## Commutative Algebra 57

Continuing from the previous article, A denotes a noetherian ring and all A-modules are finitely generated. As before all completions are taken to be $\mathfrak a$-stable for a fixed ideal $\mathfrak a \subseteq A$.

# Noetherianness

We wish to prove that the $\mathfrak a$-adic completion of a noetherian ring is noetherian. First we express:

Lemma 1.

If $\mathfrak a = (a_1, \ldots, a_n)$, then

$\hat A \cong A[[X_1, \ldots, X_n]] / (X_1 - a_1, \ldots, X_n - a_n).$

Proof

Let $B = A[X_1, \ldots, X_n]$, still noetherian, with ideal $\mathfrak a' = (X_1 - a_1, \ldots, X_n - a_n)$. We have a ring isomorphism $f : B/\mathfrak a' \stackrel \cong\to A$ taking $X_i \mapsto a_i$. Let $\mathfrak b = (X_1, \ldots, X_n)$ be an ideal of B; we will take the $\mathfrak b$-adic completion on both sides of f, treated as B-modules.

• On the LHS, $\hat B = A[[X_1, \ldots, X_n]]$ and $\hat{\mathfrak a}'$ is generated by $X_1 - a_1, \ldots, X_n - a_n$ by proposition 5 here.
• On the RHS, we get the $f(\mathfrak{b})$-adic completion on A; but $f(\mathfrak b) = (a_1, \ldots, a_n)$, so we get the $\mathfrak a$-adic completion.

This completes our proof. ♦

Now all it remains is to prove this.

Proposition 1.

If A is a noetherian ring, so is $A[[X]]$.

Proof

For a formal power series $f\in A[[X]]$ where $f = b_n X^n + b_{n+1} X^{n+1} + \ldots$ with $b_n \ne 0$, we say the lowest coefficient of f is $b_n$ and its lowest term is $b_n X^n$. We also write $\deg f = n$.

Now suppose $\mathfrak b\subseteq A[[X]]$ is a non-zero ideal.

Step 1: find a finite set of generators of 𝔟.

Now for each n = 0, 1, …, let $\mathfrak a_n \subseteq A$ be the set of all $b\in A$ for which $b=0$ or $b X^n$ occurs as a lowest term of some $f \in \mathfrak b$. We get an ascending chain of ideals $\mathfrak a_0 \subseteq \mathfrak a_1 \subseteq \ldots$. Since A is noetherian, for some n we have $\mathfrak a_n = \mathfrak a_{n+1} = \ldots$.

For each of $0\le i \le n$, pick a finite generating set $S_i$ of $\mathfrak a_i$ comprising of non-zero elements; for each $b\in S_i$ pick $f \in \mathfrak b$ whose lowest term is $bX^i$. This gives a finite subset $T_i \subseteq \mathfrak b$ of degree-i power series whose lowest coefficients generate $\mathfrak a_i$. Note that if $\mathfrak a_i = 0$ then $T_i = \emptyset$.

Let $T := \cup_{i=0}^n T_i$.

Step 2: prove that T generates 𝔟.

Now suppose $f\in \mathfrak b$ has lowest term $bX^m$ so $b\in \mathfrak a_m$. By our choice of T we can find $g_1, \ldots, g_k \in T$ such that

$f - (a_1 X^{d_1}) g_1 -\ldots - (a_k X^{d_k})g_k = b' X^{m+1} + (\text{higher terms})$,

for some $a_i \in A$, $d_i = m - \deg g_i \ge 0$. Since T is finite, in fact we can assume $T = \{g_1, \ldots, g_k\}$, setting $a_i = 0$ for unneeded $g_i$. Repeating the process with the RHS polynomial, we obtain

$f - (a_1 X^{d_1} + b_1 X^{d_1 + 1}) g_1 - \ldots - (a_k X^{d_k} + b_k X^{d_k + 1}) g_k = b'' X^{m+2} + (\text{higher terms})$.

Repeating this inductively, we obtain formal power series $h_1, \ldots, h_k \in A[[X]]$ such that $f - h_1 g_1 - \ldots - h_k g_k = 0$. ♦

Hence, by proposition 1 and lemma 1 we have:

Corollary 1.

The $\mathfrak a$-adic completion of A is noetherian.

# Completion of Local Rings

Next suppose $(A,\mathfrak m)$ is a noetherian local ring.

Definition.

The completion of A is its $\mathfrak m$-adic completion.

Since $\hat A/\hat {\mathfrak m} \cong A/\mathfrak m =: k$ is a field, $\hat{\mathfrak m}$ is a maximal ideal of $\hat A$. Also we have:

Lemma 2.

$(\hat A, \hat {\mathfrak m})$ is a local ring.

Proof

By lemma 2 here, $\hat{\mathfrak m}$ is contained in the Jacobson radical of $\hat A$, so it is contained in all maximal ideals of $\hat A$. But $\hat{\mathfrak m}$ is already maximal. ♦

Hence, we see that the noetherian local ring $(\hat A, \hat {\mathfrak m})$ inherits many of the properties of $(A, \mathfrak m)$. E.g. they have the same Hilbert polynomial

$P(r) = \dim_{A/\mathfrak m} \mathfrak m^r/\mathfrak m^{r++1}$

since $\hat{\mathfrak m}^n / \hat{\mathfrak m}^{n+1} \cong \mathfrak m^n /\mathfrak m^{n+1}$ as k-vector spaces.

# Hensel’s Lemma

Here is another key aspect of complete local rings, which distinguishes them from normal local rings.

Proposition (Hensel’s Lemma).

Suppose $(A, \mathfrak m)$ is a complete local ring. Let $f(X) \in A[X]$ be a polynomial. If there exists $\alpha\in A$ such that $f(\alpha) \equiv 0 \pmod {\mathfrak m}$ and $f'(\alpha) \not\equiv 0 \pmod {\mathfrak m}$, then there is a unique $a\in A$ such that

$a\equiv \alpha \pmod {\mathfrak m}, \quad f(a) = 0$.

Note

Hence most of the time, if we can find a root $\alpha$ for $f(X) \in A[X]$ in the residue field $A/\mathfrak m$, then $\alpha$ lifts to a root $a\in A$.

There are more refined versions of Hensel’s lemma to consider the case where $f'(\alpha) \equiv 0 \pmod {\mathfrak m}$. One can even generalize it to multivariate polynomials. For our purpose, we will only consider the simplest case.

Proof

Fix $y\in A$ such that $f'(\alpha)y \equiv 1 \pmod {\mathfrak m}$.

Set $a_1 = \alpha$. It suffices to show: we can find $a_2, a_3, \ldots \in A$ such that

$i\ge 1 \implies a_{i+1} \equiv a_i \pmod {\mathfrak m^i}, \ f(a_i) \equiv 0 \pmod {\mathfrak m^i}$,

so that $\lim_{n\to\infty} a_n \in A$ gives us the desired element. We construct this sequence recursively; suppose we have $a_1, \ldots, a_n$ ($n\ge 1$). Write

$f(X) = c + d(X - a_n) + (X - a_n)^2 g(X),\quad g(X) \in A[X],\ c = f(a_n), \ d = f'(a_n)$.

Since $a_n \equiv \alpha \pmod {\mathfrak m}$ we have $f'(a_n) \equiv f'(\alpha) \pmod {\mathfrak m}$ so $f'(a_n)y \equiv 1 \pmod {\mathfrak m}$. Hence setting $a_{n+1} := a_n + x$ with $x = -f(a_n)y \in \mathfrak m^n$ gives

\begin{aligned} f(a_{n+1}) &\equiv \overbrace{f(a_n)}^c + \overbrace{f'(a_n)}^d x \pmod {\mathfrak m^{2n}} \\ &= f(a_n) - f'(a_n) f(a_n) y \\ &\equiv 0 \pmod {\mathfrak m^{n+1}}.\end{aligned},

which gives us the desired sequence. ♦

# Applications of Hensel’s Lemma

Now we can justify some of our earlier claims.

Examples

1. In the complete local ring $\mathbb C[[X, Y]]$ with maximal ideal $\mathfrak m = (X, Y)$, consider the equation $f(T) = T^2 - (1+X)$. Modulo $\mathfrak m$, we obtain $f(T) \equiv T^2 - 1$ which has two roots: +1 and -1. Since $f'(1) = 2 \ne 0$, by Hensel’s lemma there is a unique $g \in \mathbb C[[X, Y]]$ with constant term 1 such that $g^2 = 1+X$.

2. Similarly, consider the ring $\mathbb Z_p$ of p-adic integers with p > 2. Let $f(X) = X^2 - a$ for $a \in \mathbb Z_p$ outside $p\mathbb Z_p$. If a mod p has a square root b, then there is a Hensel lift of b to a square root of a in $\mathbb Z_p$.

3. Next, we will prove an earlier claim that the canonical map

$\mathbb C[[Y]] \longrightarrow \mathbb C[[X, Y]]/(Y^2 - X^3 + X)$

is an isomorphism. Consider the polynomial $f(X) = X^3 - X - Y^2$ as a polynomial in X with coefficients in $\mathbb C[[Y]]$. Modulo $\mathfrak m$, we have $f(X) \equiv X^3 - X$ which has roots -1, 0, +1. Hence

$X^3 - X - Y^2 = (X - \alpha_1) (X - \alpha_2)(X - \alpha_3)$ where $\alpha_i \in \mathbb C[[Y]]$

with $\alpha_1, \alpha_2, \alpha_3 \equiv -1, 0, +1 \pmod Y$ respectively. But $X-\alpha_1$ and $X-\alpha_3$ are invertible in $\mathbb C[[X, Y]]$ so

$\mathbb C[[X, Y]]/(Y^2 - X^3 +X) \cong \mathbb C[[X, Y]]/(X - \alpha_2) \cong \mathbb C[[Y]].$

4. In $\mathbb Z_p$, consider the equation $f(X) = X^{p-1} - 1$. Modulo p, this has exactly p – 1 roots; in fact any $a \in \mathbb F_p - \{0\}$ is a root of f. Now $f'(X) = (p-1)X^{p-2}$ so $f'(a) \ne 0$ in $\mathbb F_p$ for all $a \in \mathbb F_p - \{0\}$.

Hence by Hensel’s lemma, for each $1 \le a \le p-1$, there is a unique lift of a to an $\omega_a \in \mathbb Z_p$ which is a (p – 1)-th root of unity. We call $\omega_a$ the Teichmuller lift of $a\in \mathbb F_p - \{0\}$.

# Analysis in Completed Rings

Warning: the purpose of this section is to give the reader a flavour of the subject matter. It is not meant to be comprehensive. In particular, there are no proofs here.

Hensel’s lemma gives us an effective criterion to determine if a polynomial over a complete local ring have roots. Although its proof gives us a method to effectively compute these roots to arbitrary precision, there are other techniques we can borrow from real analysis.

Example 1: Binomial Expansion

In $\mathbb C[[X]]$, we can compute the square root of (1+X) with the binomial expansion:

\begin{aligned}(1+X)^{\frac 1 2} &= 1+ \tfrac 1 2 X + \tfrac{\frac 1 2 (\frac 1 2 - 1)}{2!} X^2 + \tfrac{\frac 1 2 (\frac 1 2 -1)(\frac 1 2 - 2)}{3!} X^3 + \ldots\\ &= 1 + \tfrac 1 2 X - \tfrac 1 8 X^2 + \tfrac 1 {16} X^3 + \ldots \in \mathbb C[[X]].\end{aligned}

By the same token, we can compute square roots in $\mathbb Z_p$ by taking binomial expansion of $(1+\alpha)^n$, as long as the convergence is “fast enough”. For example, to compute $\sqrt 2 \in \mathbb Z_7$, binomial expansion gives

$2\sqrt 2 = \sqrt 8 = (1 + 7)^{\frac 1 2} = 1 + \frac 1 2 (7) + \frac{\frac 1 2 (\frac 1 2 - 1)}{2!} (7^2) + \frac{\frac 1 2 (\frac 1 2 -1)(\frac 1 2 - 2)}{3!} (7^3) + \ldots \in \mathbb Z_7.$

Taking the first four terms we have $2\sqrt 2 \equiv 470 \pmod {7^4}$ so that $\sqrt 2 \equiv 235 \pmod {7^4}$. Indeed, we can easily check that $m = 235$ is a solution to $m^2 \equiv 2 \pmod {7^4}$.

Example 2: Fixed-Point Method

While solving equations of the form $x = f(x)$ in analysis, it is sometimes effective to start with a good estimate $x_0$ then iteratively compute $x_{n+1} = f(x_n)$. We can do this in complete local rings too.

For example, let us solve $X^3 - X = Y^2$ as a polynomial in X with coefficients in $\mathbb C[[Y]]$. We saw above there is a unique root $x \equiv 0 \pmod Y$. Start with $x_0 = 0$ then iteratively compute $x_{n+1} = x_n^3 - Y^2$. This gives

\begin{aligned} x_0 &= 0, \\ x_1 &= -Y^2,\\ x_2 &= -Y^6 - Y^2, \\ x_3 &= -Y^{18} - 3Y^{14} - 3Y^{10} - Y^6 - Y^2,\end{aligned}

where $x_3$ is accurate up to $Y^{13}$.

Example 3: Newton Method

To solve an equation of the form $f(x) = 0$, one starts with a good estimate $x_0$ then iterate $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.

Exercise A

Use the Newton root-finding method to obtain $\sqrt 2 \in \mathbb Z_7$ to high precision (in Python).

