# Commutative Algebra, Notes 2: Localization

We will mostly adopt the Bourbaki definitions for the preliminary material. An ideal in a ring is *prime* if is an integral domain. Two ideals are *relatively prime* if A *multiplicative subset* of a set is a submagma of In a ring, we assume it is a submagma with respect to multiplication of course. We then have the canonical extension of by formal inverses of written (although I will use the notation to reinforce the conceptualization of a ring extension).

Now suppose is an -module. A natural question to ask is “can we induce some -module”? The natural choice will be the -module where we define

for

**Proposition 1.** Let be a ring and be a multiplicative subset. Then there exists a ring and homomorphism such that

- Elements of are invertible in
- For every homomorphism such that elements in are invertible in there exists a unique homomorphism such that

It naturally turns out that and for some Similarly we have:

**Proposition 2.** Let be a ring, be a multiplicative subset, and be an -module. Then there exists an -module and homomorphism such that

- For every the map defined by is bijective.
- For every -module such that the map is bijective for every and if we have a homomorphism there exists a unique homomorphism such that

Here we can show that and is called the **ring of fractions of ** by while often denoted for short, is called the **module of fractions of** by

**Definition 3.** Let be an ideal in We define the **radical** of in as

We call the **nilradical** of is a **reduced ring** if

Let be an -algebra and be a multiplicative subset of Then since is an -module, we can make into an -module

But we can also turn it into an -algebra. We need a homomorphism

But the domain collapses to and then to under the product of So we have the canonical product we need.

[1] Bourbaki, N. *Commutative Algebra, Ch 1-7*. Springer-Verlag. 1989.

# Commutative Algebra, Notes 1: Flat Modules

We will assume all rings are commutative and unital.

**Definition 1**. Let be a ring, a right -module, and a left -module. is **-flat** if for every left -module and monomorphism the homomorphism is injective.

Some basic results can be shown:

**Proposition 2.** If is a right -module and is a left -module such that is -flat, then if we have that is -flat and -flat. Also if is a family of left -modules such that is -flat for all then is -flat.

**Proposition 3.** The following are equivalent for a right -module

- The canonical map is injective (where for some multiplicative subset ).
- is -flat for every left -module
- For every exact sequence of left -modules and homomorphisms

the sequence

is exact.

**Definition 4.** A right -module satisfying the equivalent conditions above is **flat**. (A dual definition exists for left modules)

**Proposition 5.** Let be a right -module, then the following are equivalent.

1. The sequence of left -modules is exact iff

is exact.

2. is flat and for a left -module

3. is flat and for every homomorphism

4. is flat and for every maximal left ideal of

**Definition 6.** A right -module satisfying the above conditions is **faithfully flat**.

Recall that for a subset of an -module the **annihilator of** is defined as

An -module is **faithful** if

**Definition 7.** Let be a left -module and suppose

is an exact sequence such that each is free and finitely generated. Then is said to be **finitely presented**, and the exact sequence is called a **finite presentation of** **of length**

Note that if is a ring homomorphism, then a finite presentation of a left -module induces a finite presentation of where the th term of the first sequence, becomes with (We will consider to be the term of the original sequence)

**Proposition 8.** Let be a left -module.

- If admits a finite presentation, then is finitely generated.
- If is left Noetherian, then if is finitely generated, it admits a finite presentation.
- If is projective and finitely generated, then it admits a finite presentation.

**Definition 9.** Let be a left -module. is **pseudo-coherent** if every finitely generated submodule of is admits a finite presentation. is **coherent** if it is pseudo-coherent and finitely generated.

[1] Bourbaki, N. *Commutative Algebra, Ch 1-7*. Springer-Verlag. 1989.