Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 08.1   Let $R$ be a ring. A cochain complex of $R$ -modules is a sequence of $R$ -modules

$$C^\bullet: \dots \overset{d^{n-1}}{\to} C^n \overset{d^{n}}{\to} C^{n+1} \overset{d^{n+1}}{\to} \dots$$

such that $d^{n}\circ d^{n-1}=0$ . The $n$ -th cohomology of the cochain complex is defined to be the $R$ -module

$$H^{n}(C^\bullet)=\operatorname{Ker}(d^{n})/\operatorname{Image}(d^{n-1}).$$

Elements of $\operatorname{Ker}(d^{n})$ (respectively, $\operatorname{Image}(d^{n-1})$ ) are often referred to as cocycles (respectively, coboundaries).

DEFINITION 08.2   Let $R$ be a ring.

1. An $R$ -module $I$ is said to be injective if it satisfies the following condition: For any $R$ -module homomorphism $f:M \to I$ and for any monic $R$ -module homomorphism $\iota:N \to M$ , $f$ ``extends'' to an $R$ -module homomorphism $\hat f: M\to I$ .
   $$\begin{CD} M @>\hat f >>I \\ @A \iota AA @\vert \\ N @>f >> I \end{CD}$$

2. A $R$ -module $P$ is said to be projective if it satisfies the following condition: For any $R$ -module homomorphism $f: P \to N$ and for any epic $R$ -module homomorphism $\pi:M \to N$ , $f$ ``lifts'' to a morphism $\hat f: M\to I$ .
   $$\begin{CD} P @>\hat f >>M \\ @\vert @V\pi VV \\ P @>f >> N \end{CD}$$

EXERCISE 08.1   Let $R$ be a ring. Let

$$0\to M_1 \to M_2 \to M_3 \to 0$$

be an exact sequence of $R$ -modules. Assume furthermore that $M_3$ is projective. Then show that the sequence

$$0 \to \operatorname{Hom}_R (N,M_1) \overset{\operatorname{Hom}_R(... ..._2) \overset{\operatorname{Hom}_R(N,g)}{\to} \operatorname{Hom}_R(N,M_3) \to 0$$

is exact.

LEMMA 08.3   Let $R$ be a (unital associative but not necessarily commutative) ring. Then for any $R$ -module $M$ , the following conditions are equivalent.

1. $M$ is a direct summand of free modules.
2. $M$ is projective

COROLLARY 08.4   For any ring $R$ , the category $(R \operatorname{-modules})$ of $R$ -modules have enough projectives. That means, for any object $M \in (R\operatorname{-modules})$ , there exists a projective object $P$ and a surjective morphism $f: P \to M$ .

DEFINITION 08.5   Let $R$ be a commutative ring. We assume $R$ is a domain (that means, $R$ has no zero-divisors except for 0 .)

An $R$ -module $M$ is said to be divisible if for any $r \in R\setminus \{0\}$ , the multplication map

$$M \overset{r \times }{\to} M$$

is surjective.

DEFINITION 08.6   Let $R$ be a commutative ring. We assume $R$ is a domain (that means, $R$ has no zero-divisors except for 0 .)

An $R$ -module $M$ is said to be divisible if for any $r \in R\setminus \{0\}$ , the multplication map

$$M \overset{r \times }{\to} M$$

is epic.

LEMMA 08.7   Let $R$ be a (commutative) principal ideal domain (PID). Then an $R$ -module $I$ is injective if and only if it is divisible.

PROPOSITION 08.8   For any (not necessarily commutative) ring $R$ , the category $(R \operatorname{-modules})$ of $R$ -modules has enough injectives. That means, for any object $M \in (R\operatorname{-modules})$ , there exists an injective object $I$ and an monic morphism $f:M \to I$ .

A bit of category theory:

DEFINITION 08.9   A category $\mathcal{C}$ is a collection of the following data

1. A collection $\operatorname{Ob}(\mathcal{C})$ of objects of $\mathcal{C}$ .
2. For each pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$ , a set
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)$$
   of morphisms.
3. For each triple of objects $X,Y,Z \in \operatorname{Ob}(\mathcal{C})$ , a map(``composition (rule)'')
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)\times \operatorname{Hom}_{\mathcal{C}}(Y,Z)\to \operatorname{Hom}_{\mathcal{C}}(X,Z)$$

satisfying the following axioms

1. $\operatorname{Hom}(X,Y)\cap \operatorname{Hom}(Z,W) =\emptyset$ unless $(X,Y)=(Z,W)$ .
2. (Existence of an identity) For any $X\in \operatorname{Ob}(\mathcal{C})$ , there exists an element $\operatorname{id}_X\in \operatorname{Hom}(X,X)$ such that
   $$\operatorname{id}_X \circ f=f,\quad g \circ \operatorname{id}_X=g$$
   holds for any $f \in \operatorname{Hom}(S,X), g \in \operatorname{Hom}(X,T)$ ( $\forall S,T \in \operatorname{Ob}(\mathcal{C})$ ).
3. (Associativity) For any objects $X,Y,Z,W \in \operatorname{Ob}(\mathcal{C})$ , and for any morphisms $f\in \operatorname{Hom}(X,Y), g\in \operatorname{Hom}(Y,Z), h \in \operatorname{Hom}(Z,W)$ , we have
   $$(f\circ g)\circ h =f\circ (g\circ h).$$

Morphisms are the basic actor/actoress in category theory.

An additive category is a category in which one may ``add'' some morphisms.

DEFINITION 08.10   An additive category $\mathcal{C}$ is said to be abelian if it satisfies the following axioms.

A4-1.

Every morphism $f:X\to Y$ in $\mathcal{C}$ has a kernel $\ker(f):\operatorname{Ker}(f)\to X$ .

