## Elements in a localization - category theoretic approach

This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the universal property of $S^{-1} A$? In order to make this question clear enough I will have to digress a little bit.

Examples. a) Let's start with a related example. If $C$ is a category of algebras of some type, then every continuous functor $C \to \mathrm{Set}$ is representable (special case of SAFT). For example for $C=\mathbf{Ab}$ the functor $\mathrm{Hom}(A,-) \times \mathrm{Hom}(B,-)$ is representable for all abelian groups $A,B$, showing the existence of the coproduct $A + B$. I claim that we can find a description of its elements via the universal property. Namely, choose the coproduct injections $i : A \to A + B$ and $j : B \to A + B$. Then $C:=\mathrm{im}(i) + \mathrm{im}(j)$ is an abelian group such that $i,j$ factor through $C$ and still satisfy the universal property - this shows $A = \mathrm{im}(i) + \mathrm{im}(j)$, i.e. every element has the form $i(a) + j(b)$ for $a \in A, b \in B$. I claim that $a,b$ are unique: By the universal property there is some $f : A + B \to A$ with $fi = \mathrm{id}$ and $fj=0$. It follows $f(i(a) + j(b))=a$ and therefore $a$ is unique, similarly $b$.

Conclusion: We defined $A + B$ via some universal property (whose existence follows from general category theory) and found the structure of its elements - only by applying the universal property. We didn't need to know any specific constructed model of $A + B$ in advance for this!

Of course for abelian groups and coproducts this isn't really exciting. But what about more complicated (algebraic) structures (e.g. groups, rings, affine spaces, lie algebras, etc.) ? There is a general construction of $A + B$ with generators and relations of $A$ and $B$ (see e.g. Durov, 4.16.14/15), but this does not give us a criterion when two given elements of $A + B$ are equal. It gets even more complicated for other universal properties aka representable functors: For example you can write down a group defined by $10$ generators and $27$ explicitly (Wikipedia) with an unsolvable word problem. Let me mention two (of the many) positive results in this direction:

b) Let $M,N$ be modules over a commutative ring $k$ and define the tensor product $M \otimes_k N$ as the classifying object of $k$-bilinear maps on $M \times N$. If $E$ is a generating set of $M$, then the universal property implies that every element has the form $\sum_{e \in E} e \otimes n_e$ for some $n_e \in N$ (which vanish for almost all $e$). There is a criterion when this element is zero and it can be proved with the universal property of the tensor product, without using its explicit construction (Pierre Mazet, Caracterisation des Epimorphismes par relations et generateurs, online).

c) If $H$ is a subgroup of a group $G_1$ as well as of a group $G_2$, then it is clear how to represent elements in the amalgamated sum $G_1 *_H G_2$, which is defined as a pushout. The uniqueness of the representation is shown in Serre's Trees, Section 1.2, by application of the universal property to the symmetric group over all reduced words. There is no need to impose a group operation on the set of reduced words (which would be very tedious) - after this proof you get it for free!

Localization. Let's consider a commutative ring $A$ and a map $i : S \to |A|$ into the underlying set of $A$. Then we can consider the subfunctor of $\mathrm{Hom}(A,-)$ which is given by homomorphisms $g : A \to B$ such that $gi$ factors through $B^*$. By general theorems a representing object exists and is usually denoted the localization $S^{-1} A$ when $i$ is understood. If $S'$ denotes the multiplicative closure of the image of $i$, then $S'^{-1} A = S^{-1} A$ (they satisfy the same universal property), thus we always may assume that $S$ is just a multiplicative closed subset of $A$.

Let $\tau : A \to S^{-1} A$ be the universal homomorphism which maps $S$ to units. Then $S^{-1} A = \{\tau(a) \tau(s)^{-1} : a \in A, s \in S\}$ (they satisfy the same universal property). Let es write $\frac{a}{s} = \tau(a) \tau(s)^{-1}$. Thus every element in $S^{-1} A$ is some fraction $\frac{a}{s}$. Now when are two such fractions equal? Of course we know this from the usual construction using equivalence classes of pairs $(a,s)$, but I want to avoid this construction and derive it only with the universal property.

It is clear that $s'a = sa'$ implies $\frac{a}{s} = \frac{a'}{s'}$. From this one easily derives more generally that $ts'a = tsa'$ also implies $\frac{a}{s} = \frac{a'}{s'}$; here $a,a' \in A$ and $s,s',t \in S$.

Question. Is there a proof of the converse, i.e. $\frac{a}{s} = \frac{a'}{s'} \Rightarrow \exists t \in S : ts'a = tsa'$, which only uses the universal property of the localization?

