(also nonabelian homological algebra)
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The general concept of localization applied to the (derived) category of abelian groups yield the concept of localization of abelian groups.
The two main examples are
classical localization at/aways from primes;
completion at a prime
at prime numbers $p$. Here “classical $p$-localization” is localization at the morphism $0 \to \mathbb{Z}/p\mathbb{Z}$, while $p$-completion is localization at the morphism $0 \to \mathbb{Z}[p^{-1}]$.
Recall that Ext-groups $\Ext^\bullet(A,B)$ between abelian groups $A, B \in$ Ab are concentrated in degrees 0 and 1 (prop.). Since
is the plain hom-functor, this means that there is only one possibly non-vanishing Ext-group $Ext^1$, therefore often abbreviated to just “$Ext$”:
Let $K$ be an abelian group.
Then an abelian group $A$ is called $K$-local if all the Ext-groups from $K$ to $A$ vanish:
hence equivalently (remark ) if
A homomorphism of abelian groups $f \colon B \longrightarrow C$ is called $K$-local if for all $K$-local groups $A$ the function
is a bijection.
(Beware that here it would seem more natural to use $Ext^\bullet$ instead of $Hom$, but we do use $Hom$. See (Neisendorfer 08, remark 3.2).
A homomorphism of abelian groups
is called a $K$-localization if
$L_K A$ is $K$-local;
$\eta$ is a $K$-local morphism.
We now discuss two classes of examples of localization of abelian groups
For $n \in \mathbb{N}$, write $\mathbb{Z}/n\mathbb{Z}$ for the cyclic group of order $n$.
For $n \in \mathbb{N}$ and $A \in Ab$ any abelian group, then
the hom-group out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is the $n$-torsion subgroup of $A$
the first Ext-group out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is
Regarding the first item: Since $\mathbb{Z}/p\mathbb{Z}$ is generated by its element 1 a morphism $\mathbb{Z}/p\mathbb{Z} \to A$ is fixed by the image $a$ of this element, and the only relation on 1 in $\mathbb{Z}/p\mathbb{Z}$ is that $p \cdot 1 = 0$.
Regarding the second item:
Consider the canonical free resolution
By the general discusson of derived functors in homological algebra this exhibits the Ext-group in degree 1 as part of the following short exact sequence
where the morphism on the left is equivalently $A \overset{n \cdot (-)}{\to} A$.
An abelian group $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if the operation
of multiplication by $p$ is an isomorphism, hence if “$p$ is invertible in $A$”.
By the first item of lemma we have
By the second item of lemma we have
Hence by def. $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if
and if
Both these conditions are equivalent to multiplication by $p$ being invertible.
For $J \subset \mathbb{N}$ a set of prime numbers, consider the direct sum $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ of cyclic groups of order $p$.
The operation of $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization of abelian groups according to def. is called inverting the primes in $J$.
Specifically
for $J = \{p\}$ a single prime then $\mathbb{Z}/p\mathbb{Z}$-localization is called localization away from $p$;
for $J$ the set of all primes except $p$ then $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization is called localization at $p$;
for $J$ the set of all primes, then $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localizaton is called rationalization..
For $J \subset Primes \subset \mathbb{N}$ a set of prime numbers, then
denotes the subgroup of the rational numbers on those elements which have an expression as a fraction of natural numbers with denominator a product of elements in $J$.
This is the abelian group underlying the localization of a commutative ring of the ring of integers at the set $J$ of primes.
If $J = Primes - \{p\}$ is the set of all primes except $p$ one also writes
Notice the parenthesis, to distinguish from the notation $\mathbb{Z}_{p}$ for the p-adic integers (def. below).
The terminology in def. is motivated by the following perspective of arithmetic geometry:
Generally for $R$ a commutative ring, then an $R$-module is equivalently a quasicoherent sheaf on the spectrum of the ring $Spec(R)$.
In the present case $R = \mathbb{Z}$ is the integers and abelian groups are identified with $\mathbb{Z}$-modules. Hence we may think of an abelian group $A$ equivalently as a quasicoherent sheaf on Spec(Z).