A4-2.

Every morphism $f:X\to Y$ in $\mathcal{C}$ has a cokernel $\operatorname{coker}(f):Y\to \operatorname{Coker}(f)$ .

A4-3.

For any given morphism $f:X\to Y$ , we have a suitably defined isomorphism

$$l: \operatorname{Coker}(\ker(f))\cong \operatorname{Ker}(\operatorname{coker}(f))$$

in $\mathcal{C}$ . More precisely, $l$ is a morphism which is defined by the following relations:

$$\ker(\operatorname{coker}(f))\circ \... ...exists \overline{f}),\quad \overline{f}=l \circ \operatorname{coker}(\ker(f)).$$

DEFINITION 08.11   A (covariant) functor $F$ from a category $\mathcal C$ to a category $\mathcal D$ consists of the following data:

1. An function which assigns to each object $C$ of $\mathcal C$ an object $F(C)$ of $\mathcal D$ .
2. An function which assigns to each morphism $f$ of $\mathcal C$ an morphism $F(f)$ of $\mathcal D$ .

The data must satisfy the following axioms:

functor-1.

$F(1_C)=1_{F(C)}$ for any object $C$ of $\mathcal C$ .

functor-2.

$F(f\circ g)=F(f)\circ F(g)$ for any composable morphisms $f,g$ of $\mathcal C$ .

By employing the following axiom instead of the axiom (functor-2) above, we obtain a definition of a contravariant functor:

(functor-$2'$ ) $F(f\circ g)=F(g)\circ F(f)$ for any composable morphisms

DEFINITION 08.12   Let $F: \mathcal{C}_1 \to \mathcal{C}_2$ be a functor between additive categories. We call $F$ additive if for any objects $M, N$ in $\mathcal{C}_1$ ,

$$\operatorname{Hom}(M,N)\to \operatorname{Hom}(F(M),F(N))$$

is additive.

DEFINITION 08.13   Let $F$ be an additive functor from an abelian category $\mathcal{C}_1$ to $\mathcal{C}_2$ .

1. $F$ is said to be left exact (respectively, right exact ) if for any exact sequence
   $$0 \to L\to M\to N\to 0,$$
   the corresponding map
   $$0\to F(L)\to F(M)\to F(N)$$
   (respectively,
   $$F(L)\to F(M)\to F(N) \to 0)$$
   is exact
2. $F$ is said to be exact if it is both left exact and right exact.

LEMMA 08.14   Let $R$ be a (unital associative but not necessarily commutative) ring. Then for any $R$ -module $M$ , the following conditions are equivalent.

1. $M$ is a direct summand of free modules.
2. $M$ is projective

COROLLARY 08.15   For any ring $R$ , the category $(R \operatorname{-modules})$ of $R$ -modules have enough projectives. That means, for any object $M \in (R\operatorname{-modules})$ , there exists a projective object $P$ and a surjective morphism $f: P \to M$ .

DEFINITION 08.16   Let $R$ be a commutative ring. We assume $R$ is a domain (that means, $R$ has no zero-divisors except for 0 .)

An $R$ -module $M$ is said to be divisible if for any $r \in R\setminus \{0\}$ , the multplication map

$$M \overset{r \times }{\to} M$$

is surjective.

DEFINITION 08.17   Let $R$ be a commutative ring. We assume $R$ is a domain (that means, $R$ has no zero-divisors except for 0 .)

An $R$ -module $M$ is said to be divisible if for any $r \in R\setminus \{0\}$ , the multplication map

$$M \overset{r \times }{\to} M$$

is epic.

DEFINITION 08.18   Let $(K^\bullet,d_K)$ , $(L^\bullet, d_L)$ be complexes of objects of an additive category $\mathcal{C}$ .

1. A morphism of complex $u:K^\bullet \to L^\bullet$ is a family
   $$u^j: K^j \to L^j$$
   of morphisms in $\mathcal{C}$ such that $u$ commutes with $d$ . That means,
   $$u^{j+1} \circ d^j_K = d^j_K \circ u^j$$
   holds.
2. A homotopy between two morphisms $u,v: K^\bullet \to L\bullet$ of complexes is a family of morphisms
   $$h^j: K^j \to L^{j-1}$$
   such that $u-v = d \circ h + h \circ d$ holds.

LEMMA 08.19   Let $\mathcal{C}$ be an abelian category that has enough injectives. Then:

1. For any object $M$ in $\mathcal{C}$ , there exists an injective resolution of $M$ . That means, there exists an complex $I^\bullet$ and a morphism $\iota_M:M \to I^0$ such that
   $$% latex2html id marker 1710H^j(I^\bullet) = \begin{cases} ... ...iota_M) &\text{ if } j=0 \\ 0 &\text{ if } j\neq 0 \end{cases}$$

2. For any morphism $f:M\to N$ of $\mathcal{C}$ , and for any injective resolutions $(I^\bullet,\iota_M)$ , $(J^\bullet,\iota_N)$ of $M$ and $N$ (respectively), There exists a morphism $\bar f:I^\bullet \to J^\bullet$ of complexes which commutes with $f$ . Forthermore, if there are two such morphisms $\bar f$ and $f'$ , then the two are homotopic.

DEFINITION 08.20   Let $\mathcal{C}_1$ be an abelian category which has enough injectives. Let $F: \mathcal{C}_1 \to \mathcal{C}_2$ be a left exact functor to an abelian category. Then for any object $M$ of $\mathcal{C}_1$ we take an injective resolution $I^\bullet_M$ of $M$ and define

$$R^i F(M)=H^i(I^\bullet_M).$$

and call it the derived functor of $F$ .

LEMMA 08.21   The derived functor is indeed a functor.