# Commutative Algebra, Notes 6: Valuations

**Definition 1.** Let be a field. An **absolute value** on is a map such that for all

- and iff

We will call the pair an **absolute field**. The absolute value that sends all nonzero elements to is called the trivial absolute value. One can also verify that the function defines a metric on Two absolute values are **dependent** if their induced topologies are the same, and **independent** otherwise. Note that

and

Hence and thus since Similarly

and

**Proposition 2.** Let and be two nontrivial absolute values on Then and are dependent iff If they are dependent, then there exist some such that

**Theorem 3. (Approximation Theorem) (Artin-Whaples)**. Let be a field and be nontrivial pairwise independent absolute values on Then there exist elements such that for any there exists an such that

for all

An absolute field is **complete** if all of its Cauchy sequences converge to a point in the field. One can show that each absolute field has a unique completion field up to isometry.

**Definition 4.** An abelian group is an **ordered group** if it has a partial ordering such that for all

**Proposition 5.** is an ordered group iff for a multiplicative subset where for all and

For an ordered group we will hereafter use to denote where and for all

**Definition 6.** Let be a field and be an ordered group. A **valuation** on is a map such that

- iff

We will call the triple a **valuation field**. The image of on is an ordered subgroup of Two valuations are **equivalent** if there is an isomorphism that respects order and value. One can also verify that and that if then

**Definition 7.** A subring of a field is a **valuation ring** if for all or

**Proposition 8.** If is a valuation ring, then it induces a valuation field.

*Proof.* Note that is a local ring, for, if are not units in then if we have

So if is a unit in , then which is a contradiction. Hence is not a unit. Furthermore if is not a unit, then for all we have that is not a unit. So the nonunits form an ideal in which is necessarily maximal (and uniquely so). So is a local ring. Let us denote this ideal by Thus we can write

where is the group of units of We now give a valuation. We will define our group as the quotient group In turn, we define with When then define iff and iff Observe that

and

Hence is a valuation field.

We thus hereafter refer to a *valuation ring* as the restriction of the induced valuation field to the ring.

**Proposition 9.** Let be fields where is a valuation field. Then there exists an extension making a valuation field.

*Proof Idea.* Let and We will call the **valuation ring** of One first takes the morphism and extends it to a valuation ring in One can construct an order preserving monomorphism

where is the maximal ideal in We then define as before, which is seen to agree with

**Proposition 10.** Let be fields and Let be a -valued valuation field and be the extension group to the induced valuation on Then

We will call the **ramification index** of in

**Definition 11.** A valuation is **discrete** if its codomain is a cyclic group.

It turns out that if is a valuation field and is the maximal ideal of its valuation ring, then there exists an element such that is a generator of Such an element is called a **local parameter** of One can also show that is a principal ideal generated by Moreover every element can be written as

for some unit integer is called the **order** of at If we say has a **zero** of order and if then we say has a **pole** of order

**Proposition 12.** Let be fields where is a finite extension of Further suppose that is a complete discrete valuation field (i.e. induced metric space is complete) and that and are the corresponding valuation rings and maximal ideals after extending the valuation. Then

[1] Lang, Serge. *Algebra*. Revised Third Edition. Springer-Verlag. 2000.

# Commutative Algebra, Notes 5: Topologies

**Topology Induced by Filtration:**

Let be a descending filtration of a group such that for all Consider the collection of all cosets of subgroups in the filtration together with the empty set; we will denote this collection by The fact that for subgroups that is linearly ordered (which, together with the first fact, gives (where without loss of generality ) and ), and that is maximal in the collection of cosets (which, with Zorn’s lemma, gives closure under abritrary unions) makes a topology on . In particular, group multiplication can be shown to be continuous with respect to this topology–making a topological group. We will call this the **filtrated topology (or topology induced by filtration) of** by

Hence (linearly ordered) filtrations on a structure of “at least group-type” induce a topology. We can also induce gradings from filtrations of groups with the assumption that elements of the filtration are normal in We then define

in the same way we previously did for ideals in a ring. As a partition, a grading can also induce a topology, where the open sets are generated by elements of the partition. For example, in the grading of a group, let open sets be terms in the sum. Note that unions of terms are terms, the empty set can trivially be considered a term, and finite intersections of terms are terms.