The “ring of functions” on Spec(Z) is the integers, and a point in $Spec(\mathbb{Z})$ is labeled by a prime number $p$, thought of as generating the ideal of functions on Spec(Z) which vanish at that point. The residue field at that point is $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
Inverting a prime means forcing $p$ to become invertible, which, by this characterization, it is precisely away from that point which it labels. The localization of an abelian group at $\mathbb{Z}/p\mathbb{Z}$ hence corresponds to the restriction of the corresponding quasicoherent sheaf over $Spec(\mathbb{Z})$ to the complement of the point labeled by $p$.
Similarly localization at $p$ is localization away from all points except $p$.
See also at function field analogy for more on this.
For $J \subset \mathbb{N}$ a set of prime numbers, a homomorphism of abelian groups $f \;\colon\; A \longrightarrow B$ is local (def. ) with respect to $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ (def. ) if under tensor product of abelian groups with $\mathbb{Z}[J^{-1}]$ (def. ) it becomes an isomorphism
Moreover, for $A$ any abelian group then its $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization exists and is given by the canonical projection morphism
(e.g. Neisendorfer 08, theorem 4.2)
We have seen above in remark that classical localization of abelian groups at a prime number is geometrically interpreted as restricting a quasicoherent sheaf over Spec(Z) to a single point, the point that is labeled by that prime number.
Alternatively one may restrict to the “infinitesimal neighbourhood” of such a point. Technically this is called the formal neighbourhood, because its ring of functions is, by definition, the ring of formal power series (regarded as an adic ring or pro-ring). The corresponding operation on abelian groups is accordingly called formal completion or adic completion or just completion, for short, of abelian groups.
It turns out that if the abelian group is finitely generated then the operation of p-completion coincides with an operation of localization in the sense of def. , namely localization at the p-primary component $\mathbb{Z}(p^\infty)$ of the group $\mathbb{Q}/\mathbb{Z}$ (def. below). On the one hand this equivalence is useful for deducing some key properties of p-completion, this we discuss below. On the other hand this situation is a shadow of the relation between localization of spectra and nilpotent completion of spectra, which is important for characterizing the convergence properties of Adams spectral sequences.
For $p$ a prime number, then the p-adic completion of an abelian group $A$ is the abelian group given by the limit
where the morphisms are the evident quotient morphisms. With these understood we often write
for short. Notice that here the indexing starts at $n = 1$.
The p-adic completion (def. ) of the integers $\mathbb{Z}$ is called the p-adic integers, often written
which is the original example that gives the general concept its name.
With respect to the canonical ring-structure on the integers, of course $p \mathbb{Z}$ is a prime ideal.
Compare this to the ring $\mathcal{O}_{\mathbb{C}}$ of holomorphic functions on the complex plane. For $x \in \mathbb{C}$ any point, it contains the prime ideal generated by $(z-x)$ (for $z$ the canonical coordinate function on $\mathbb{z}$). The formal power series ring $\mathbb{C}[ [(z.x)] ]$ is the adic completion of $\mathcal{O}_{\mathbb{C}}$ at this ideal. It has the interpretation of functions defined on a formal neighbourhood of $X$ in $\mathbb{C}$.
Analogously, the p-adic integers $\mathbb{Z}_p$ may be thought of as the functions defined on a formal neighbourhood of the point labeled by $p$ in Spec(Z).
There is a short exact sequence
Consider the following commuting diagram
Each horizontal sequence is exact. Taking the limit over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.
We now consider a concept of $p$-completion that is in general different from def. , but turns out to coincide with it in finitely generated abelian groups.
For $p$ a prime number, write
for the colimit (in Ab) over iterative applications of multiplication by $p$ on the integers.
This is the abelian group generated by formal expressions $\frac{1}{p^k}$ for $k \in \mathbb{N}$, subject to the relations
Equivalently it is the abelian group underlying the ring localization $\mathbb{Z}[1/p]$.
For $p$ a prime number, then localization of abelian groups (def. ) at $\mathbb{Z}[1/p]$ (def. ) is called $p$-completion of abelian groups.
An abelian group $A$ is $p$-complete according to def. precisely if both the limit as well as the lim^1 of the sequence
vanishes:
and
By def. the group $A$ is $\mathbb{Z}[1/p]$-local precisely if
Now use the colimit definition $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}_n \mathbb{Z}$ (def. ) and the fact that the hom-functor sends colimits in the first argument to limits to deduce that
This yields the first statement. For the second, use that for every cotower over abelian groups $B_\bullet$ there is a short exact sequence of the form
(by this lemma).
