M′)is not closed. This finishes the proof of (2).
We prove (3). Let x∈M belong to the closure of {0} inM. That is, there are S ∈ BM and a null sequence (δn)n∈N inV withx∈δn⋅S for all n∈ N. Then g(x) ∈δn⋅g(S)for alln∈N. This impliesg(x) =0 becauseM′′is separated. So there is y∈M′ withf(y) =x. Andf(y) =x∈δn⋅S. Choosexn ∈S withf(y) =δn⋅xn. We may assume δn ≠0 for all n∈ Nbecause otherwise x∈δn⋅S is 0. Since M′′
is torsion-free,δn⋅xn∈f(M′)impliesg(xn) =0. So we may writexn=f(yn)for some yn ∈M′. Sincef is a bornological embedding, the set {yn∶n∈N} inM′ is bounded. SinceM′is separated and y=δn⋅yn for alln∈N, we gety=0. Hence
x=0. So{0}is closed inM.
The quotientM/{0}of a bornologicalV-moduleM by the closure of 0 is called the separated quotient of M. It is separated by Lemma 2.1, and it is the largest separated quotient ofM. Even more, the quotient mapM →M/{0}is the universal arrow to a separated bornologicalV-module, that is, any bounded V-linear map fromM to a separated bornological V-module factors uniquely throughM/{0}.
The following example shows that Lemma 2.1.(3) fails without the torsion-freeness assumption.
Example2.2. LetM′=V and letM =V[x]/S, whereS is theV-submodule of V[x]generated by 1−πnxn for all n∈N. We embedM′ =V as multiples of 1=x0. Then
M/M′=⊕∞
n=1
V/(πn),
We endow M,M′and M/M′with the bornologies where all subsets are bounded.
We get an extension of bornologicalV-modulesV ↣M ↠ ⊕∞n=1V/(πn). HereV and⊕∞n=1V/(πn)areπ-adically separated, butM is not: the constant sequence 1 inM converges to 0 because 1=1−πnxn+πnxn≡πnxn inM.
2.2. Completeness. We call a bornological V-module M complete if it is separated and for any S ∈ BM there is T ∈ BM so that all S-Cauchy sequences areT-convergent. Equivalently, anyS∈ BM is contained in aπ-adically complete bounded V-submodule (see [11, Proposition 2.8]). By definition, any Cauchy sequence in a complete bornologicalV-module has a unique limit.
Theorem 2.3. LetM′Ð→f M Ð→g M′′ be an extension of bornologicalV-modules.
(1) If M is complete and f(M′)is closed inM, thenM′ is complete.
(2) IfM′is complete,M separated, andM′′torsion-free, thenf(M′)is closed inM.
(3) LetM be complete. ThenM′′ is complete if and only if f(M′)is closed inM.
(4) If M′ andM′′ are complete and M is separated, thenM is complete. If M′ andM′′ are complete andM′′ is torsion-free, then M is complete.
Proof. Statement (1) is [11, Lemma 2.13], and there is no need to repeat the proof here. It is somewhat similar to the proof of (4). Next we prove (2). Assume
2. BASIC NOTIONS 25
thatM′ is complete, thatM′′ is torsion-free, and thatf(M′)is not closed in M. We are going to prove thatM is not separated. There is a sequence(xn)n∈Nin M′ for whichf(xn)n∈N converges inM towards somex∉f(M′). So there is a bounded setS⊆M and a sequence(δk)k∈NinV with lim∣δk∣ =0 andf(xn) −x∈δn⋅Sfor all n∈N. We may assume without loss of generality thatS is a boundedV-submodule and that the sequence of norms ∣δn∣ is decreasing: let δ∗n be the δm for m ≥ n with maximal norm. Then f(xn) −x∈ δn⋅S ⊆δn∗⋅S and still lim∣δn∗∣ = 0. We may write f(xn) −x=δn∗yn with yn ∈ S. Let m< n. Then δm∗g(ym) = −g(x) = δ∗ng(yn)andδn∗/δm∗ ∈V; this implies firstδ∗m⋅ (g(ym) −g(yn) ⋅δn∗/δm∗) =0 and then g(ym) = g(yn) ⋅δn∗/δm∗ because M′′ is torsion-free. So there is zm,n ∈ M′ with ym+f(zm,n) =yn⋅δ∗n/δ∗m. We even havezm,n∈f−1(S)becauseS is aV-submodule.
The subset f−1(S) ⊆M′is bounded becausef is a bornological embedding. We get f(xn) −f(xm) =δn∗yn−δ∗mym=f(δ∗mzm,n)and hencexn−xm=δ∗mzm,nfor n>m.