### Completion in Geometry

As another application of completion, consider $A = \mathbb C[X, Y]/(Y^2 - X^3 - X^2)$ with $\mathfrak m = (X, Y)$. Taking the $\mathfrak m$-adic completion, we obtain

$\hat A = \mathbb C[[X, Y]]/(Y^2 - X^3 - X^2)$

by proposition 5 here. Note that since $1+X$ has a square root in $\mathbb C[[X, Y]]$, the ring $\hat A$ is no longer an integral domain, but a “union of two lines” since $Y^2 - X^3 - X^2 = (Y - \alpha X)(Y + \alpha X)$ where $\alpha = \sqrt{1+X} \in \mathbb C[[X, Y]]$ is a unit.

This reflects the geometrical fact that when we magnify at the origin, we obtain a union of two lines.

[ Image edited from GeoGebra plot. ]

Exercise B

Let $(A,\mathfrak m)$ be a local ring. Prove that if the $\mathfrak m$-adic completion of A is an integral domain, then so is A.

[ Hint: use a one-line proof. ]

Posted in Advanced Algebra | | 1 Comment

## Commutative Algebra 56

Throughout this article, A denotes a noetherian ring and $\mathfrak a \subseteq A$ is a fixed ideal. All A-modules are finitely generated.

# Consequences of Artin-Rees Lemma

Suppose we have an exact sequence of finitely generated A-modules

$0 \longrightarrow N \longrightarrow M \longrightarrow P \longrightarrow 0.$

Let M be given the 𝔞-adic filtration; the induced filtration on P is 𝔞-adic so its completion is the 𝔞-adic completion. By the Artin-Rees lemma, the induced filtration on N is 𝔞-stable and by proposition 2 here its completion is also the 𝔞-adic completion. Hence we have shown:

Proposition 1.

The following functor is exact:

From general properties of exact functors, this has the following properties.

1. If $N \subseteq M$ is a submodule, then $\hat N$ can be identified as a submodule of $\hat M$.

2. If $N_1, N_2 \subseteq M$ are submodules, then

$(N_1 \cap N_2)\hat{} \cong \hat N_1 \cap \hat N_2, \quad (N_1 + N_2)\hat{} \cong \hat N_1 + \hat N_2$.

3. If $f:N\to M$ is a map of A-modules, then $\hat f : \hat N \to \hat M$ satisfies

$\mathrm{ker } \hat f = (\mathrm{ker } f)\hat{}, \quad \mathrm{im } \hat f = (\mathrm{im } f)\hat{}$.

In particular, for a fixed $m\in M$, take the A-linear map $f : A\to M, 1 \mapsto m$. Taking the $\mathfrak a$-adic completion gives $\hat f : \hat A \to \hat M, 1 \mapsto i(m)$ as well, where $i : M\to \hat M$ is the canonical map. Hence

$\hat A \cdot i(m) = \mathrm{im } \hat f = (\mathrm{im } f)\hat{} = (Am)\hat{}.$

From property 2, we obtain, for $m_1, \ldots, m_n \in M$,

$\hat A \cdot i(m_1) + \ldots + \hat A \cdot i(m_n) = (Am_1 + \ldots + Am_n)\hat{}.$

Thus we have shown:

Proposition 2.

Identifying M with its image in $\hat M$,

$\hat A \cdot M = \hat M.$

In particular, if M is finitely generated, so is $\hat M$.

We also have:

Corollary 1.

For any ideal $\mathfrak b\subseteq A$ and A-module M

$(\mathfrak b M)\hat{} = \hat{\mathfrak b} \hat M$.

Proof

By proposition 2, $\hat {\mathfrak b}\hat M = (\hat A \mathfrak b)(\hat A M) = \hat A(\mathfrak b M) = (\mathfrak b M)\hat{}$. ♦

# Krull’s Intersection Theorem

Another interesting consequence of the Artin-Rees lemma is as follows.

Krull’s Intersection Theorem.

Suppose $(A,\mathfrak m)$ is local and noetherian. If M is a finitely generated A-module, then $\cap_n \mathfrak m^n M = 0$.

In particular, the canonical map $M \to \hat M$ is injective where $\hat M$ is the $\mathfrak m$-adic completion of M.

Proof

Let $N = \cap_n \mathfrak m^n M$, a submodule of M. By the Artin-Rees lemma, the $\mathfrak m$-adic filtration on M induces a $\mathfrak m$-stable filtration on N so for some n,

$\mathfrak m (N \cap \mathfrak m^n M) = N \cap \mathfrak m^{n+1} M \implies \mathfrak m N = N \implies N = 0$

by Nakayama’s lemma. ♦

In particular, $A \to \hat A$ is an injective ring homorphism when we take the $\mathfrak m$-adic completion of a local ring $(A, \mathfrak m)$.

We give an example where Krull’s intersection theorem fails when A is not noetherian. Take the set of all infinitely differentiable functions $f : I\to \mathbb R$, where $I$ is an open interval containing 0; let A be the set of equivalence classes under the relation: $f : I \to \mathbb R$ and $g : I' \to \mathbb R$ are equivalent if $f|_J = g|_J$ for some $J\subseteq I\cap I'$ containing 0.

Now A is a ring with addition and product given by pointwise addition and product. Its unique maximal ideal is $\mathfrak m = \{f \in A : f(0) = 0\}$. Then $\cap_n \mathfrak m^n \ne 0$ since it contains $\exp(-\frac 1 {x^2})$.

Exercise A

1. Find a noetherian ring A and a proper ideal $\mathfrak a \subsetneq A$ such that $\cap_n \mathfrak a^n \ne 0$.

2. Prove that if A is a noetherian integral domain, then any proper ideal $\mathfrak a\subsetneq A$ satisfies $\cap_n \mathfrak a^n = 0$. [ Hint: follow the proof of Krull’s Intersection Theorem; use the “adjugate matrix” trick. ]

# Tensoring with Â

Proposition 3.

For any finitely generated M, we have a natural isomorphism $\hat A \otimes_A M \cong \hat M$.

Note

In short, for finitely generated module M, taking its completion is the same as taking the induced Â-module of M.

Proof

Since $\hat M$ is an $\hat A$-module with a canonical A-linear $M\to \hat M$, by universal property of induced modules we have a map $\hat A \otimes_A M \to \hat M$ which is natural in M. And since M is a noetherian module, it is finitely presented so we can find an exact sequence of the form $A^m \to A^n \to M \to 0$. This gives a commutative diagram of maps:

where the top row is exact because tensor product is right-exact and the bottom row is exact from proposition 1. Since the first two vertical maps are isomorphisms, so is the third one. ♦

Hence the functor $\hat A \otimes_A -$ is exact when restricted to the category of finitely generated A-modules. To see that $\hat A$ is A-flat, we apply:

Lemma 1.

Let A be any ring (possibly non-noetherian) and M be an A-module.

M is A-flat if and only if for any injective map of finitely generated A-modules $N_1 \to N_2$, the resulting $N_1 \otimes_A M \to N_2 \otimes_A M$ is also injective.

Proof

(⇒) Obvious. (⇐) Let $P\subseteq Q$ be a submodule of any module. We need to show that $P\otimes_A M\to Q\otimes_A M$ is injective. Let $\Sigma$ be the set of all pairs $(N_1, N_2)$ where $N_2 \subseteq Q$ and $N_1 \subseteq P\cap N_2$ are finitely generated A-submodules, ordered by inclusion (in both terms). Clearly $\Sigma$ is a directed set; since $N_1$ runs through all finitely generated submodules of P, we have direct limits

$\varinjlim_{(N_1, N_2) \in \Sigma} N_1 \cong P, \quad \varinjlim_{(N_1, N_2)\in \Sigma} N_2 \cong Q.$

By the given condition, $N_1\otimes_A M \to N_2 \otimes_A M$ is injective for each $(N_1, N_2) \in \Sigma$. By proposition 3 here, taking the direct limit gives an injective

$\varinjlim_{(N_1, N_2)} (N_1 \otimes_A M) \to \varinjlim_{(N_1, N_2)} (N_2 \otimes_A M).$

By exercise B.4 here, the LHS is isomorphic to $(\varinjlim_{(N_1, N_2)} N_1) \otimes_A M \cong P\otimes_A M.$ Likewise the RHS is isomorphic to $Q\otimes_A M$ so $P\otimes_A M \to Q\otimes_A M$ is injective. ♦

Corollary 2.

$\hat A$ is a flat A-algebra.

Exercise B

Prove that in lemma 1, we can weaken the flatness condition to:

1. For each ideal $\mathfrak a\subseteq A$, $\mathfrak a\otimes_A M \to A\otimes_A M \cong M$ is injective.
2. For each finitely generated ideal $\mathfrak a\subseteq A$, $\mathfrak a\otimes_A M \to A\otimes_A M \cong M$ is injective.

# Completion and Quotients

Recall that for any submodule $N\subseteq M$ we have $(M/N)\hat{} \cong \hat M / \hat N$. In particular if $\mathfrak b\subseteq A$ is an ideal then

$\hat A / \hat {\mathfrak b} \cong (A/\mathfrak b)\hat{}$ as $\hat A$-modules.

But $(A/\mathfrak b)\hat{}$ also has a ring structure! Indeed by definition it is the completion obtained from the $\mathfrak a$-adic filtration as an A-module

$A/\mathfrak b = A_0' \supseteq A_1' \supseteq \ldots$, where $A'_n := (\mathfrak a^n + \mathfrak b)/\mathfrak b$

which is also a filtration of $A/\mathfrak b$ as a ring since $A_i' A_j' \subseteq A_{i+j}'$. The construction which gives us $(A/\mathfrak b)\hat{} = \varprojlim [(A/\mathfrak b)/A_n']$ as A-modules also gives us the inverse limit as rings. One easily verifies that $\hat A \to (A/\mathfrak b)\hat{}$ is a ring homomorphism so:

Proposition 4.

We have an isomorphism of rings

$\hat A / \hat {\mathfrak b} \cong (A/\mathfrak b)\hat{}$,

where $(A/\mathfrak b)\hat{}$ is its $(\mathfrak a + \mathfrak b)/\mathfrak b$-adic completion as a ring.

Furthermore, by proposition 2, if $\mathfrak b$ is generated (as an ideal) by $a_1, \ldots, a_n$, then $\hat{\mathfrak b}$ is generated by the images of $a_i$ in $\hat A$. Thus we have shown:

Proposition 5.

Suppose $\mathfrak b \subseteq A$ is an ideal generated by $a_1, \ldots, a_n$. Then the completion of $A/\mathfrak b$ is the quotient of $\hat A$ by the ideal generated by (the images of) $a_1, \ldots, a_n$.

Example

Take the example $A = \mathbb C[X, Y]/(Y^2 - X^3 + X)$ with $\mathfrak m = (X, Y)$ from an earlier example; we wish to compute the $\mathfrak m$-adic completion Â of A. By the proposition, Â is the quotient of $\mathbb C[X, Y]^\wedge$ (the $(X, Y)$-adic completion) by $(Y^2 - X^3 + X)$. But we clearly have $\mathbb C[X, Y]^\wedge \cong \mathbb C[[X, Y]]$ so

$\hat A \cong \mathbb C[[X, Y]]/(Y^2 - X^3 + X)$

as we had claimed. In the next article, we will show that this ring is isomorphic to $\mathbb C[[Y]]$.

## Completion of Completion

As a special case, we have

$\hat A / (\hat {\mathfrak a})^n = \hat A / (\mathfrak a^n)\hat{} \cong A / \mathfrak a^n$,

where the equality is from corollary 1 and the isomorphism from lemma 1 here.

Hence, the $\hat a$-adic completion of $\hat A$ is isomorphic to $\hat A$. We also have the following.

Lemma 2.

For each $x \in \hat{\mathfrak a}$, $1-x$ is invertible in $\hat A$.

In particular, (by proposition 4 here) $\hat{\mathfrak a}$ is contained in the Jacobson radical of $\hat A$.

Proof

Since $x^n \in \hat{\mathfrak a}^n$, we can take the infinite sum

$y = 1 + x + x^2 + \ldots \in \hat A$.

Then $(1-x)y \in \cap_n \hat{\mathfrak a}^n$ so $(1-x)y = 0$ in $\hat A$. ♦

# Exactness of Completion

Proposition 1.

Let $0 \to N \to M \to P \to 0$ be a short exact sequence of A-modules. Suppose M is filtered, inducing filtrations on N and P. Then

$0 \longrightarrow \hat N \longrightarrow\hat M \longrightarrow \hat P \longrightarrow 0$

is also exact as $\hat A$-modules.

Proof

Without loss of generality, assume N is a submodule of M and PM/N. Each term in the filtration gives a short exact sequence

$0 \longrightarrow \overbrace{N/(M_i \cap N)}^{N/N_i} \longrightarrow M/M_i \longrightarrow \overbrace{M/(M_i + N)}^{P/P_i} \longrightarrow 0$

since $N/(M_i \cap N) \cong (M_i + N)/M_i$ by the second isomorphism theorem. By proposition 1 here, taking (inverse) limit is left-exact so we obtain an exact sequence

$0\longrightarrow \hat N \longrightarrow \hat M \longrightarrow \hat P$.