In the present case all $B_n \simeq \mathbb{Z}$, which is a free abelian group, hence a projective object, so that all the Ext-groups out of it vannish. Hence by exactness there is an isomorphism
This gives the second statement.
For $p$ a prime number, the p-primary cyclic groups of the form $\mathbb{Z}/p^n \mathbb{Z}$ are $p$-complete (def. ).
By lemma we need to check that
and
For the first statement observe that $n$ consecutive stages of the tower compose to the zero morphism. First of all this directly implies that the limit vanishes, secondly it means that the tower satisfies the Mittag-Leffler condition (def.) and this implies that the $\lim^1$ also vanishes (prop.).
For $p$ a prime number, write
(the p-primary part of $\mathbb{Q}/\mathbb{Z}$), where $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}(\mathbb{Z}\overset{p}{\to} \mathbb{Z} \overset{p}{\to} \mathbb{Z} \to \cdots )$ from def. .
Since colimits commute over each other, this is equivalently
For $p$ a prime number, the $\mathbb{Z}[1/p]$-localization
of an abelian group $A$ (def. , def. ), hence the $p$-completion of $A$ according to def. , is given by the morphism
into the first Ext-group into $A$ out of $\mathbb{Z}(p^\infty)$ (def. ), which appears as the first connecting homomorphism $\delta$ in the long exact sequence of Ext-groups
that is induced (via this prop.) from the defining short exact sequence
e.g. (Neisendorfer 08, p. 16)
If $A$ is a finitely generated abelian group, then its $p$-completion (def. , for any prime number $p$) is equivalently its p-adic completion (def. )
By theorem the $p$-completion is $Ext^1(\mathbb{Z}(p^\infty),A)$. By def. there is a colimit
Together this implies, by this lemma, that there is a short exact sequence of the form
By lemma the lim^1 on the left is over the $p^n$-torsion subgroups of $A$, as $n$ ranges. By the assumption that $A$ is finitely generated, there is a maximum $n$ such that all torsion elements in $A$ are annihilated by $p^n$. This means that the Mittag-Leffler condition (def.) is satisfied by the tower of $p$-torsion subgroups, and hence the lim^1-term vanishes (prop.).
Therefore by exactness of the above sequence there is an isomorphism
If $A$ is a $p$-divisible group in that $A \overset{p \cdot (-)}{\longrightarrow} A$ is an isomorphism, then its $p$-completion (def. ) vanishes.
By theorem the localization morphism $\delta$ sits in a long exact sequence of the form
Here by def. and since the hom-functor takes colimits in the first argument to limits, the second term is equivalently the limit
But by assumption all these morphisms $p \cdot (-)$ that the limit is over are isomorphisms, so that the limit collapses to its first term, which means that the morphism $\phi$ in the above sequence is an isomorphism. But by exactness of the sequence this means that $\delta = 0$.
Let $p$ be a prime number. If $A$ is a finite abelian group, then its $p$-completion (def. ) is equivalently its p-primary part.
By the fundamental theorem of finite abelian groups, $A$ is a finite direct sum
of cyclic groups of ordr $p_i^{k_1}$ for $p_i$ prime numbers and $k_i \in \mathbb{N}$ (thm.).
Since finite direct sums are equivalently products (biproducts: Ab is an additive category) this means that
By theorem the $i$th factor here is the $p$-completion of $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$, and since $p \cdot(-)$ is an isomorphism on $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$ if $p_i \neq p$ (because its kernel evidently vanishes), prop. says that in this case the factor vanishes, so that only the factors with $p_i = p$ remain. On these however $Ext^1(\mathbb{Z}(p^\infty),-)$ is the identity by example .
Joseph NeisendorferA Quick Trip through Localization, in Alpine perspectives on algebraic topology, Third Ariolla Conference 2008 (pdf)
Peter May, Kate Ponto, section 10.1 of More concise algebraic topology: Localization, completion, and model categories (pdf)
Last revised on December 10, 2020 at 00:10:32. See the history of this page for a list of all contributions to it.