# 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.