To show that $\hat M \to \hat P$ is surjective, we pick an element of $\hat P$. Since $P/P_k \cong M/(M_k + N)$, the element is represented by a sequence $(m_k)$ in M such that $m_{k+1} - m_k \in M_k + N$. We need to show there is a sequence $(x_k)$ in M such that

$k\ge 0 \implies x_{k+1} - x_k \in M_k, x_k - m_k \in M_k + N.$

When k = 0, just pick any $x_0$. Suppose we have $x_0, \ldots, x_k$; we need $x_{k+1} \in M$ such that $x_{k+1} - x_k \in M_k$ and $x_{k+1} - m_{k+1} \in M_{k+1} + N$. But observe that $m_{k+1} - x_k = (m_{k+1} - m_k) + (m_k - x_k) \in M_k + N$. If we write $m_{k+1} - x_k = m + n$ for $m\in M_k, n\in N$, then $x_{k+1} := m_{k+1} - n$ works. ♦

# Completion of Completion

Lemma 1.

We have $\hat M /\hat M_n \cong M/M_n$, where $M_n$ has the filtration induced from M.

Proof

Let $P = M/M_n$. From proposition 1 we get an exact sequence

$0 \to \hat M_n \to \hat M \to \hat P \to 0.$

But we also have $P/P_m = M/(M_m + M_n)$ which is $M/M_n$ for all $m\ge n$. Thus $\hat P = M/M_n$ and we are done. ♦

Hence if we let $\hat M$ take the filtration given by

$\hat M = \hat M_0 \supseteq \hat M_1 \supseteq \ldots$

then by lemma 1, the completion of $\hat M$ with respect to this filtration is still $\hat M$.

If $m_1, m_2, \ldots \in \hat M$ is a Cauchy sequence, from the previous article we have its limit

$(\lim_{n\to \infty} m_n) \in \hat{\hat M} = \hat M$

Since the map from $\hat M$ to its completion is injective, we have $\cap_n \hat M_n = 0$ so as shown in exercise A.3 here, we can define an (ultra)metric on $\hat M$ such that the resulting topology has a basis comprising of the set of all cosets $\{m + \hat M_n\}$. From the above, every Cauchy sequence converges in $\hat M$. Thus:

Summary.

$\hat M$ is a complete metric space.

Furthermore, the image of $M \to \hat M$ is dense; indeed any basic open subset of $\hat M$ is of the form $m + \hat M_n$ for $m\in \hat M$ and $n\ge 0$. Since $\hat M / \hat M_n\cong M/M_n$, we see that m can be represented by an element of M. Thus any non-empty open subset of $\hat M$ contains an element of M.

Thus $\hat M$ is the completion of M even in the topological sense.

Note

For visualization, one can show that $\mathbb Z_2$ is homeomorphic to the Cantor set:

E.g. the point above corresponds to a 2-adic integer ending at $(\ldots 0010)_2$.

Now instead of arbitrary filtrations on M, we will focus our attention to the 𝔞-adic filtrations on A and M for a fixed ideal $\mathfrak a$:

$M_n = \mathfrak a^n M \implies \hat M = \varprojlim M/\mathfrak a^n M.$

Clearly if M is given the 𝔞-adic filtration, so is any quotient, because $\mathfrak a^n(M/N) = (\mathfrak a^n M + N)/N$, so the induced filtration on M/N is also 𝔞-adic. On the other hand, the induced filtration on a submodule N is $\mathfrak a^n M \cap N\ne \mathfrak a^n N$.

But the situation is salvageable when A is noetherian. Instead of the 𝔞-adic filtration, let us loosen our definition a little.

Definition.

A filtration $(M_n)$ of M is said to be 𝔞-stable if for some n, we have $M_{n+k} = \mathfrak a^k M_n$ for all $k\ge 0$.

In other words, an 𝔞-stable filtration is “eventually 𝔞-adic”. When we take the completion, we get the same thing.

Proposition 2.

Suppose M is an A-module with an 𝔞-stable filtration. Its completion is canonically isomorphic to the 𝔞-adic completion of M.

Proof

Since $(M_n)$ is a filtration for M we have $A_i M_j \subseteq M_{i+j}$, i.e. $\mathfrak a^i M_j \subseteq M_{i+j}$. Now fix an n such that $M_{n+k} = \mathfrak a^k M_n$ for all $k\ge 0$. We get

$k\ge 0 \implies M_k \supseteq \mathfrak a^k M \supseteq \mathfrak a^k M_n = M_{n+k} \supseteq \mathfrak a^{n+k}M$

and hence maps $M/\mathfrak a^{n+k}\to M/M_{n+k} \to M/\mathfrak a^k M \to M/M_k$. Taking the inverse limit:

$\varprojlim_k M/\mathfrak a^{n+k}M \to \varprojlim_k M/M_{n+k} \to \varprojlim M/\mathfrak a^k M \to \varprojlim M/M_k$.

By explicitly writing out elements of inverse limits, we see that the above give isomorphisms $\varprojlim_k M/M_{n+k} \cong \varprojlim M/M_k$ and $\varprojlim_k M/\mathfrak a^{n+k} \cong \varprojlim M/\mathfrak a^k$; thus

$\hat M \cong \varprojlim M/\mathfrak a^k M$. ♦

Exercise A

1. Fill in the last step of the proof.

2. Show that in any category, the inverse limit of the diagram

remains the same when we drop finitely many terms on the right.

# Artin-Rees Lemma

The main result we wish to prove is the following.

Artin-Rees Lemma.

Let A be a noetherian ring with the $\mathfrak a$-adic filtration, and N a submodule of a finitely generated A-module M. If M has an $\mathfrak a$-stable filtration, the induced filtration on N is also $\mathfrak a$-stable.

Proof

Step 1: define blowup algebra and module.

Definition.

Given any filtered module M over a filtered ring A, the blowup algebra and blowup module are defined by

$B(A) := A_0 \oplus A_1 \oplus \ldots, \quad B(M) := M_0 \oplus M_1 \oplus \ldots.$

We define a product operation $A_i \times A_j \to A_{i+j}$ from multiplication in A. Hence, B(A) has a canonical structure of a graded ring.

Similarly, since M is a filtered module, we obtain a product operation $A_i \times M_j \to M_{i+j}$ which gives B(M) a structure of a graded B(A)-module. When A and M are given the 𝔞-adic filtration, we write $B_{\mathfrak a}(A)$ and $B_{\mathfrak a}(M)$ for their blowup algebra and module.

Step 2: if A is a noetherian ring, so is B𝔞(A).

Since A is noetherian, we can write $\mathfrak a = x_1 A + \ldots + x_k A$ for some $x_1, \ldots, x_k \in \mathfrak a$. It follows that $\mathfrak a^n$ is a sum of $x_1^{d_1}\ldots x_k^{d_k} A$ where $\sum_{i=1}^k d_i = n$. Hence the map

$A[X_1, \ldots, X_k] \longrightarrow B_{\mathfrak a}(A), \quad X_i \mapsto (x_i \in A_1)$

is a surjective ring homomorphism so $B_{\mathfrak a}(A)$ is also noetherian.

Now we suppose A is noetherian and is given the 𝔞-adic filtration. Let M be a finitely generated filtered A-module.

Step 3: B(M) is finitely generated if and only if the filtration on M is 𝔞-stable.

(⇐) For some n we have $B(M) = M_0 \oplus M_1 \oplus \ldots \oplus M_n \oplus \mathfrak a M_n \oplus \mathfrak a^2 M_n \oplus \ldots.$ Since M is a noetherian A-module, each $M_i$ ($0\le i \le n$) is finitely generated as an A-module by, say $m_{i1}, \ldots, m_{iN}$. Now we take the set of $m_{ij}$, as homogeneous elements of B(M) of degree i.

In the above, each homogeneous element of $M_0, \ldots, M_n$ is an A-linear combination of these generators. Furthermore, $M_{n+k} = \mathfrak a^k M_n = A_k M_n$ so $m_{n1}, \ldots, m_{nN} \in B(M)_n$ generate (over $B_{\mathfrak a}(A)$) the homogeneous elements in B(M) of degree n and higher.

(⇒) Suppose $B(M)$ is finitely generated over $B_{\mathfrak a}(A)$ by homogeneous elements $x_1, \ldots, x_k$; let $d_i = \deg x_i$ and $N = \max d_i$. We claim that $M_{n+1} = \mathfrak a M_n$ for all $n\ge N$. Since M is filtered, we have $\mathfrak a M_n \subseteq M_{n+1}$

Conversely take $y\in M_{n+1}$, regard it as an element of $B(M)_{n+1}$ and write $y = a_1 x_1 + \ldots +a_k x_k$ with $a_i \in B_{\mathfrak a}(A)$. Since y and $x_i$ are homogeneous, we may assume $a_i$ is homogeneous of degree $e_i := n+1 - d_i > 0$. So $a_i \in B_{\mathfrak a}(A)_{e_i} = \mathfrak a^{e_i}$. Write

$a_i = b_{i1} c_{i1} + b_{i2} c_{i2} + \ldots + b_{ik} c_{ik}, \quad b_{ij} \in \mathfrak a, c_{ij} \in \mathfrak a^{e_i-1} \subseteq A.$

Now y is a sum of $b_{ij}c_{ij} x_i$, with $c_{ij} x_i \in M_n$ so $y \in \mathfrak a M_n$.

Step 4: prove the Artin-Rees lemma.

By step 2, $B_{\mathfrak a}(A)$ is a noetherian ring; since M has an $\mathfrak a$-stable filtration, by step 3 B(M) is a noetherian $B_{\mathfrak a}(A)$-module. And since $B(N) \subseteq B(M)$ is a $B_{\mathfrak a}(A)$-submodule it is also noetherian. By step 3 again, this says the induced filtration on N is $\mathfrak a$-stable. ♦

# Filtered Rings

Definition.

Let A be a ring. A filtration on A is a sequence of additive subgroups

$A = A_0 \supseteq A_1 \supseteq A_2 \supseteq \ldots$

such that $A_i A_j \subseteq A_{i+j}$ for any $i, j\ge 0$. A filtered ring is a ring with a designated filtration.

Note

Since $A\cdot A_i = A_0 \cdot A_i \subseteq A_i$, in fact each $A_i$ is an ideal of A.

Examples

1. If $A = \oplus_{i=0}^\infty A_i$ is a grading, we can form a filtration by taking $n \mapsto A_n \oplus A_{n+1} \oplus \ldots$ (where $A_n, A_{n+1}, \ldots$ refers to the grading).

2. Let $\mathfrak a\subseteq A$ be an ideal. The $\mathfrak a$adic filtration is given by $A_i = \mathfrak a^i$. E.g. we can take $A = \mathbb Z$ and $\mathfrak a = 2\mathbb Z$, then $A_i$ is the set of integers divisible by $2^i$. More generally, in a dvr with uniformizer $\pi$, we can take $A_i = (\pi^i)$.

Definition.

Suppose A is a filtered ring. A filtration on an A-module M is a sequence of additive subgroups

$M = M_0 \supseteq M_1 \supseteq M_2 \supseteq \ldots$

such that $A_i M_j \subseteq M_{i+j}$ for any $i, j\ge 0$. A filtered module is a module with a designated filtration.

Note

Since $A\cdot M_i = A_0 M_i \subseteq M_i$, each $M_i$ is an A-submodule of M.

Also, we need a fixed filtration on the base ring A before we can talk about filtrations on A-modules.

Example

Again, for an ideal $\mathfrak a \subseteq A$, we obtain the $\mathfrak a$adic filtration of M, where $M_i = \mathfrak a^i M$.

# Induced Filtrations

Definition.

Let M, N be filtered A-modules. A linear map $f:M\to N$ is said to be filtered if $f(M_i) \subseteq N_i$ for each i.

We also have:

Definition.

Let $f:M\to N$ be a linear map of A-modules.

• If $(M_i)$ is a filtration of M, the induced filtration on N via f is given by $N_i = f(M_i)$.
• If $(N_i)$ is a filtration of N, the induced filtration on M via f is given by $M_i = f^{-1}(N_i)$.

Note

Let us show that the induced filtrations are legitimate. In the first case,

$A_i N_j = A_i f(M_j) = f(A_i M_j) \subseteq f(M_{i+j}) = N_{i+j}$.

And in the second,

$f(A_i M_j) = A_i\cdot f(M_j) \subseteq A_i N_j \subseteq N_{i+j} \implies A_i M_j \subseteq M_{i+j}.$

In particular, if M is a filtered A-module and $N\subseteq M$ is a submodule, the induced filtrations on N and M/N are given by:

$N_i = M_i \cap N, \quad (M/N)_i = (M_i + N)/N.$

So far everything is natural, but beneath all this a danger lurks.

If $f:M\to N$ is a filtered A-linear map, then $M/\mathrm{ker} f$ and $\mathrm{im }f$ have induced filtrations via quotient module of M and submodule of N. Although $M/\mathrm{ker } f \cong \mathrm{im }f$ as A-modules, it is not an isomorphism in the category of filtered A-modules!

Let us write everything out explicitly. The filtration on the LHS and RHS are given respectively by

$(M / \mathrm{ker } f)_i = (M_i + \mathrm{ker } f)/\mathrm{ker } f, \quad (\mathrm{im } f)_i = (\mathrm{im} f) \cap N_i$

which gives

$(M_i + \mathrm{ker } f)/\mathrm{ker } f \cong M_i / (M_i \cap \mathrm{ker } f) = M_i / \mathrm{ker} (f|_{M_i}) \cong \mathrm{im} (f|_{M_i})$

which is not isomorphic to $(\mathrm{im} f) \cap N_i$ in general.

# Completion

Definition.

Let M be a filtered A-module, the completion of M is the following (inverse) limit of A-modules:

$\hat M := \varprojlim M/M_n.$

By the description of inverse limits in the category of A-modules, $\hat M$ comprises of the set of all $(\ldots, x_n, \ldots, x_2, x_1) \in \prod_{i=1}^\infty (M/M_i)$ such that for each i, $x_{i+1}$ maps to $x_i$ under the canonical map $M/M_{i+1}\to M/M_i$.