This witnesses that the sequence(xn)n∈Nis Cauchy in M′. SinceM′is complete, it converges towards somey∈M′. Thenf(xn)converges both towardsf(y) ∈f(M′) and towardsx∉f(M′). SoM is not separated. This finishes the proof of (2).
Next we prove (3). Iff(M′)is not closed, then Lemma 2.1 shows thatM′′ is not separated and hence not complete. Conversely, we claim thatM′′is complete iff(M′)is closed. Lemma 2.1 shows that M′′ is separated. LetS′′∈ BM′′. There is S ∈ BM with g(S) =S′′ because g is a bornological quotient map. And there is T ∈ BM so that any S-Cauchy sequence is T-convergent. We claim that any S′′-Cauchy sequence isg(T)-convergent. So let(x′′n)n∈N be anS′′-Cauchy sequence.
Thus there is a null sequence(δn)n∈NinV with x′′n−x′′m∈δj⋅S′′ for alln, m, j∈N withn, m≥j. As above, we may assume without loss of generality that the sequence of norms ∣δn∣ is decreasing. Choose anyx0∈M withg(x0) =x′′0. For each n∈N, chooseyn∈S withx′′n+1−x′′n=δn⋅g(yn). Let
xn∶=x0+δ0⋅y0+ ⋯ +δn−1⋅yn−1.
Then g(xn) = x′′n. And xn+1−xn = δn⋅yn ∈δn⋅S. Since∣δn∣ is decreasing, this impliesxm−xn∈δn⋅S for allm≥n. So the sequence(xn)n∈N isS-Cauchy. Hence it isT-convergent. Thus g(xn) =x′′n isg(T)-convergent as asserted. This finishes the proof of (3).
Finally, we prove (4). So we assume M′ andM′′ to be complete. If M′′ is torsion-free, thenM is separated by Lemma 2.1. Hence the second statement in (4) is a special case of the first one. Let S ∈ BM. We must find T ∈ BM so that everyS-Cauchy sequence isT-convergent. SinceM is separated, this says that it is complete. Since M′′ is complete, there is aπ-adically complete V-submodule T0 ∈ BM′′ that contains g(S). Since g is a bornological quotient map, there is T1∈ BM withg(T1) =T0. Replacing it byT1+S, we may arrange, in addition, that S⊆T1. Sincef is a bornological embedding,T2∶=f−1(T1)is bounded inM′. AsM′ is complete, there isT3∈ BM′ so that everyT2-Cauchy sequence isT3-convergent.
We claim that anyS-Cauchy sequence isT1+f(T3)-convergent. The proof of this claim will finish the proof of the theorem.
Let (xn)n∈N be an S-Cauchy sequence. So there are δn∈V andyn ∈S with lim∣δn∣ =0 andxn+1−xn=δn⋅yn. As above, we may assume that∣δn∣is decreasing and that δ0 = 1. Since g(yn+k) ∈ g(S) ⊆ T0 and T0 is π-adically complete, the
following series converges inT0:
(2.4) w˜n∶= −∑∞
k=0
δn+k
δn g(yn+k). Since ˜wn∈T0, there is wn∈T1 withg(wn) =w˜n. So
δng(wn) = lim
N→∞−g(∑N
k=0
δn+kyn+k)
= lim
N→∞g(xn−xN+n+1) =g(xn) − lim
N→∞g(xN). In particular,g(w0) =g(x0) −limN→∞g(xN). Now let
˜
xk ∶=xk−δkwk+w0−x0. Then
g(x˜k) =g(xk) −g(xk) + lim
N→∞g(xN) +g(x0) − lim
N→∞g(xN) −g(x0) =0.
So ˜xk∈f(M′)for allk∈N. And
(2.5) x˜n+1−x˜n=xn+1−xn−δn+1wn+1+δnwn
=δnyn+δnwn−δn+1wn+1=δn⋅ (yn+wn−δn+1
δn
wn+1). Letzn∶=yn+wn−δn+1δn wn+1. A telescoping sum argument shows that
(2.6) g(zn) =g(yn) +w˜n−δn+1 δn
˜ wn+1=0.
So zn ∈ f(M′). And zn ∈ S+T1+T1 = T1. Thus there is ˆzn ∈ f−1(T1) = T2 with zn =f(zˆn). Equation (2.5) means that the sequence f−1(x˜n) is T2-Cauchy.
Hence it isT3-convergent. So(x˜n)isf(T3)-convergent. Then(xn)isT1+f(T3)
-convergent.
The following examples show that the technical extra assumptions in (2) and (4) in Theorem 2.3 are necessary. They only involve extensions ofV-modules with the bornology where all subsets are bounded. For this bornology, bornological com-pleteness and separatedness are the same asπ-adic completeness and separatedness, respectively, and any extension ofV-modules is a bornological extension.
