Setup
Following page 32 of Vakil's notes,let S be a multiplicative subring of a ring A; i.e.,
$1 ∈ S ∧ x,y ∈ S ⇒ x · y ∈ S$. Then we consider “formal fractions”,
S⁻¹A ≔ { a / s ∣ a ∈ A , s ∈ S }
The property we're interested in is
𝒫 : A-algebra → Bool
𝒫 f ≔ for every e in S, f e ∈ B is invertible
Want to show:
S⁻¹A is initial among A-algebras B where every element of S is sent to an invertible element in B
So we're interested in the category whose objects are pairs
(B, f) where B is a ring and f : A → B is a ring morphism satisfying 𝒫.
Note that $S⁻¹A$ is such an algebra due to the emebeding
$i : a ↦ a / 1$ since we know $1 ∈ S$. Moreover we have that $𝒫 i$ holds since
$(s / 1)⁻¹ = 1 / s$ in $S⁻¹A$ for $s \in S$ : it is clear that $1/1$ is the unit of multiplication and so
we need to show $(s / 1) · (1 / s) = 1 / 1$. Indeed,
(s / 1) · (1 / s) = 1 / 1
≡{ multiplication in S⁻¹A }
s · 1 / 1 · s = 1 / 1
≡{ ring arithmetic }
s / s = 1 / 1
≡{ equality in S⁻¹A }
∃ s' : S • s' · (1 · s - s · 1) = 0
≡{ ring arithmetic }
∃ s' : S • s' · 0 = 0
≡{ ring arithmetic }
true
Now the natural definition of an arrow between two such objects is a ring morphism
of the underlying rings that makes the obvious diagram commute:
f : (B, g) ⟶ (C, h) iff f : B → C is a ring map with f ∘ g = h.
A --g--->B
| /
| /
h / f
| /
| /
v /
C</
Defining the needed morphism
Given an A-algebra $f : A → B$ satisfying $𝒫$, let us show that there is an arrow
$(S⁻¹A, i) ⟶ (B, f)$.
Given an element $a / s : S⁻¹A$, we know that $f \ s$ is invertible by assumption $𝒫 \ f$ and so we can form
the expression $f a · (f s)⁻¹ : B$. Let $⟨ f ⟩$ be the name of this operation; i.e.,
$⟨ f ⟩ : S⁻¹A → B$ with $⟨ f ⟩ (a / s) = f a · (f s)⁻¹$. Since / is just formal notation for pairs,
in this context, that gets reified into an actual division by ⟨_⟩, let us write ÷ for a legitimate
division in a ring: $x ÷ y = x · y ⁻¹$. Then, $⟨ f ⟩ (a / s) = f a ÷ f s$.
Notice that we do not know if $s⁻¹$ exists and so cannot use ring morphism properties to
rewrite this last expression as $f (a / s)$.
Anyhow, for ⟨ f ⟩ to be an arrow in our category we must show that is
- satisfies 𝒫
- is a ring morphism
- satisfies the triangle diagram
- and is unique, so that we have intiality.
Satisfies 𝒫
Let $a, s$ be arbitrary, then
⟨ f ⟩ (a / s) ⁻¹ = ⟨ f ⟩ (s / a)
Indeed,
⟨ f ⟩ (a / s) · ⟨ f ⟩ (s / a)
={ definition of angels }
(f a ÷ f s) · (f s ÷ f a)
={ division notation }
f a · (f s)⁻¹ · f s · (f a)⁻¹
={ ring arithmetic }
f a · 1 · (f a)⁻¹
={ ring arithmetic }
1
Ring morphism
For example, for additivity:
⟨ f ⟩ (a/s + b/t)
={ definition of + in S⁻¹A }
⟨ f ⟩ ( (t · a + s · b)/(s · t) )
={ definition of angels }
f (t · a + s · b) ÷ f (s · t)
={ f is a ring morphism }
(f t · f a + f s · f b) ÷ (f s · f t)
={ ring arithmetic }
(f t · f a) ÷ (f s · f t)
+ (f s · f b) ÷ (f s · f t)
={ ring arithmetic:
(f t · f a) ÷ (f s · f t)
={ definition of ÷ }
f t · f a · (f s · f t)⁻¹
={ ring arithmetic }
f t · f a · f t ⁻¹ · f s ⁻¹
={ f ring homomorphism }
f (t · a) · f t ⁻¹ · f s ⁻¹
={ definition }
⟨ f ⟩ ( t · a / t) · f s ⁻¹
={ lemma; see below; and ring arithmetic using f ring morphism }
f a · f s ⁻¹
={ notation }
f a ÷ f s
lemma:
t · a / t = a / 1
≡{ equality in S⁻¹A }
∃ s : S • s · (1 · t · a - t · a) = 0
≡{ ring arithemtic }
∃ s : S • s · 0 = 0
≡{ ring arithmetic }
true
Similarly for the right argument to +.