The canonical maps $M\to M/M_n$ induce an A-linear map $M \to \hat M$ by the universal property of inverse limits. This can be described as follows: for each $m\in M$, let $m_i$ be its image in $M/M_i$; then the map takes $(m \in M) \mapsto (\ldots, m_n, \ldots, m_2, m_1) \in \hat M$. From this we see that:

Lemma 1.

The map $i:M\to \hat M$ is injective if and only if $\cap_n M_n = 0$, in which case we say the filtration is Hausdorff.

Next, by setting MA, we also have $\hat A = \varprojlim A/A_n$. Although we defined $\hat A$ as an A-module, one sees by the explicit construction that $\hat A$ has a ring structure. To be specific,

$(a_n)_{n=1}^\infty, (b_n)_{n=1}^\infty \in \hat A \implies (a_n) \times (b_n) := (a_n b_n)_{n=1}^\infty.$

Exercise A

1. Prove that $\hat M$ has a canonical structure as an $\hat A$-module.

2. Prove that $\hat A$ is the inverse limit of $A/A_n$ in the category of rings. In particular, the canonical map $i : A\to \hat A$ is a ring homomorphism.

3. Prove that if the filtration on M is Hausdorff, we can define a metric on M via:

$d(x, y) := 2^{-|x-y|}$, where $|z| := \sup\{ n : z\in M_n\},$

such that the collection of all cosets $\{ m + M_n : m\in M, n \ge 0\}$ forms a basis for the resulting topology. In fact, d is an ultrametric.

# Limits in Completion

The completion $\hat M$ enables us to take limits of infinite sequences and sums of infinite series in modules.

Definition.

Let $m_1, m_2, \ldots \in M$ be a sequence in a filtered module M. We say the sequence is Cauchy if: for any i, there exists n such that

$m_n \equiv m_{n+1} \equiv m_{n+2} \equiv \ldots \pmod {M_i}$.

The astute reader would note that if the filtration is Hausdorff, then $(m_n)$ is Cauchy here if and only if it is Cauchy with respect to the metric of exercise A.3.

Definition.

The limit of a Cauchy sequence $m_1, m_2, \ldots \in M$ is defined as follows. For each i, pick n such that $m_n \equiv m_{n+1} \equiv \ldots$ modulo $M_i$, with image $y_i \in M/M_i$. Now define:

$\lim_{n\to\infty} m_n := (\ldots, y_n, \ldots, y_2, y_1) \in \hat M$.

From the definition, it is clear that if $(m_n), (m_n')$ (resp. $(a_n)$) are Cauchy sequences in M (resp. in A), then $(m_n + m_n')$ and $(a_n m_n)$ are also Cauchy and we have

\begin{aligned}\lim_{n\to\infty} (m_n + m_n') &= (\lim_{n\to\infty} m_n) + (\lim_{n\to\infty} m_n') \in \hat M, \\ \lim_{n\to\infty} (a_n m_n) &= (\lim_{n\to\infty} a_n) (\lim_{n\to\infty} m_n) \in \hat M.\end{aligned}

Pick a ring A with maximal ideal $\mathfrak m$; we will take the $\mathfrak m$-adic filtration $A_n = \mathfrak m^n$. Given a sequence $x_i \in A_i$ for $i=0, 1, \ldots$, let

$y_i := x_0 + \ldots + x_{i-1} \pmod {\mathfrak m^i}$, an element of $A/A_i$.

Then $(y_i)$ is a Cauchy sequence in A and we write:

$\sum_{i=0}^\infty x_i := \lim_{n\to\infty} y_n \in \hat A$.

Arithmetic Example.

Let $A = \mathbb Z$ and $A_n = p^n \mathbb Z$ for a prime p. The completion $\hat A$ is called the ring of p-adic integers and denoted by $\mathbb Z_p$. Let us take p = 2 and the element:

$y = 1 + 2^2 + 2^4 + 2^6 + \ldots \in \mathbb Z_2.$

Then $3y = 3 + 3(2^2) + 3(2^4) + \ldots$ is congruent to -1 modulo any $2^n$. Thus $y = -\frac 1 3$. In general, an element of $\mathbb Z_p$ can be regarded as having an infinite base-p expansion. Thus the above $y \in \mathbb Z_2$ would have base-2 expansion $(\ldots 1010101)_2$. One easily checks that three times this value is $(\ldots 1111)_2$, which is -1.

Geometric Example

Let A be a ring, $B = A[X, Y]$ and $B_n = \mathfrak m^n$ where $\mathfrak m = (X, Y)$. Again, we can take the infinite sum $\alpha = 1 + XY + (XY)^2 + \ldots$ and check that $\alpha(1-XY) = 1$. Note that $\hat B \cong A[[X,Y]]$, the ring of formal power series with coefficients in A.

Definition.

Let A be any ring. A formal power series in X with coefficients in A is an expression

$f(X) = a_0 + a_1 X + a_2 X^2 + \ldots$, where $a_i \in A$.

Unlike polynomials, we allow infinitely many $a_i$ to be non-zero. Addition and multiplication of formal power series are defined as follows. For $f(X) = \sum_{i=0}^\infty a_n X^n$ and $g(X) = \sum_{j=0}^\infty b_n X^m$,

\begin{aligned} f(X) + g(X) &=(a_0 + b_0) + (a_1 + b_1)X + (a_2 + b_2)X^2 + \ldots \\ f(X)\times g(X) &= (a_0 b_0) + (a_0 b_1 + a_1 b_0)X + (a_0 b_2 + a_1 b_1 + a_2 b_0)X^2 + \ldots \end{aligned}

This gives a ring structure on the set $A[[X]]$ of formal power series. To define rings of formal power series in multiple variables, we set recursively

$A[[X_1, \ldots, X_n]] := (A[[X_1, \ldots, X_{n-1}]])[[X_n]]].$

As another example, let us take the $\mathfrak m$-adic completion for $A = \mathbb C[X, Y]/(Y^2 - X^3 + X)$ and $\mathfrak m = (X, Y)$. We will prove later that

$\hat A \cong \mathbb C[[X, Y]]/(Y^2 - X^3 + X)$

and that the map $\mathbb C[[Y]] \to \hat A$ is an isomorphism. Geometrically, this means when we project $E: Y^2 = X^3 - X$ to the Y-axis, the map is locally invertible at the origin.

[ Image edited from GeoGebra plot. ]

Functorially, the ring $A[[X, Y]]$ behaves quite differently from $A[X, Y]$, because as an A-module, it is a countably infinite direct product of copies of A, unlike $A[X, Y]$ which is a direct sum. If we follow the earlier guideline, it is (for example) generally false that $B \otimes_A A[[X, Y]] \cong B[[X, Y]]$ for any A-algebra B.

Exercise B

1. Prove that the canonical map $\mathbb C[[X]] \to \mathbb C[[X, Y]]/(Y^2 - X^3 + X)$ is not an isomorphism.

2. Let A be a filtered ring. Prove that if $f(X) \in A[[X]]$ is $a_0 + a_1 X + a_2 X^2 + \ldots$, then f defines a map

$A_1 \to \hat A, \quad f(\alpha) := \sum_{n=0}^\infty a_n \alpha^n \in \hat A.$

Prove that if we fix $\alpha \in A_1$, we get a ring homomorphism $A[[X]] \to \hat A$, $f\mapsto f(\alpha)$.

## Commutative Algebra 53

Definition.

A grading on a ring A is a collection of additive subgroups $A_0, A_1, \ldots \subseteq A$ such that

$A = A_0 \oplus A_1 \oplus A_2 \oplus \ldots$

as abelian groups, and $A_i A_j \subseteq A_{i+j}$ for any $i, j\ge 0$, i.e..

$a\in A_i, b\in A_j \implies ab \in A_{i+j}.$

Note

The notation $A = \oplus_i A_i$ means every $a\in A$ can be uniquely written as a finite sum $a_0 + a_1 + \ldots + a_n$ for some $a_i \in A_i$ (uniqueness holds up to appending or removal of $0\in A_m$). Then $a_d$ is called the degree-d component of $a$.

An $a\in A$ is said to be homogeneous of degree d if $a \in A_d$; we write $\deg a = d$. Note that d is unique if $a\ne 0$.

Example

The standard example of a graded ring is the ring of polynomials $B = A[X_1, \ldots, X_n]$ over some ring A, where $B_d$ has A-basis given by the set of monomials $X_1^{m_1} \ldots X_n^{m_n}$ satisfying $m_i \ge 0$ and $\sum_i m_i = d$.

Lemma 1.

$A_0$ is a subring of A and each $A_i$ is a module over $A_0$.

Proof

For the first statement, since $A_0 A_0 \subseteq A_0$ it suffices to show $1 \in A_0$; we may assume $1\ne 0$. Write $1 = a_0 + \ldots + a_n$ with $a_i \in A_i$ and $a_n \ne 0$. Suppose n > 0. For any $b_m \in A_m$ we have

$b_m = b_m\cdot 1 = b_m(a_0 + \ldots + a_n) = b_m a_0 + \ldots + b_m a_n.$

Since the LHS is homogeneous of degree m, we have $b_m a_n = 0$. Thus $a_n A_m = 0$ for any m so we have $a_n A = 0$. This gives $a_n = 0$, a contradiction.

The second statement is clear. ♦

By definition if $a, b\in A$ are homogeneous of degrees m and n, then $ab$ is homogeneous of degree m+n. We also have:

Lemma 2.

Suppose A is an integral domain with grading. If $a,b\in A$ are non-zero elements such that $ab$ is homogeneous, then so are a and b.

Proof

Write $a = \sum_m a_m$ and $b = \sum_n b_n$ as sums of their components. Let m (resp. m’) be the minimum (resp. maximum) degree for which $a_m \ne 0$ (resp $a_{m'} \ne 0$). Similarly, let n (resp. n’) be the minimum (resp. maximum) degree for which $b_n \ne 0$ (resp $b_{n'} \ne 0$). By definition $a_m b_n, a_{m'}b_{n'} \ne 0$. Since ab is homogeneous we have $m+n = m'+n'$ and thus $m = m', n = n'$. So a and b are homogeneous. ♦

Exercise A

Find a graded ring A, with $a,b \in A-\{0\}$ such that ab is homogeneous but a and b are not. [ Hint: come back to this exercise after finishing the whole article. ]

Definition.

A grading on an A-module M is a collection of additive subgroups $M_0, M_1, \ldots \subseteq M$ such that

$M = M_0 \oplus M_1 \oplus M_2 \oplus \ldots$

as additive groups, and $A_i M_j \subseteq M_{i+j}$ for any $i,j \ge 0$.

Note

We need a fixed grading on the base ring A before we can talk about grading on A-modules.

• As before if we write $m\in M$ as a sum $m_0 + m_1 + \ldots + m_n$ with $m_i \in M_i$, then $m_d$ is called the degree-d component of m.
• Also $m\in M$ is homogeneous of degree d if $m \in M_d$; again write $\deg m = d$.

The following result is quite important for grading of submodules.

Proposition 1.

Let M be a graded A-module and $N\subseteq M$ be a submodule. The following are equivalent.

1. There is a generating set for N comprising of homogeneous elements.
2. If $n\in N$, all homogeneous components of n lie in N.
3. We have $N = (N \cap M_0) \oplus (N \cap M_1) \oplus \ldots$.

Proof

(1⇒3) It suffices to show $N = \sum_i (N \cap M_i)$. Pick a generating set S for N comprising of non-zero homogeneous elements. For each $n\in N$, write $n = \sum_{i=1}^k a_i m_i$ with $a_i \in A$ and $m_i \in S$ with $\deg m_i = d_i$. Fix $d\ge 0$ and let n’ be the degree-d component of n. Then n’ is the sum of the degree-d components of $a_i m_i$. Hence

$n' = \sum_{i=1}^k a_i' m_i$, where $a_i'$ is the degree-$(d-d_i)$ component of $a_i$

so $n'\in N \cap M_d$.

(3⇒2) Let $n\in N$; we can write $n = n_0 + n_1 + \ldots + n_d$ where $n_i \in N \cap M_i$. Since $n_i \in M_i$, it is homogeneous of degree i; thus $n_i$ is the degree-i component of n and it lies in N.

(2⇒1) Pick any generating set S of N, then take the homogeneous components of all $m\in S$ to obtain a homogeneous generating set.

Definition.

Let M be a graded A-module. A submodule $N\subseteq M$ is said to be graded if it satisfies the conditions of proposition 1.

An ideal $\mathfrak a\subseteq A$ is said to be graded if it is graded as a submodule.

Proposition 1 immediately gives the following.

Corollary 1.

Given a graded module M, with collection of graded submodules $(N_i)$, graded submodule N, and graded ideal $\mathfrak a\subseteq A$,

$\sum_i N_i, \quad \cap_i N_i, \quad \mathfrak a N$

are all graded submodules of M.

Proof

Since each $N_i$ is generated by homogeneous elements, so is $\sum N_i$; thus $\sum N_i$ is graded. Suppose $n\in \cap_i N_i$, if n’ is a homogeneous component of n, then for each i, $n \in N_i \implies n' \in N_i$ and hence $n' \in \cap N_i$. Thus $\cap N_i$ is graded. Finally, if S (resp. T) is a generating set of $\mathfrak a$ (resp. N) comprising of homogeneous elements, then $\{an : a\in S, n\in T\}$ is a generating set of $\mathfrak aN$ comprising of homogeneous elements. ♦

Exercise B

Decide if each statement is true.