First observe that $\frac{a}{s} = \frac{a'}{s'}$ iff $\frac{sa'-sa'}{1}=0$. Thus it is enough to show $\frac{a}{1} = 0 \Rightarrow sa=0$ for some $s \in S$, i.e. that the kernel of $\tau : A \to S^{-1} A$ equals $I=\cup_{s \in S} \mathrm{Ann}(s)$. Since $S$ is multiplicative, $I$ is an ideal, and we may replace $A$ by $A/I$, where localization commutes with quotients because of universal properties. Thus we may assume that $I=0$, i.e. that $S$ consists of regular elements. Our task is then to show that $A \to S^{-1} A$ is injective. But a priori we only know by the universal property that the kernel of $A \to S^{-1} A$ equals the intersection of all kernels of homomorphisms $A \to B$ which map $S$ to units. Of course we have to use this and construct some specific $A \to B$, but I hope that we can either avoid some nasty element construction of $B$ or that we can just use a ring which is built up out of $S^{-1} A$, but can be used to show that the kernel is zero - therefore to find a kind of self-referential proof. The examples above suggest that this might be possible after all.

Another approach is to use $S^{-1} A = A[\{X_s\}_{s \in S}]/(s X_s - 1)$ (which is clear since both sides satisfy the same universal property). Here the polynomial algebra should be defined by its universal property, as well as the quotient ring. Assuming that we know the structure of its elements, then one can also show that the kernel of $\tau$ is as desired; see the note "Rings of fractions the hard way" by José Felipe Voloch. I think this is quite interesting, but it basically just gives another construction of the localization in terms of polynomial algebras and quotients, and then computes the kernel of $\tau$ from this specific construction. So this is not really what I'm after.

Moosbrugger suggest to talk about the category of $A$-modules. Here is a basic observation which follows from the universal property: If $R$ is a possibly noncommutative $A$-algebra, then there is at most one algebra homomorphism $S^{-1} A \to R$, and it exists iff all elements of $S$ become invertible in $R$. This follows easily from the universal property of $S^{-1} A$ applied to the center of $R$. Now let $M$ be an $A$-module and apply the above to $R=\mathrm{End}_A(M)$. It follows that the category of $S^{-1} A$-modules is equivalent to the category of $A$-modules, on which the elements of $S$ act as isomorphisms.

Therefore a possible formalization of my question might be the following: Let $S \subseteq A$ as above and let $B$ be a commutative $A$-algebra such that scalar restriction $\mathrm{Mod}(B) \to \mathrm{Mod}(A)$ is fully faithful and whose image consists of those $A$-modules on which the elements of $S$ act as isomorphisms. Can we then compute the kernel of $A \to B$ (without refering to the explicit construction of the localization $S^{-1} A$, but only using this statement about module categories, which of course yields $B \cong S^{-1} A$ by Morita).

Motivation. I hope that a categorical proof makes the usual construction of the localization (via equivalence classes of pairs) redundant. Note also that it is rather nasty to prove all the details (equivalence relation, well-defined addition, well-defined multiplication, universal property) with the usual construction. Also the definition $(a,s) \sim (a',s') \Leftrightarrow \exists t : ts'a=tsa'$ is not motivated at all there. A categorical proof should show in particular that this is the right choice, and not some random definition which turns out to be correct only after some computation. But of course I don't claim that a categorical construction of the localization is the easiest or best one.

Although the universal property is probably the most important aspect of localization (which we get, as already mentioned, by abstract nonsense), the criterion for the equality of elements is essential even for basic properties in commutative algebra, for example in order to prove that every integral domain embeds into its field of quotients.

On the other hand, localization could be seen just as a toy example for other, more involved examples, where it is not clear at all how to understand some representing object of some functor, which exists my general nonsense.

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Meta-discussion: meta.mathoverflow.net/discussion/1269/… – Martin Brandenburg Jan 28 2012 at 21:32
Martin: I corrected the LaTeX to be able to read it, but you were probably doing the same at the same time. I did not mean to mess with your post! (Also, I did not change anything!) – Sándor Kovács Jan 28 2012 at 21:54
Two comments. First, in the second paragraph of your section on motivation, you say the criterion for equality in a localization is essential to prove any domain embeds into its fraction field. That's not really true, since everyone learns (or at least should learn) about the fraction field construction before learning general localization, and for fraction fields of domains one can use the more naive notion of equivalence for $a/s$ and $a'/s'$ (i.e., no need for mult. by an extra element of $S$ after clearing denominators). My second comment is that you say the definition of [to be contd.] – KConrad Jan 28 2012 at 22:17