Anyhow, back to the main calculation.
}
f a ÷ f s + f b ÷ f t
={ definition }
⟨ f ⟩ (a / s) + ⟨ f ⟩ (b / t)
I will not check the other properties ─I'm not as vested in this problem as you!
Arrow
It is also an arrow in the category; indeed: for arbitrary a,
⟨ f ⟩ (i a)
={ definition of i }
⟨ f ⟩ (a / 1)
={ definition of ⟨⟩ }
f a ÷ f 1
={ f ring morphism }
f a ÷ 1
={ ring arithmetic }
f a
Whence $⟨ f ⟩ ∘ i = f$
Uniqueness
It remains to show that $⟨ f ⟩$ is unique with these properties.
That is, given any ring morphism $G : S⁻¹A → B$ with $G ∘ i = f$, let us show that it equals $⟨ f ⟩$.
⟨ f ⟩ (a / s)
={ definition }
f a ÷ f s
={ assumption G ∘ i = f }
G (i a) ÷ G (i s)
={ definition of i}
G (a / 1) ÷ G (s / 1)
={ notation }
G (a / 1) · G (s / 1) ⁻¹
={ G ring morphism }
G ( (a / 1) · (s / 1)⁻¹ )
={ first lemma about inverses of formal fractions }
G ( (a / 1) · (1 / s) )
={ multiplication in S⁻¹A }
G (a · 1 / 1 · s)
={ ring arithmetic }
G (a / s)
Hence, by extensionality, G = ⟨ f ⟩ sweetums :-)
Conclusion
At the end of the problem statement they give rephrasings of the desideratum,
which we'll restate for fun.
Using the above formalisation and usual category theory definition of initiality, we have
S⁻¹A is initial among A-algebras sending every element of S to an invertible element
≡
∀ B ring, f : A → B ring-map satisfying 𝒫 • ∃₁ f' : S⁻¹A → B • f' ∘ i = f ∧ 𝒫 f'
But by construction this f' is obtained by the angle brackets; we're doing a process known as skolemisation. So this equivales
∀ B ring, ring maps f : A → B and g : S⁻¹A → B •
g = ⟨ f ⟩ ≡ 𝒫 f ∧ g ∘ i = f
This' loosey-goosey true since:
for ⇒ to hold we need ⟨ f ⟩ to be defined which means we need 𝒫 f to hold
and we need to know that g is an arrow in the category which means g ∘ i = f;
conversely, we know that g ∘ i = f has the unique solution ⟨ f ⟩ which is defined
since we have 𝒫 f. A formal proof is not difficult either:
𝒫 f ∧ g ∘ i = f
⇒{ “Defining the needed morphism” and “uniqueness” }
g = ⟨ f ⟩
⇒{ composition and “arrow” }
g ∘ i = f
≡{ logic }
g ∘ i = f ∧ g ∘ i = f
⇒{
claim: g ∘ i = f ⇒ 𝒫 f
Proof: let s : S be arbitrary, then
we show (f s)⁻¹ = g (1 / s).
f s · g (1 / s)
={ assumption and definition of i }
g (s / 1) · g (1 / s)
={ g ring morphism }
g ( s/1 · 1/s )
={ very first calculation in “setup” }
g (1 / 1)
={ g ring morphism }
1
}
𝒫 f ∧ g ∘ i = f
Anyhow,
∀ B ring, ring maps f : A → B and g : S⁻¹A → B •
g = ⟨ f ⟩ ≡ 𝒫 f ∧ g ∘ i = f
Read ⇒:
a ring map S⁻¹A → B is the “same thing as” (ie corresponds to) a ring map A → B satisfying 𝒫
Read ⇐:
any ring map A → B sending elements of S to invertible elements
factors through i : A → S⁻¹A and does so uniquely.
Intiality really just says that $i$ is the best map satisfying 𝒫 and any other such map must
be “further away” (in that it factors through $i$).
Hope this helps! :-)
PS. Notice that since $g ∘ i = f ⇒ 𝒫 f$, we could simplify the equivalence to just
$g = ⟨ f ⟩ ≡ g ∘ i = f$, but then those English renditions are no longer immediate.
Best Answer
The assumption that $M\to S^{-1}M$ is initial among maps where elements of $S$ act invertibly on the target includes the assumption that the elements of $S$ act invertibly on $S^{-1}M$ (if it didn't, $M$ would always be the appropriate universal object, which is clearly wrong). This takes care of all issues relating to (1) and you can apply the standard technique about universal properties uniquely determining objects up to unique isomorphism.
As for (2), an $S^{-1}A$-module is an $A$-module where every element of $S$ acts invertibly. Once you know that $S$ acts invertibly on $S^{-1}M$, this is clear.