In the case of an ideal of a ring the topology induced by the filtration

is called the **Krull topology** (or –**adic topology**) of by

**Spectral/Zariski Topology**:

Let be the set of prime ideals of a unital ring If is an ideal of let

Note that and Hence the ‘s form a basis of closed sets of The corresponding topology is called the **spectral (or Zariski) topology** of which we also denote by

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

# Commutative Algebra, Notes 4: Graduations and Filtrations

**Graduations:**

**Definition 1.** Let be a monoid. An –**graded ring** is a ring where

and where are abelian groups. A **graded ring** will mean an -graded ring. is an –**graded** –**algebra** if it is -graded as a ring. If is a graded -algebra and is graded, then we say is **graded with respect to** if and

**Example 2.** The tensor algebra of a module is trivially graded with its concatenation product. A group ring is a -graded ring, which can be seen with its decomposition A -graded ring (algebra) is called a **super ring** (**super algebra**). Note that -graded rings and graded rings induce a grading as follows

Note we could slightly generalize the above definition of being graded with respect to as follows: suppose is -graded and is -graded for monoids and that there exists a monoid homomorphism Then we can say that is an –**graded** –**algebra** if

and

A similar definition exists for an -graded -module. Note an -graded -algebra has an -grading where is the trvial monoid grading and is of course the trivial morphism. An element of an -graded -algebra is called **homogeneous (of order** **)** if it has the form for some Let be -graded -algebras and be an -algebra homomorphism. Then is called a **graded homomorphism** if In more generality, we could have be an -graded -algebra together with a module homomorphism and require

**Definition 3.** A subset is **homogeneous** if for every element of the component elements (which are homogeneous) are in If is an ideal, then we call it a **homogeneous (or graded) ideal**. Note it has a grading as it can be written as a direct sum of the ideals generated by the homogeneous elements.

**Proposition 4.** Let be an -graded -algebra and be a homogeneous ideal in Then is an -graded -algebra.

*Proof.* Since is homogeneous, it is graded (since elements give component elements). So we have

Cosets of in the quotient thus have the form

which gives us a decomposition. Hence

**Filtrations:**

**Definition 5.** Let be a poset. A subset is called a **filter** if the following hold

- with and
- If and then

**Definition 6.** Let be a structure and be a filter. An **descending (ascending) filtration** on ** with respect to** is a collection of substructures of such that (). A **filtration on** will mean a filtration with respect to the filter (either ascending/descending). We can similarly define filtrations of modules and filtrations of modules that respect the filtration of their ring.

**Definition 7.** If is an ideal in and is an -module with a descending filtration, then the filtration is called an –**filtration** if for all It is called –**stable** if for all for some

Hence we can view multiplication by as increasing/decreasing the degree (depending upon preferred terminology) of elements in

**Induced Graduations:**

Let be an ideal of Then has an -filtration:

We can define the **Rees algebra** (which Lang calls the “first associated graded ring”) as

This is clearly a graded -algebra. We could also consider

This is easily verified as a graded -algebra with a product defined componentwise:

Definition 8. Let be a graded -module with grading We define the Hilbert polynomial by We define the of as

[1] Lang, Serge. *Algebra*. Revised Third Edition. Springer-Verlag. 2000.

[2] Dummit, David and Richard Foote. *Abstract Algebra.* Third Edition. John Wiley and Sons, Inc. 2004.

# Commutative Algebra, Notes 3: Local Rings

**Definition 1.** The **Jacobson radical** of a commutative ring is defined as the intersection of all maximal ideals of

**Proposition 2.** Let be a ring and be the set of all nonunits of Then the following are equivalent.

- is an ideal.
- has a unique maximal ideal.

*Proof.* If is an ideal, then it is maximal since any bigger ideal would contain a unit, and hence equal the whole ring. Uniqueness follows by adjoining nonunits. Conversely, note that the product of a unit and a nonunit, as well as the product of two nonunits, must be a nonunit. If has a unique maximal ideal, then note that if and is a unit, then there is a unit such that Hence the ideals generated by and are relatively prime. Hence no maximal ideal can contain both and so uniqueness is violated.

**Definition 3.** is a **local ring** if it satisfies the above conditions.

If is a local ring, then of course is the Jacobson radical of Note that if is a prime ideal in then its complement is a multiplicative subset.

**Definition 4.** We define We may call the **localization of** **at** We define the **globalization of** as the ring

where is a maximal ideal in

So if is a local ring, then its globalization is just since the one factor would invert already invertible elements.

**Proposition 5.** The set of maximal ideals of is finite iff is a finite direct sum of fields (where is the Jacobson radical of ).

*Proof.* is a field for each maximal ideal in Moreover if then maps onto via the canonical mapping. The kernel is clearly so we have the quotient as a finite sum of fields. Conversely, if is a finite direct sum of fields, then it has a finite number of ideals, and hence a finite number of maximal ideals. Since each maximal ideal in contains these are just the preimages of the maximal ideals of under the map of which there are finitely many.

**Definition 6.** A ring is **semilocal** if it is a finite direct sum of local rings.

Hence if then is semilocal, as fields are local. Now suppose is a semilocal ring. So each is local, with a maximal ideal We have a canonical epimorphism

whose kernel is Hence if is semilocal, then is semilocal (and in particular semisimple (as an -module), which is the traditional definition of being semilocal).

**Proposition 7.** Let be a finite set of prime ideals of Let us define

Then if is semilocal (i.e. is semilocal in the traditional sense), and has maximal ideals where are the maximal elements of

*Proof.* A maximal ideal in must be contained in otherwise it contains a unit. Hence it must be contained in one of the and hence it must be a So are the maximal ideals of Hence by Proposition 5 is semilocal.

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

# 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.