• If $\mathfrak a\subseteq A$ is a graded ideal of A, then $r(\mathfrak a)$ is graded.
• If $N, P \subseteq M$ are graded submodules, then $(N : P) = \{a \in A: aP \subseteq N\}$ is a graded ideal.

# Quotient

Proposition 2.

Let N be a graded submodule of a graded module M. Then M/N has a canonical grading given by

$(M/N)_i := M_i / (N\cap M_i) \hookrightarrow M/N$.

Proof

Note that we have $A_i (M/N)_j \subseteq (M/N)_{i+j}$. It remains to show M/N is a direct sum of $(M/N)_i$.

Let $m\in M$. Write $m = \sum_i m_i$ with $m_i \in M_i$. Then $m+N \in M/N$ is the sum of images of $m_i + (N \cap M_i) \in M/(N\cap M_i)$ in $M/N$. So $M/N = \sum_i (M/N)_i$.

Suppose $m + N$ can be written as $\sum_i [m_i + (N\cap M_i)]$ and $\sum_i [m_i' + (N\cap M_i)]$. Then

$m+N = (\sum_i m_i) + N = (\sum_i m_i') + N \implies \sum_i (m_i - m_i') \in N.$

By condition 2 of proposition 1, each $m_i - m_i' \in N$ so $m_i + (N\cap M_i) = m_i' + (N\cap M_i)$. ♦

Corollary 2.

If $\mathfrak a\subseteq A$ is a graded ideal, then $A/\mathfrak a$ is a graded ring under the above grading.

Proof

Since $A_i A_j \subseteq A_{i+j}$, multiplication gives

$A_i / (\mathfrak a \cap A_i) \times A_j / (\mathfrak a \cap A_j) \longrightarrow A_{i+j} / (\mathfrak a \cap A_{i+j}).$

Example

1. Suppose A is the coordinate ring of a variety $V\subseteq \mathbb A^n$, so that $A \cong k[X_1, \ldots, X_n]/I(V)$. If I(V) is homogeneous, then A is a graded ring. In particular, the group of units in A is $k^*$, the multiplicative group $k - \{0\}$. This does not work for non-homogeneous I(V), e.g. the unit group of $k[X, Y]/(XY - 1)$ contains $\{ c X^n : n \in \mathbb Z, c \in k^*\}$ (does equality hold?).

2. We will do exercise C here, i.e. show that $A = \mathbb C[X, Y, Z]/(Z^2 - X^2 - Y^2)$ is not a UFD. Indeed we have the equality $Z \cdot Z = (X+iY)(X - iY)$ in A; we claim that $Z, X+iY, X-iY \in A$ are all irreducible. By lemma 2, since Z is homogeneous of degree 1, we can only factor it as a product of a degree-0 and a degree-1 element. But all non-zero degree-0 elements of A are units (see example 1). Similarly $X+iY, X-iY$ are irreducible and $Z, X+iY, X-iY$ are clearly not associates.

3. By the same reasoning, $\mathbb C[W, X, Y, Z]/(Z^2 - W^2 - X^2 - Y^2)$ is not a UFD.

Note: however $\mathbb C[X_1, \ldots, X_n]/(X_1^2 + \ldots + X_n^2)$ is a UFD for all $n\ge 5$, as we will see later.

Definition.

Let M and N be graded A-modules, and $f:M\to N$ be A-linear. We say f is graded if $f(M_i) \subseteq N_i$ for each i.

Immediately we have:

Lemma 3.

If $f:M\to N$ is a graded map of graded A-modules, then $\mathrm{ker} f$ is a graded submodule of M and $\mathrm{im} f$ is a graded submodule of N.

Proof

If $m \in M$ and $m = m_0 + \ldots + m_d$ with $m_i \in M_i$, then $f(m_i) \in N_i$ and hence $f(m) = f(m_0) + \ldots + f(m_d)$ is the unique decomposition of f(m) into its homogeneous components. The rest is an easy exercise. ♦

Proposition 3 (First Isomorphism Theorem).

For a graded map $f:M\to N$ of graded modules, we have an isomorphism

$g : M/\mathrm{ker} f \longrightarrow \mathrm{im } f, \quad m + \mathrm{ker } f \mapsto f(m).$

in the category of graded A-modules.

Proof

Let us show that g is graded. The grading on the LHS is given by $i\mapsto M_i / (\mathrm{ker} f \cap M_i)$. Since f is graded, g takes the LHS into $f(M_i) = N_i$.

Since g is bijective, it remains to show that $g^{-1}$ is also graded. For $n\in N_i$, let $m = g^{-1}(n)$. Write $m = m_0 + \ldots + m_d$ as a sum of homogeneous components; since $g(m_i)$ is the degree-i homogeneous component of $g(m)$ we have $g(m) = g(m_i)$ so $m = m_i$. ♦

Exercise C

State and prove the remaining two isomorphism theorems.

# Direct Limits of Rings

Let $((A_i), (\beta_{ij}))$ be a directed system of rings. Regard them as a directed system of abelian groups (i.e. ℤ-modules) and take the direct limit A.

Proposition 1.

The abelian group A has a natural structure of a commutative ring.

Note

General philosophy of the direct limit: “if something happens at index i and another thing happens at index j, then by picking k greater than i and j, both things may be assumed to happen at the same index”. The cautious reader is advised to fill in the gaps in the following proof.

Sketch of Proof

For each $i\in J$, let $\epsilon_i : A_i \to A$ be the canonical map.

Given $a,b \in A$, by proposition 2 here there exists $j\in J$ and $a', b' \in A_j$ such that $a = \epsilon_j(a')$, $b = \epsilon_j(b')$. Now define multiplication in A by $a\times b := \epsilon_j(a'b')$. This does not depend on our choice of j and $a',b' \in A_j$. Indeed, if we have another index $i\in J$ and $a_1, b_1 \in A_i$ such that $a = \epsilon_i(a_1)$, $b = \epsilon_i(b_1)$, by proposition 2 here again pick $k\in J$ greater than i and j such that

$\beta_{ik}(a_1) = \beta_{jk}(a'), \ \beta_{ik}(b_1) = \beta_{jk}(b') \implies \beta_{ik}(a_1 b_1) = \beta_{jk}(a'b').$

This gives us the desired equality

$\epsilon_i(a_1 b_1) = \epsilon_k(\beta_{ik}(a_1 b_1)) = \epsilon_k (\beta_{jk}(a'b')) = \epsilon_j(a'b').$

Clearly product in A is commutative. To show associativity, given $a,b,c\in A$, pick $j\in J$ and $a',b',c' \in A_j$ such that $a = \epsilon_j(a')$, $b = \epsilon_j(b')$ and $c = \epsilon_j(c')$. Then $a(bc) = \epsilon_j(a'(b'c')) = \epsilon_j((a'b')c') = (ab)c$.

To define $1\in A$, we pick any index $i\in J$ and set $1_A := \epsilon_i(1_{A_i})$. ♦

Exercise A

1. Prove that $1_A$ in the above proof is well-defined, and $1_A \times a = a$ for all $a\in A$.

2. Prove that the resulting ring A with the canonical $\epsilon_i : A_i \to A$ gives the direct limit of $A_i$ in the category of rings.

3. Prove that if the direct limit of rings $A_i$ is zero, then $A_i = 0$ for some i. [ Hint: a ring is zero if and only if 1 = 0. ]

4. Suppose $(B_i)_{i\in I}$ is an arbitrary collection of A-algebras. For each finite subset $L = \{i_1, \ldots, i_n\} \subseteq I$, define $B_L := B_{i_1} \otimes_A \ldots \otimes_A B_{i_n}$. Define a directed system of $B_L$ over the directed set of all finite subsets of I, ordered by inclusion $L\subseteq L'$. The tensor product of $B_i$ over A is defined to be the direct limit of this system. Prove that this gives the coproduct of $(B_i)_{i\in I}$ in the category A-algebras.

Note

Since direct limits are denoted by $\varinjlim$, we will write $\varprojlim$ for the earlier limits and call them inverse limits. For most cases of interest, inverse limits will be taken over J such that $J^{op}$ is directed.

Even over directed sets, taking the inverse limit is not exact. A useful criterion for determining exactness is given by the Mittag-Lefler condition, which we will not cover (for now).

# Taking Stock

We have seen many constructions which commute and some which do not. In the following examples, $(M_i)$ is an arbitrary collection of modules; M and N are modules, B is an A-algebra and $S\subseteq A$ is a multiplicative subset.

Case 1 (proposition 1 here): $S^{-1} (\oplus_i M_i) = \oplus_i S^{-1} M_i$ but $S^{-1}(\prod_i M_i) \ne \prod_i S^{-1}M_i$ in general.

Case 2 (exercise A here): $\mathfrak a (\oplus_i M_i) = \oplus_i \mathfrak a M_i$ but $\prod_i \mathfrak a M_i \ne \mathfrak a (\prod_i M_i)$.

Case 3 (corollary 1 here): a direct sum of projective modules is projective. A direct product of projective modules is not projective in general, but a counter-example is not too easy to construct.

Case 4 (exercise B here): for any collection $(N_i)$ of submodules of M, we have $S^{-1}(\sum_i N_i) = \sum_i S^{-1}N_i$ but $S^{-1}(\cap_i N_i) \ne \cap_i S^{-1} N_i$ in general.

Case 5 (exercise A here): if M is a flat A-module, then $M^B$ is a flat B-module.

Case 6 (proposition 1 here): we have $(\oplus_i M_i) \otimes_A N \cong \oplus_i (M_i \otimes_A N)$.

Case 7: more generally, we have $(\mathrm{colim}_{i\in J} M_i) \otimes_A N \cong \mathrm{colim}_{i\in J} (M_i \otimes_A N)$ if $(M_i)_{i\in J}$ is a diagram of A-modules of type J.

Case 8: hence we have an isomorphism $(\mathrm{colim}_{i\in J} M_i)^B \cong \mathrm{colim}_{i\in J}(M_i)^B$ of B-modules; in particular $S^{-1}(\mathrm{colim}_{i\in J} M_i) \cong \mathrm{colim}_{i\in J} (S^{-1} M_i)$.

Case 9 (proposition 3 here): if $f: M_1 \to M_2$ is surjective, then $f\otimes_A 1_N : M_1 \otimes_A N \to M_2 \otimes_A N$ is surjective; however, if f is injective $f\otimes_A 1_N$ is not injective in general.

Exercise B

Prove that there is always a canonical map between $(\prod_i M_i) \otimes_A N$ and $\prod_i (M_i \otimes_A N)$. Find an example where the map is not an isomorphism.

If M is a projective A-module, must $M^B = B\otimes_A M$ be a projective B-module?

# Duality Principle

Remembering all the above relations may seem like a pain: in general if we have n constructions, we have about $O(n^2)$ relations to learn. It turns out most of these constructions can be classified as either “left-adjoint-like” or “right-adjoint-like”, which saves us a whole lot of effort in remembering them.

In the following table, constructions on the same side tend to commute or have consistent properties. Constructions on different sides may commute under specific additional conditions (e.g. finiteness, noetherianness).

 Left-adjoint-like Right-adjoint-like Sum of submodules Intersection of submodules Coproducts Products Right-exact functors Left-exact functors Pushouts Pullbacks, fibre products Direct sum of modules Direct product of modules Tensor products of modules Hom modules HomA(-, M) HomA(M, -) Colimits / Direct limits Limits / Inverse limits Injective maps Surjective maps Quotient modules Submodules Induced modules $M\mapsto M^B$. (Coinduced modules) Projective / free modules (Injective modules) Localization Multiplying ideal by module: $\mathfrak a M$

Do consider this table as a very rough guide. For example, if $0 \to N \to M \to P \to 0$ is a short exact sequence of A-modules, we do not get a right-exact sequence $\mathfrak a N \to \mathfrak a M \to \mathfrak a P \to 0$. [ Take $0\to 2\mathbb Z \to \mathbb Z \to \mathbb Z / 2\mathbb Z \to 0$ and $\mathfrak a = 2\mathbb Z$. ]

Also note that the terms in brackets have not been defined yet.

### Further Examples

1. We have $\mathrm{Hom}_A(\oplus_i N_i, M) \cong \prod_i \mathrm{Hom}_A(N_i, M)$.

2. The functor $\mathrm{Hom}_A(-, M)$ takes a right-exact sequence to a left-exact sequence.

3. Recall that the colimit of a diagram of A-modules was constructed by taking a quotient of the direct sum of these modules. Dually, its limit can be constructed by taking a submodule of the direct product.

4. The tensor product was constructed by taking a quotient of a free (hence projective) module.

Exercise C (Coinduced Modules)

1. Let B be an A-algebra. Prove that for an A-module M, $M_B := \mathrm{Hom}_A(B, M)$, the set of all A-linear maps $B\to M$, has a natural structure of a B-module.

2. Prove that we get a functor

$F :A\text{-}\mathbf{Mod} \longrightarrow B\text{-}\mathbf{Mod}, \quad M \mapsto M_B$

such that there is a natural bijection

$\mathrm{Hom}_A(N, M) \cong \mathrm{Hom}_B(N, M_B).$

for any B-module N. Thus $F = \mathrm{Hom}_A(B, -)$ is right-adjoint to the forgetful functor $B\text{-}\mathbf{Mod} \to A\text{-}\mathbf{Mod}$.

We call $M_B$ the coinduced B-module from M.