Example 2.7. Let M′∶= {0} andM ∶=F with the bornology of all subsets.
Then M′ is bornologically complete, but not closed in M, and M/M′ = M is torsion-free. So Theorem 2.3.(2) needs the assumption thatM be separated.
Example 2.8. Let M be the V-module of all power series ∑∞n=0cnxn with lim∣cn∣ =0 and with the bornology where all subsets are bounded; this is theπ-adic completion of the polynomial algebraV[x]. Let M′=M and define f∶M′→M, f(∑∞n=0cnxn) ∶= ∑∞n=0cnπnxn. This is a bornological embedding simply because all subsets inM =M′ are bounded. Letpn∶= ∑nj=0xj. This sequence in M′=M does not converge. Nevertheless, the sequencef(pn) = ∑nj=0πjxj converges inM to
∑∞j=0πjxj. Thusf(M′)is not closed inM, althoughM andM′are complete andf is a bornological embedding. So Theorem 2.3.(2) needs the assumption thatM′′ be torsion-free.
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Example2.9. We modify Example 2.2 to produce an extension of V-modules N′↣N↠N′′ whereN′ andN′′are π-adically complete, butN is notπ-adically separated and hence notπ-adically complete. We let N′∶=V/(π) =F. We letN′′
be theπ-adic completion of theV-moduleM′′of Example 2.2. That is, N′′∶= {(cn)n∈N∈ ∏∞
n=0
V/(πn) ∶lim∣cn∣ =0}. This is indeed π-adically complete. So is
N1∶= {(cn)n∈N∈∏∞
n=0
V/(πn+1) ∶lim∣cn∣ =0}.
The kernel of the quotient map q∶N1↠N′′ is isomorphic to ∏∞n=0V/(π) = ∏NF. This is aF-vector space, and it contains theF-vector space∑∞n=0F. Since anyF-vector space has a basis, we may extend the linear functional ∑∞n=0F → F, (cn)n∈N ↦
∑∞n=0cn, to a F-linear functional σ∶ ∏NF → F. Let L ∶= kerσ ⊆ kerq and let N∶=N1/L. The mapqdescends to a surjectiveπ-linear mapN ↠N′′. Its kernel is isomorphic to∏NF/kerσ≅F=N′. The functionalσ∶ ∏NF→F vanishes onδ0−δk
for allk∈N, but not onδ0. When we identify∏NF≅kerq, we mapδk toπkδk∈N1. Soδ0 andπkδk get identified inN, butδ0does not become 0: it is the generator of N′=V/(π)insideN. Since[δ0] =πk[δk]inN, theV-moduleN is notπ-adically separated.
Thecompletion M of a bornological V-moduleM is a complete bornological V-module with a boundedV-linear mapM →M that is universal in the sense that any boundedV-linear map fromM to a complete bornologicalV-moduleX factors uniquely throughM. Such a completion exists and is unique up to isomorphism (see [11, Proposition 2.15]). We shall describe it more concretely later when we
need the details of its construction.
2.3. Vector spaces over the fraction field. Recall that F denotes the quotient field ofV. AnyV-linear map between twoF-vector spaces is alsoF-linear.
SoF-vector spaces withF-linear maps form a full subcategory in the category of V-modules. AV-moduleM comes from aF-vector space if and only if the map
(2.10) πM∶M→M, m↦π⋅m,
is invertible. We could define bornologicalF-vector spaces without reference toV. Instead, we realise them as bornologicalV-modules with an extra property:
Definition2.11. A bornologicalV-moduleM is a bornological F-vector space if the mapπM in (2.10) is a bornological isomorphism, that is, an invertible map with bounded inverse.
Given a bornologicalV-moduleM, the tensor productF⊗M∶=F⊗V M with the tensor product bornology (see [11, Lemma 2.18]) is a bornological F-vector space because multiplication byπis a bornological isomorphism onF.
Lemma 2.12. The canonical bounded V-linear map ιM∶M →F⊗M, m↦1⊗m,
is the universal arrow fromM to a bornologicalF-vector space, that is, any bounded V-linear mapf∶M →N to a bornologicalF-vector spaceN factors uniquely through a boundedV-linear mapf#∶F⊗M →N, and this map is alsoF-linear.
Proof. A V-linear map f#∶F⊗M →N must be F-linear. Hence the only possible candidate is theF-linear map defined byf#(x⊗m) ∶=x⋅f(m)form∈M, x∈F. Any bounded submodule ofF⊗M is contained inπ−kV⊗Sfor some bounded submoduleS⊆M and somek∈N, andf#(π−kV⊗S) =πN−k(f(S))is bounded inN becauseπN is a bornological isomorphism. Thusf#is bounded.