# Limits Are Left-Exact

By example 6 and proposition 2 in the previous article, one is inclined to conclude that taking the colimit in $\mathcal C = A\text{-}\mathbf{Mod}$ is a right-exact functor, but there is a rather huge issue here: the functors are between $\mathcal C$ and $\mathcal D = \mathcal C^J$, the category of diagrams in $\mathcal C$ while we only defined exactness of functors between categories of modules. The proper way to do this is to introduce the framework of abelian categories and extend our concept of additive functors and exact functors there. However, doing this will take us too far afield so we will prove it directly (which is, admittedly, a bit of a cop out).

Proposition 1.

Let J be an index category, and $D', D, D'' : J\to A\text{-}\mathbf{Mod}$ be diagrams of type J. For concreteness, write these diagrams as

$((N_i), (\beta_e^N : N_i \to N_j)), \ ((M_i), (\beta_e^M : M_i \to M_j)),\ ((P_i), (\beta_e^P : P_i \to P_j))$

where $i\in J$ and $e:i\to j$. Let $D'\to D \to D''$ be morphisms, written as a collection of $N_i\stackrel {\phi_i} \to M_i \stackrel {\psi_i} \to P_i$ over $i\in J$. Then

\left( \begin{aligned}0\to N_i\stackrel {\phi_i} \to M_i \stackrel {\psi_i} \to P_i\\ \text{exact for each } i\in J\end{aligned}\right) \implies 0 \to \lim N_i \to \lim M_i \to \lim P_i \text{ exact.}

\left( \begin{aligned}N_i\stackrel {\phi_i} \to M_i \stackrel {\psi_i} \to P_i \to 0\\ \text{exact for each } i\in J\end{aligned}\right) \implies \mathrm{colim} N_i \to \mathrm{colim} M_i \to \mathrm{colim} P_i \to 0\text{ exact.}

Note

In summary, taking the limit is left-exact while taking the colimit is right-exact.

Proof

We prove the second claim, leaving the first as an exercise. By proposition 1 here, $\mathrm{colim} M_i$ is concretely described as follows. Take the quotient of $\oplus_{i\in J} M_i$ by all $m_i - \beta_e^M(m_i)$, where $e:i\to j$ is an arrow in J, $m_i \in M_i$ and $\beta_e^M(m_i)\in M_j$ are identified with their images in $\oplus_i M_i$.

With this description, clearly $\mathrm{colim} M_i \to \mathrm{colim} P_i$ is surjective. Also, composing $\mathrm{colim} N_i \to \mathrm{colim} M_i \to \mathrm{colim} P_i$ is the zero map so $\mathrm{im}(\mathrm{colim} \phi_i) \subseteq \mathrm{ker}(\mathrm{colim} \psi_i)$. Now write $\phi : \oplus_i N_i \to \oplus_i M_i$ for $\oplus \phi_i$ and $\psi : \oplus_i M_i \to \oplus_i P_i$ for $\oplus \psi_i$.

Conversely, let $x\in \oplus_i M_i$ represent an element in the kernel of $\mathrm{colim} \psi_i$. Thus $\psi(x) \in\oplus_i P_i$ is a finite sum of $p_i - \beta_e^P(p_i)$. Since $\psi_i$ is surjective, we can write such a term as

$\psi_i(m_i') - \beta_e^P(\psi_i(m_i')) = \psi_i(m_i') - \psi_j(\beta_e^M(m_i')) = \psi(m_i' - \beta_e^M(m_i'))$

for some $m_i'\in M_i$. Since $\psi(x)$ is a finite sum of $\psi(m_i' - \beta_e^M(m_i'))$, we can replace x by another representative such that $\psi(x) = 0$. Then $x = \phi(y)$ for some $y\in \oplus N_i$. ♦

Neither the limit nor the colimit functor is exact in general. For the colimit case, consider the following commutative diagram of A-modules

where all maps $A\to A$ are identities. The rows are short exact sequences and the squares all commute, but taking the colimit of the columns gives

$0 \longrightarrow A^2 \longrightarrow A \longrightarrow 0 \longrightarrow 0$

which is not exact.

Exercise A

Find an example for the case of limits.

# Direct Limits

We will describe a special case where taking the colimit is exact.

Given a poset $(S, \le)$, we recall the category $\mathcal C(S)$ whose objects are elements of S, and between any $x,y\in S$, $|\mathrm{hom}(x, y)| \le 1$ with equality if and only if $x \le y$. Composition is the obvious one.

Definition.

A poset $(S, \le)$ is called a directed set if for any $a,b\in S$, there is a $c\in S$ such that $a\le c$ and $b\le c$.

In other words, a poset is directed if every finite set has an upper bound.

Definition.

If J is an index category obtained from $\mathcal C(S)$ for some directed set S, then a diagram in $\mathcal C$ of type J is called a directed system. The colimit of $(A_i)_{i\in J}$ is called the direct limit and denoted by

$\varinjlim_{i\in J} A_i$.

In other words, direct limit = colimit over directed set. We will abuse notation a little and regard J as the directed set itself.

To avoid set-theoretic difficulties, the directed set J is always assumed to be non-empty.

Example

In exercise C.3 here, for a multiplicative $S\subseteq A$ and A-module M, we have an isomorphism of A-modules

$\mathrm{colim}_{f\in S} M_f \cong M_S$

where $f\le g$ if g is a multiple of f. Since S is multiplicative, any {fg} has an upper bound fg. Hence $M_S$ is the direct limit of $M_f$ over $f\in S$:

$\varinjlim_{f\in S} M_f \cong M_S$.

Similarly, we have the following direct limit in the category of rings:

$\varinjlim_{f\in S} A_f \cong A_S$.

Next we will discuss the general direct limit in the categories A-Mod and Ring.

# Direct Limit of Modules

Let A be a fixed ring; the following holds for direct limits in the category of A-modules.

Proposition 2.

Suppose $((M_i)_{i\in J}, (\beta_{ij})_{i\le j})$ is a directed system of A-modules over a directed set J. Let

$M = \varinjlim_{i\in J} M_i$, with canonical $\epsilon_i : M_i \to M$ for each $i\in J$.

Then for each $m\in M$, there exists an $i\in J$ and $m_i \in M_i$ such that $\epsilon_i(m_i) = m$.

Also if $m_i\in M_i$ satisfies $\epsilon_i(m_i) = 0$, then there exists $j\ge i$ such that $\beta_{ij}(m_i) = 0 \in M_j$.

Note

The philosophy is that “whatever happens in the direct limit happens in $M_j$ for some sufficiently large index j“.

Proof

By proposition 1 here, the colimit M is described concretely by taking the quotient of $P = \oplus_{i\in J} M_i$ (with canonical $\nu_i : M_i \to P$) by relations of the form

$\nu_k(m_k) - \nu_l\beta_{kl}(m_k),\ m_k \in M_i,\ k\le l\ (k, l\in J).$

Hence any $m\in M$ can be written as $\epsilon_{i_1}(m_1) + \ldots + \epsilon_{i_N}(m_N)$ for $m_1 \in M_{i_1}, \ldots, m_N \in M_{i_N}$. But J is a directed set, so we can pick index $j\in J$ such that $j\ge i_1, \ldots, i_N$; then

$m = \epsilon_j(m_j)$, where $m_j = \beta_{i_1 k}(m_1) + \ldots + \beta_{i_N k}(m_N)$,

proving the first claim.

For the second claim, if $\epsilon_i(m_i) = 0$ then $\nu_i(m_i) \in \oplus_i M_i$ is a finite sum of the above relations. Pick an index $j\in J$ larger than i and all indices kl in the sum; then $\beta_{ij}(m_i)$ is the sum of the images of these relations in $M_k$. But each such relation has image $\beta_{kj}(m_k) - \beta_{lj}\beta_{kl}(m_k) = 0$ in $M_k$, so $\beta_{ij}(m_i) = 0$ as desired. ♦

Corollary 1.

If $((M_i), (\beta_{ij}))$ is a directed system of A-modules such that $\beta_{ij}$ are all injective, then

$\epsilon_j : M_j \longrightarrow \varinjlim_i M_i$

is also injective for each $j\in J$.

Finally we have:

Proposition 3.

Let $((M_i), (\beta_{ij}))$ and $((N_i), (\gamma_{ij}))$ be directed systems of A-modules and $\phi_i : M_i \to N_i$ be a morphism of the directed systems, i.e. for any $i\le j$, we have $\gamma_{ij} \circ \phi_i = \phi_j \circ \beta_{ij} : M_i \to N_j$.

If each $\phi_i$ is injective, so is $\phi : \varinjlim M_i \to \varinjlim N_i$.

Since taking the colimit is right-exact by proposition 1, we see that taking the direct limit is exact.

Proof

Write $\epsilon_i^M : M_i \to \varinjlim M_i$ and $\epsilon_i^N : N_i \to \varinjlim N_i$ for the canonical maps.

Suppose $\phi(m) = 0$ for $m \in \varinjlim M_i$. By proposition 2, we have $m = \epsilon_i^M(m_i)$ for some $m_i \in M_i$; then

$0 = \phi(m) = \phi(\epsilon_i^M(m_i)) = \epsilon_i^N(\phi_i(m_i))$

so by proposition 2 again, there exists $j\ge i$ such that $\gamma_{ij}(\phi_i(m_i)) = 0$, so $\phi_j(\beta_{ij}(m_i)) = 0$. Since $\phi_j$ is injective we have $\beta_{ij}(m_i) = 0$ so $m = \epsilon_i^M(m_i) = \epsilon_j^M\beta_{ij}(m_i) = 0$. ♦

Exercise B

Describe the direct limit of sets $(S_i)$ over J. State and prove an analogue of proposition 2.

## Commutative Algebra 50

Adjoint functors are a general construct often used for describing universal properties (among other things).

Take two categories $\mathcal C$ and $\mathcal D$.

Definition.

Covariant functors $F:\mathcal D\to \mathcal C$ and $G: \mathcal C \to \mathcal D$ are said to be adjoint if we have isomorphisms

$A \in \mathcal C, B \in \mathcal D \implies \mathrm{hom}_{\mathcal C}(F(B), A) \cong \mathrm{hom}_{\mathcal D}(B, G(A))$

which are natural in A and B, when we regard both sides as functors $\mathcal D^{\text{op}} \times \mathcal C \to \mathbf{Set}$.

We also say F is left adjoint to G and G is right adjoint to F.

Unwinding the definition, for any $B\in \mathcal D$ and $A\in \mathcal C$, we have a bijection

$T(B, A) : \mathrm{hom}_{\mathcal C}(F(B), A) \stackrel\cong\longrightarrow \mathrm{hom}_{\mathcal D}(B, G(A))$

such that the following diagram commutes for any $g:B' \to B$ and $f:A\to A'$.

Note

Suppose F and G are adjoint. For any $B\in \mathcal D$, the covariant functor

$\mathcal C \longrightarrow \mathbf{Set}, \quad A \mapsto \mathrm{hom}_{\mathcal D}(B, G(A))$

is representable since it is isomorphic to $\mathrm{hom}_{\mathcal C}(F(B), -)$. Similarly for each $A\in\mathcal C$, the contravariant functor $\mathcal D \to \mathbf{Set}$ taking $B\mapsto \mathrm{hom}_{\mathcal C}(-, G(A))$ is representable.

There are various equivalent ways of defining adjoint functors but we will not delve into those here. Instead, let us contend ourselves with some examples before restricting to the case of modules.

# Examples Galore

“Adjoint functors arise everywhere.” – Saunders Mac Lane, Categories for the Working Mathematician.

### Example 1

Let $F: \mathbf{Set} \to \mathbf{Gp}$ take a set S to the free group F(S) on S and $U: \mathbf{Gp} \to \mathbf{Set}$ be the forgetful functor. By the universal property of free groups, we have a natural bijection

$\mathrm{hom}_{\mathbf{Gp}}(F(S), G) \cong \mathrm{hom}_{\mathbf{Set}}(S, U(G))$

for any set S and group G.

### Example 2

Let $F : \mathbf{Set} \to \mathbf{Ring}$ take a set S to the (commutative) ring $\mathbb \mathbb Z[S] := Z[x_s : s\in S]$ and $U :\mathbf{Ring} \to \mathbf{Set}$ be the forgetful functor. Again we get

$\mathrm{hom}_{\mathbf{Ring}}(\mathbb Z[S], R) \cong \mathrm{hom}_{\mathbf{Set}}(S, U(R))$

for any set S and ring R.

In general, free objects are left adjoint to forgetful functors.

### Example 3

Let B be an A-algebra. Recall that for each A-module M, we have an induced B-module $M^B = B\otimes_A M$; this gives a functor $F: A\text{-}\mathbf{Mod} \to B\text{-}\mathbf{Mod}$. On the other hand, we have the forgetful functor $U : B\text{-}\mathbf{Mod} \to A\text{-}\mathbf{Mod}$ from the canonical ring homomorphism $A\to B$. By the universal property of induced B-modules

$\mathrm{hom}_{B\text{-}\mathbf{Mod}}(M^B, N) \cong \mathrm{hom}_{A\text{-}\mathbf{Mod}}(M, U(N)).$

Thus the induced module construction is also left adjoint to a forgetful functor.

### Example 4

Let $\mathcal C = A\text{-}\mathbf{Mod}$ and $N \in \mathcal C$. Take functors $F_N, G_N : \mathcal C \to \mathcal C$ where $F_N(M) = M\otimes_A N$ and $G_N(P) = \mathrm{Hom}_A(N, P)$. In proposition 2 here we obtained a natural isomorphism $\mathrm{Hom}_A(M\otimes N, P) \cong \mathrm{Hom}_A(M, \mathrm{Hom}_A(N, P))$. This translates to a natural isomorphism

$\mathrm{hom}_{\mathcal C}(F_N(M), P) \cong \mathrm{hom}_{\mathcal C}(M, G_N(P)).$

In short $- \otimes_A N$ is left adjoint to $\mathrm{Hom}_A(N, -)$.

### Example 5

Let $\mathcal D = \mathcal C \times \mathcal C$. Suppose the coproduct of any two objects in $\mathcal C$ exist. Let $\Sigma:\mathcal D \to \mathcal C$ take $(A, A')\mapsto A \amalg A'$ and $\Delta : \mathcal C \to \mathcal D$ take $A\mapsto (A, A)$. By definition of coproduct in categories, we have

\begin{aligned} \mathrm{hom}_{\mathcal D}((A, A'), \Delta(B)) &= \mathrm{hom}_{\mathcal C}(A, B) \times \mathrm{hom}_{\mathcal C}(A', B)\\ &\cong \mathrm{hom}_{\mathcal C}(A \amalg A', B)\\ &= \mathrm{hom}_{\mathcal C}(\Sigma(A, A'), B)\end{aligned}

for any $(A, A') \in \mathcal D$ and $B\in \mathcal C$. Hence the coproduct functor is left adjoint to the diagonal functor. Dually, the product is right adjoint to the diagonal functor.

### Example 6

More generally, let J be an index category and $\mathcal D = \mathcal C^J$ be the category of all diagrams in $\mathcal C$ of type J. Assuming all colimits of type J exist in $\mathcal C$, let $\mathrm{colim}_J : \mathcal D \to \mathcal C$ be this colimit functor. On the other hand, take the diagonal embedding $\mathcal C \to \mathcal D$ which takes an object A to the diagram where all objects are A and all morphisms are $1_A$. Then

$\mathrm{hom}_{\mathcal D}(D, \Delta(B)) \cong \mathrm{hom}_{\mathcal C}(\mathrm{colim} D, B)$

for any diagram $D\in \mathcal D$ and object $B \in \mathcal C$.

Exercise A

Let $U : \mathbf{Top} \to \mathbf{Set}$ be the forgetful functor. Find a left adjoint and a right adjoint to U (these are two different functors of course).

Let $U : \mathbf{AbGp} \to \mathbf{Gp}$ be the inclusion functor, where $\mathbf{AbGp}$ is the category of abelian groups. Find a left adjoint functor to U.

# Properties

Suppose $F:\mathcal D \to \mathcal C$ is left adjoint to $G : \mathcal C \to \mathcal D$. If coproducts exist in both categories, then $F(B) \amalg F(B') \cong F(B\amalg B')$ because we have natural isomorphisms

\begin{aligned} A \in\mathcal C \implies \mathrm{hom}_{\mathcal C}(F(B\amalg B'), A) &\cong \mathrm{hom}_{\mathcal D}(B \amalg B', G(A)) \\ &\cong \mathrm{hom} _{\mathcal D}(B, G(A)) \times \mathrm{hom}_{\mathcal D}(B', G(A)) \\ & \cong \mathrm{hom}_{\mathcal C}(F(B), A) \times \mathrm{hom}_{\mathcal C}(F(B'), A) \\ &\cong \mathrm{hom}_{\mathcal D}(F(B) \amalg F(B'), A).\end{aligned}

More generally, we have:

Proposition 1.

Let J be an index category; assume that colimits of diagrams of type J all exist $\mathcal C$ and $\mathcal D$. Then for any diagram $(B_i)_{i\in J}$ in $\mathcal D$ of type J,

$\mathrm{colim }_{i\in J} F(B_i) \cong F(\mathrm{colim}_{i\in J} B_i)$.

Proof

By the universal property of colimits and limits, we have, for any diagram $(A_i)_{i\in J}$ in $\mathcal C$ of type J, and any object $A\in \mathcal C$,

$\mathrm{hom}_{\mathcal C}(\mathrm{colim}_{i\in J} A_i, A) \cong \mathrm{lim}_{i\in J^{\text{op}}} (\mathrm{hom}_{\mathcal C}(A_i, A)).$

Now we apply this universal property twice to obtain:

\begin{aligned}A \in \mathcal C \implies \mathrm{hom}_{\mathcal C}(F(\mathrm{colim}_{i \in J} B_i), A) &\cong \mathrm{hom}_{\mathcal D}(\mathrm{colim}_{i \in J} B_i, G(A))\\ &\cong \mathrm{lim}_{i \in J^{\text{op}}} (\mathrm{hom}_{\mathcal D}(B_i, G(A))) \\ &\cong \mathrm{lim}_{i\in J^{\text{op}}} ( \mathrm{hom}_{\mathcal C} (F(B_i), A))\\ &\cong \mathrm{hom}_{\mathcal C} (\mathrm{colim}_{i\in J} F(B_i), A)\end{aligned}

and we are done. ♦

Exercise B

1. State and prove the dual of the above properties.

2. Prove that the forgetful functor $U:\mathbf{Gp} \to \mathbf{Set}$ has no right adjoint.

3. Prove that the inclusion functor $U:\mathbf{AbGp} \to \mathbf{Gp}$ has no right adjoint.

4. Prove that for any diagram of A-modules $(M_i)_{i \in J}$ and A-module N, we have

$\mathrm{colim}_{i \in J} (M_i \otimes_A N) \cong (\mathrm{colim}_{i\in J} M_i) \otimes_A N.$

As a special case, this implies tensor product is distributive over direct sums (already proved in proposition 1 here).

$F : B\text{-}\mathbf{Mod} \to A\text{-}\mathbf{Mod}, \quad G : A \text{-}\mathbf{Mod} \to B\text{-}\mathbf{Mod}$

such that F is left adjoint to G. Further, we assume that the natural $\mathrm{Hom}_B(F(N), M) \cong \mathrm{Hom}_A(N, G(M))$, for A-module M and B-module N, is an isomorphism of additive groups

Note

In fact, one can show that if F and G are adjoint functors as above, then they must be additive and $\mathrm{Hom}_B(F(N), M) \cong \mathrm{Hom}_A(N, G(M))$ must preserve the additive structure.

Proposition 2.

The functors F and G are right-exact and left-exact respectively.

In summary, left adjoint functors are right-exact and vice versa.

Proof

Suppose $M_1 \to M_2 \to M_3 \to 0$ is an exact sequence of A-modules. To prove $F(M_1) \to F(M_2) \to F(M_3) \to 0$ is exact, by proposition 4 here it suffices to show:

$0 \longrightarrow \mathrm{Hom}_B(F(M_3), N) \longrightarrow \mathrm{Hom}_B(F(M_2), N) \longrightarrow \mathrm{Hom}_B(F(M_1), N)$

is exact for every B-module N. But by adjointness, the above is naturally isomorphic to

$0 \longrightarrow \mathrm{Hom}_A(M_3, G(N)) \longrightarrow \mathrm{Hom}_B(M_2, G(N)) \longrightarrow \mathrm{Hom}_B(M_1, G(N))$

which is exact since $\mathrm{Hom}_A(-, G(N))$ is a left-exact functor by proposition 3 here. The case for G is similar. ♦

### Examples

1. By example 4, we see that for any A-module M, the functor $M\otimes -$ is right-exact.

2. By example 5, for any A-algebra B, taking the induced module $M\mapsto M^B$ is right-exact.

# Morphism of Diagrams

Throughout this article $\mathcal C$ denotes a category and J is an index category.

Definition

Given diagrams $D, D' : J\to \mathcal C$, a morphism $D \to D'$ is a natural transformation $T : D\Rightarrow D'$.

Thus we have the category of all diagrams in $\mathcal C$ of type J, which we will denote by $\mathcal C^J$.

For example if we write D and D’ as tuples:

$\left((A_i)_{i\in J}, (\beta_{e}: A_i \to A_j)_{(e : i\to j)}\right), \quad \left((A'_i)_{i\in J}, (\beta'_{e}: A'_i \to A'_j)_{(e : i\to j)}\right)$

a morphism $D\to D'$ is a collection of morphisms $\gamma_i : A_i \to A_i'$ such that

$(e:i\to j) \implies \beta'_e \circ \gamma_i = \gamma_j \circ \beta_e : A_i \to A_j'.$

In diagram form, we have the following, where all “rectangles” commute.

Exercise A

Suppose any diagram of type J in $\mathcal C$ has a colimit. Prove that we get a functor $\mathrm{colim} : \mathcal C^J \longrightarrow \mathcal C$ which takes a diagram in $\mathcal C$ to its colimit. In other words show that a morphism of two diagrams of same type in $\mathcal C$ induces a morphism of their colimits.

# Category of Modules

Proposition 1.

Colimits always exist in the category of A-modules.

Proof

Suppose $\left((M_i)_{i\in J}, (\beta_{e}: M_i \to M_j)_{(e : i\to j)}\right)$ is a diagram of type J. Let $P = \oplus_{i\in J} M_i$ with canonical embeddings $\nu_i : M_i \to P$. Let $Q\subseteq P$ be the submodule generated by all elements of the form $\nu_i(m_i) - \nu_j(\beta_e(m_i))$ over all $m_i \in M_i$ and $e : i\to j$ in J. We claim that $P/Q$ satisfies our desired universal properties. Define

$\epsilon_i : M_i \longrightarrow P/Q, \qquad M_i \stackrel {\nu_i}\longrightarrow \oplus_{i\in J} M_i = P \stackrel \pi \longrightarrow P/Q.$

By definition $\epsilon_j \circ \beta_e = \pi\circ \nu_j \circ \beta_e = \pi\circ \nu_i = \epsilon_i$ for all $e:i \to j$.

Now suppose we have a module N with linear maps $(\alpha_i : M_i \to N)_{i\in J}$ such that for any $e:i\to j$ we have $\alpha_j \circ \beta_e = \alpha_i$. The collection of $\alpha_i$ induce, by definition of direct sum, a unique map $g : \oplus_{i\in J} M_i \to N$ such that $g\circ\nu_i = \alpha_i$ for each $i\in J$. Hence

$e:i\to j \implies g\circ (\nu_j \circ \beta_e - \nu_i) = g\circ \nu_j \circ \beta_e - g\circ \nu_i = \alpha_j \circ \beta_e - \alpha_i = 0$

so $g$ factors through $f:P/Q \to N$ such that $f\circ \pi = g$. Thus for each $i\in J$ we have $f\circ \epsilon_i = f\circ \pi\circ \nu_i = g\circ \nu_i = \alpha_i$. ♦

Exercise B

Prove that colimits exist in the categories $\mathbf{Set}$, $\mathbf{Top}$ and $\mathbf{Gp}$.

# Another Functoriality

Definition.

Suppose $F: J_0 \to J$ is a morphism of index categories. Composition then gives:

$(D : J \to \mathcal C) \mapsto (D\circ F : J_0 \to \mathcal C)$.

Thus a diagram of type J gives us a diagram of type $J_0$. If $J_0$ is a subcategory of J, this is just the restriction of D to $J_0$, denoted by $D|_{J_0}$.

In fact we get a functor $F: \mathcal C^{J} \to \mathcal C^{J_0}$. Indeed, a morphism between diagrams $D_1, D_2 : J \to\mathcal C$ is a natural transformation $T : D_1\Rightarrow D_2$. We let F take this T to

$F(T) = 1_F * T : (D_1 \circ F) \Rightarrow (D_2\circ F)$,

where * is a form of “horizontal composition” of natural transformation (see the optional exercise here).

Although the abstract definition looks harrowing, the underlying concept is quite easy when $J_0$ is a subcategory of J, so it helps to keep this special case in mind. We denote the diagrams $D_1, D_2$ by the following tuples

$((A_i)_{i\in J}, (\beta_e : A_i \to A_j)_{e:i\to j}), \quad ((A'_i)_{i\in J}, (\beta'_e : A'_i \to A'_j)_{e:i\to j})$

so that a morphism $D_1 \to D_2$ is of a collection of morphisms $\gamma_i : A_i \to A_i'$ in $\mathcal C$ such that for any $e:i\to j$ we have $\beta'_e\circ \gamma_i = \gamma_j \circ \beta_e$. Now the new diagrams $D_1|_{J_0}$ and $D_2|_{J_0}$ are the same tuples but with $i\in J_0$ and $e:i\to j$ running through morphisms in $J_0$. Hence, $F(T)$ is given by the same collection of $\gamma_i$, except now i runs through $i \in J_0$.

Proposition 2.

Let $F : J_0 \to J$ be a morphism of index categories. Then for any diagram in $\mathcal C$ of type J, denoted by the pair $(A_i)$ and $(\beta_e)_{e:i\to j}$, we have an induced

$f: \mathrm{colim}_{i_0 \in J_0} A_{F(i_0)} \longrightarrow \mathrm{colim}_{i\in J} A_i$

assuming both colimits exist.

Proof

By definition $A := \mathrm{colim}_{i\in J} A_i$ comes with a collection of morphisms $(\epsilon_i : A_i \to A)_{i\in J}$ such that $\epsilon_j \circ \beta_e = \epsilon_i$ for all $e:i\to j$ in J.

Similarly $A_0 := \mathrm{colim}_{i_0 \in J_0} A_{F(i_0)}$ comes with a collection of morphisms $(\epsilon_{i_0} : A_{F(i_0)} \to A_0)_{i_0 \in J_0}$ such that $\epsilon_{j_0} \circ \beta_{F(e_0)} = \epsilon_{i_0}$ for all $e_0 : i_0 \to j_0$ in $J_0$.

From restricting the first colimit, we get a collection $(\epsilon_{F(i_0)} : A_{F(i_0)} \to A)_{i_0 \in J_0}$ such that $\epsilon_{F(j_0)} \circ \beta_{F(e_0)} = \epsilon_{F(i_0)}$ for all $e_0 : i_0 \to j_0$ in $J_0$.

By universal property of the colimit $A_0$, this induces a unique morphism $f:A_0 \to A$ such that $f\circ \epsilon_{i_0} = \epsilon_{F(i_0)}$ . ♦

Example

By restricting the following diagram

we obtain a morphism $B \coprod B' \to B \coprod_A B'$, assuming both objects exist. More generally we have $\coprod_{i \in J} A_i \to \mathrm{colim}_{i\in J} A_i$.

In fact, the proof of proposition 1 gives us a clue on how to construct general colimits. First take the coproduct, which corresponds to colimit over a diagram of vertices. Next we “add the arrow relations” by taking the coequalizer for each arrow.

# Limits

Limits are the dual of colimits.

Definition.

Take a diagram in $\mathcal C$ of type J, written as

$((A_i)_{i\in I}, (\beta_e : A_i \to A_j)_{(e:i\to j)}$

The limit of the diagram comprises of the following data:

$(A, (\pi_i : A \to A_i)_{i\in J})$

where $A = \lim_{i \in J} A_i \in \mathcal C$ is an object, $\pi_i : A \to A_i$ is a morphism in $\mathcal C$ for each $i\in J$, such that for any arrow $e:i\to j$, we have $\beta_e\circ\pi_i = \pi_j$.

We require the following universal property. For any tuple

$(B, (\alpha_i : B \to A_i)_{i\in J})$

where $B\in \mathcal C$ is an object, $\alpha_i$ is a morphism for each $i\in J$, such that for any arrow $e:i\to j$, we have $\beta_e \circ \alpha_i = \alpha_j$, there is a unique morphism $f : B \to A$ such that

$\pi_i\circ f = \alpha_i \text{ for each } i\in J.$

As before, we have the following special cases.

### Example 1: Products

If J is obtained from an index set I, the limit is the product $\prod_{i\in I} A_i$.

### Example 2: Pullbacks (Fiber Products)

If J is the following, the resulting limit is the pullback.

### Example 3: Equalizers

Definition.

The equalizer of $\beta_1, \beta_2 : A\to B$ in a category $\mathcal C$ is the limit of the following diagram.

This is a pair $(C, \pi : C\to A)$ such that $\beta_1\circ \pi = \beta_2\circ \pi$ and, for any pair $(D, \alpha : D\to A)$ such that $\beta_1 \circ \alpha = \beta_2 \circ \alpha$, there is a unique $f:D\to C$ such that $\pi\circ f = \alpha$.

Exercise C

Prove that limits always exist in the category of A-modules.

# Initial and Terminal Objects

Definition.

An object $A\in \mathcal C$ is said to be initial (resp. terminal) if for any object $B\in \mathcal C$, there is a unique morphism $A\to B$ (resp. $B\to A$).

Note

• If A is initial or terminal, there is a unique morphism $A\to A$, i.e. the identity.
• A is initial if and only if it is a (colimit / limit) of the empty diagram. [ Exercise: pick the right option and write the dual statement. ]

Exercise D

Prove that if A and A’ are initial, there is a unique isomorphism $A\to A'$. Dually, the same holds for terminal objects. In summary, initial (resp. terminal) objects are unique up to unique isomorphism.

The phrase “unique up to unique isomorphism” has been used multiple times while looking at universal properties. This is not a coincidence, for initial and terminal objects can be used to describe universal properties of various constructions. Here is an example.

Lemma 1.

Let M, N be A-modules. Consider the category $\mathcal C(M, N)$, whose objects are pairs

$(P, B : M\times N \to P)$

where P is an A-module, B is an A-bilinear map. The morphisms

$(B : M\times N \to P) \longrightarrow (B' : M\times N \to P')$

are A-linear maps $f:P\to P'$ such that $f\circ B = B'$. Then $(M\otimes_A N, (m, n)\mapsto m\otimes n)$ is an (initial / terminal) object in $\mathcal C$. [ Exercise: pick the right option. ]

Proof

Follows directly from the definition. ♦

# Introduction

For the next few articles we are back to discussing category theory to develop even more concepts. First we will look at limits and colimits, which greatly generalize the concept of products and coproducts and cover loads of interesting cases.

As a starting example, recall that for A-algebras B and C, we have $B\otimes_A C$ which is the coproduct of B and C in the category of A-algebras. But the category of A-algebras corresponds to the coslice category $A\downarrow \mathcal C$ whose objects are morphisms $A\to B$ (as B runs through objects of $\mathcal C$), and morphisms are just morphisms $B\to B'$ in $\mathcal C$ making the diagram commute. If we unwind the definition, coproduct in the coslice category means the following.

Definition.

Let $\beta: A\to B$, $\beta':A \to B'$ be morphisms in the category $\mathcal C$. The pushout of $\beta$ and $\beta'$ is a triplet:

$(C, \epsilon : B \to C, \epsilon' : B'\to C)$,

where $C := B \coprod_A B'\in \mathcal C$ is an object, $\epsilon$, $\epsilon'$ are morphisms in $\mathcal C$ satisfying $\epsilon\circ \beta = \epsilon' \circ \beta'$ such that for any triplet:

$(D, \alpha : B \to D, \alpha' : B'\to D)$

of object $D\in \mathcal C$ and morphisms $\alpha$, $\alpha'$ satisfying $\alpha \circ \beta = \alpha' \circ \beta'$, there is a unique morphism $f : C\to D$ such that $f\circ \epsilon = \alpha$ and $f\circ \epsilon' = \alpha'$.

Pictorially, the pushout gives a correspondence as follows.

The idea is that $B\coprod_A B'$ classifies all morphisms “from” the diagram in red.

The pushout may not exist; if it does, it is unique up to unique isomorphism.

## Examples

1. We already saw that in the category of rings, the pushout of $A\to B$ and $A\to B'$ is $B\otimes_A B'$.

2. Take $\beta : S\to T$ and $\beta' : S\to T$ in the category $\mathbf{Set}$. Then:

$T \coprod_S T' = \{T \coprod T' \} / (\beta(s) \sim \beta'(s), s\in S)$

where $T\coprod T'$ is the disjoint union of T and T’.

3. Let $\beta : H\to G$ and $\beta' : H\to G'$ be injective group homomorphisms in $\mathbf{Gp}$. The pushout is called the amalgamation of G and G’ over H and can be concretely described in terms of words in H, coset representatives of G/H and those of G’/H. A description is given in Trees by Jean-Pierre Serre.

Exercise A

1. Prove example 2.

2. Find the pushforward for:

• $\alpha : X\to Y$ and $\alpha' :X \to Y'$ in the category $\mathbf{Top}$ of topological spaces;
• $\alpha : M\to N$ and $\alpha' : M\to N'$ in the category $A\text{-}\mathbf{Mod}$.

# Pullbacks (Fibre Products)

Dually, we can define the following.

Definition.

Let $\beta :B\to A$, $\beta' :B' \to A$ be morphisms in $\mathcal C$. The pullback (or fibre product) of $\beta$ and $\beta'$ corresponds to the pushforward of $\beta^{\text{op}}$ and $\beta'^{\text{op}}$ in the opposite category $\mathcal C^{\text{op}}$.

The underlying object of the pullback is denoted by $B\times_A B'$.

Easy exercise

Write out the details of the definition (see below diagram).

## Examples

1. Let $\gamma : T \to S$ and $\gamma' : T' \to S$ be functions in $\mathbf{Set}$. The pullback is given by

$T\times_S T' = \{(t, t') \in T\times T' : \gamma(t) = \gamma'(t') \in S\}$

with projection maps $T\times_S T' \to T$, $T\times_S T' \to T'$ taking $(t,t')$ to $t, t'$ respectively.

2. As a special case suppose $T\subseteq S$ and $\gamma : T\to S$ is the inclusion map. Then $T\times_S T' = \gamma'^{-1}(T)$, i.e. the fibre space of $\gamma' : T' \to S$ over the subset T.

3. As a special case of special case, suppose $T, T'\subseteq S$ and $\gamma, \gamma'$ are both inclusions. Then $T\times_S T' = T\cap T'$.

Exercise B

Do the same constructions work for $\mathbf{Gp}$, $\mathbf{Ring}$, $\mathbf{Top}$, $A\text{-}\mathbf{Mod}$, $A\text{-}\mathbf{Alg}$?

Explain how the three constructions at the end of Chapter 30 are all fibre products in the category of k-schemes (or pushouts, if we take the dual category of finitely generated k-algebras).

# Colimits

Now we will generalize the above constructions, by taking limits and colimits over a diagram in a category.

Definition.

An index category J is a category such that its class of objects is a set.

If you are concerned about the difference between classes and sets, please read the note at the end.

In summary, an index category consists of a set of vertices and arrows between them such that we can compose arrows. For convenience, we will employ this terminology for an index category: objects are called vertices and edges are called arrows.

Now an index set I may be regarded as an index category J, where we pick a vertex for each $i\in I$ and the only morphisms are the identities. E.g. the index set {1,2,3} has 3 vertices and one identity arrow for each object.

Definition.

Let $J$ be an index category; a diagram of type $J$ in the category $\mathcal C$ is a covariant functor $D: J \to \mathcal C$.

Thus to each vertex $i \in J$ we assign an object $A_i \in \mathcal C$ and to each arrow $e : i \to j$ in J, we assign a morphism $\beta_{e} : A_i \to A_j$.

[ Note: when representing a diagram in pictorial form, we often remove the arrows which are implied, e.g. the arrow corresponding to $\beta_{35}\circ \beta_{13}$ is not drawn. In particular, identity maps are not drawn. ]

In particular, if J is obtained from an index set I, a diagram of type J is just a collection of objects $(A_i)$ indexed by $i\in I$ (with the identity arrows mapped to the identity morphisms).

Definition.

Let $D : J\to \mathcal C$ be a diagram, written as

$\left((A_i)_{i\in J}, (\beta_{e}: A_i \to A_j)_{(e : i\to j)}\right)$.

The colimit of this diagram comprises of the tuple

$(A, (\epsilon_i : A_i\to A)_{i\in J})$

where $A = \mathrm{colim }_{i \in J} A_i \in \mathcal C$ is an object, $\epsilon_i$ is a morphism for each $i\in J$, such that for any arrow $e: i\to j$, we have $\epsilon_j \circ \beta_e = \epsilon_i$ as morphisms $A \to A_j$.

We require the following universal property to hold: for any tuple

$(B, (\alpha_i : A_i \to B)_{i\in I})$

where $B\in \mathcal C$ is an object, $\alpha_i$ is a morphism for each $i\in J$, such that for any arrow $e:i\to j$ in $J$ we have $\alpha_j \circ\beta_e = \alpha_i$, there is a unique morphism $f:A\to B$ in $\mathcal C$ such that

$f\circ \epsilon_i = \alpha_i \text{ for each } i\in J$.

Clearly, the colimit is unique up to unique isomorphism if it exists.

### Example 1: Coproducts

If J is obtained from an index set I, the resulting colimit is the coproduct $\coprod_{i\in I} A_i$.

### Example 2: Pushout

If J is the following index category, the resulting colimit is the pushout.

### Example 3: Coequalizers

Definition.

The coequalizer of $\beta_1, \beta_2 :A\to B$ in a category $\mathcal C$ is the colimit of the following diagram.

This is a pair $(C, \epsilon : B\to C)$ where $C\in \mathcal C$ is an object such that $\epsilon \circ \beta_1 = \epsilon\circ \beta_2$ and, for any pair $(D, \alpha : B\to D)$ satisfying $\alpha\circ \beta_1 = \alpha\circ \beta_2$, there exists a unique $f : C \to D$ such that $f\circ \epsilon = \alpha$.

Exercise C

1. Describe the colimits for each of the following two diagrams.

2. Compute the coequalizer in the categories $\mathbf{Set}$, $\mathbf{Top}$ and $\mathbf{Gp}$.

3. Let $S\subseteq A$ be a multiplicative subset and M an A-module. Prove that we have an isomorphism of A-modules:

$\mathrm{colim}_{f\in S} M_f \cong M_S$,

with morphisms $M_f \to M_g$ defined as follows. If g = af we take the canonical map $M_f \to M_g$ via $\frac m f \mapsto \frac{am}g$. Otherwisee there is no map between $M_f$ and $M_g$. Prove also that we get a ring isomorphism $\mathrm{colim}_{f\in S} A_f \cong A_S$.

In particular for any prime $\mathfrak p \subset A$ we have

$\mathrm{colim}_{f\in A-\mathfrak p} M_f \cong M_{\mathfrak p}, \quad \mathrm{colim}_{f\in A-\mathfrak p} A_f \stackrel{\text{rings}}\cong A_S$.

## Note

For readers who are unfamiliar with the set-theoretic notion of class and set, try not to fret too much over this. Roughly, a set is a class which is “not too huge”. If we allow arbitrary collections to be sets, then Russell’s paradox occurs (where one can take the set of all sets not containing themselves). A common way out of this paradox is to restrict the type of collections which can be considered as sets.

As a very rough guide, in Zermelo-Fraenkel set theory, a collection is a set if it can be “built from the set of natural numbers ℕ”. Thus the collection of real numbers ℝ is a set because its cardinality is the same as the power set of ℕ. Similarly, the set of open subsets of ℝ forms a set because it is a subset of the power set of ℝ, so we can do topology on ℝ. On the other hand, the collection of all groups does not form a set.