Modules of fractions, Primary decomposition

Rings of fractions

Rings of fractions, and the associated localization process, correspond to the algebro-geometric picture of studying behavior of an open set or near a point. Using the way we construct from , we can construct equivalence classes of , where , defined as follows:

but this definition only works if is an integral domain. A more general definition of equivalence involves using a multiplicative subset of and defining over :

Let denote the equivalence class of and the set of all equivalence classes. We can endow with a ring structure by defining addition and multiplication as operations on fractions, as in elementry algebra:

We also have the ring homomorphism defined by . It is not injective in general. is termed the ring of fractions of with respect to .

Let be a ring homomorphism where is unit in for all . Then there exists a unique ring homomorphism such that .

The ring and the homomorphism have the following properties:

1. is unit in .
2. for some .
3. Every element of has the form for some .

Conversely, these properties determine uniquely upto isomorphism.

Some examples follow:

1. Let be a prime ideal in . Then is multiplicatively closed and is written as , the localization of at . with form an ideal in . If , and , then is unit. is the only maximal ideal in , and is a local ring. If is the set of prime numbers, then is the set of all rationals with denominators as powers of .
2. is the zero ring .
3. Let . Then we write for in this case. Let be a polynomial ring where is a field, and a prime ideal in . Then, is the ring of all rational functions . Let be a variety defined by , that is to say the set such that whenever . Then, can be identified with the ring of all rational functions in which are defined at almost all points of ; it is the local ring of defined along variety . This is the prototype of local rings, as they arise in algebraic geometry.
4. Let be an ideal in , such that . Then, is multiplicatively closed.

Modules of fractions

The same construction involving can be carried out in modules instead of rings. Let be an -module homomorphism. This gives rise to an -module homomorphism , such that maps to . We have .

preserves exactness, in that if the following sequence is exact at :

then, this sequence is exact at :

This can proved by noticing that if , then so is . Hence, . To prove inclusion in the other direction, let , then , hence in . But , since is an -module homomorphism, hence . Therefore, . Hence, in , we have . Hence, .

If is a submodule of , then the mapping is injective, and is a submodule of .

We have the following properties of fractions of modules, given that are submodules of -module :

1. .
2. .
3. .

The first is obvious. To prove (ii), consider . Then, . Hence, , and therefore, . Consequently, . The reverse inclusion is obvious.

To prove (iii), apply to the exact sequence .

Let be an -module. Then, there exists a unique isomorphism such that:

To prove this, notice that the mapping given by is -bilinear, and by universal property of the tensor product, induces the -homomorphism:

is clearly surjective, and it remains to be proved that it is injective as well. Notice that every element of is of the form . Suppose that , then and hence . Therefore:

Hence, is injective, and therefore an isomorphism.

is a flat -module, and this is immediately obvious.

If are -modules, there is an isomorphism of -modules such that:

In particular, if is a prime ideal:

Local properties

A ring , or a module is said to have a local property if, has has , or has has , for prime ideal . Now, for example, the following statements are equivalent:

1. .
2. , for all prime ideals of .
3. , for all maximal ideals of .

Since (i) (ii) (iii) is obvious, let us prove (i), given (iii). Suppose , then , and let . is an ideal, and therefore must be contained within some maximal ideal . Let . Since , , and hence, is killed by some element of . But this is impossible since .

Let be an -module homomorphism. Then, the following statements are equivalent:

1. is injective.
2. is injective for each prime ideal .
3. is injective for each maximal ideal .

Flatness is a local property. For any -module , the following statements are equivalent:

1. is a flat -module.
2. is flat for each prime ideal .
3. is flat for each maximal ideal .

Primary decomposition

Just as prime ideals are generalizations of primes in , in some sense, powers of primes can be generalized to primary ideals: ideal of is primary if and

In other words, for to be primary,

Clearly, every prime ideal is primary. Also, the contraction of a primary ideal is primary, for if and is primary in , .

Let be a primary ideal in . Then is the smallest prime ideal containing . To prove this, it suffices to show that is prime. Let . Then , which implies that either or ; i.e. or . If , then is said to be -primary.

Examples:

1. and are primary in , where is a prime number.
2. Let and . Then, , in which all zero-divisors are multiples of , hence nilpotent. Hence, is primary and its radical is such that , so that a primary ideal is not necessarily a prime power.
3. Let . is not primary because , but and .

If is maximal, then is primary. In particular, the powers of a maximal ideal are -primary. To prove this, let . The image of in is the nilradical of , and hence only has one prime ideal. Every element of is either unit or nilpotent, so every zero-divisior of is nilpotent.

If are -primary, then so is . To prove this, notice that . Let , . Then, for some , we have and . Hence, since is primary.

Let be -primary, and . Then:

1. If , then the ideal quotient .
2. If then is -primary, and hence .
3. If , then .

Let us prove (iii), since (i) and (ii) are obvious. Let and , and hence, since , . Hence, ; taking radicals, we get . Let so that ; then , hence and .

The primary decomposition of ideal is given by finite intersection of primary ideals:

An ideal is said to be decomposable if such a decomposition exists.

First uniqueness theorem: Let be decomposable, and be its minimal primary decomposition. Let ; then prime ideals are precisely the ones that occur in the set , for , and are hence independent of a particular primary decomposition of . To prove this, let , so that . Hence . Suppose is prime; then , for some . Hence, every prime ideal of the form is one of . Conversely, for each , there exists , , since the decomposition is minimal. Hence, .

Let be a decomposable ideal. Then, any prime ideal contains a minimal prime ideal belonging to , and thus the minimal prime ideals of are exactly the minimal elements in the set of all prime ideals containing .

Let be a minimal primary primary decomposition, and . Then,

In particular, if the zero ideal is decomposable, then the set of zero-divisors of is the union of prime ideals belonging to .

Let be a multiplicatively closed subset of , and let be -primary. Then:

1. If , then .
2. If , then is -primary, and its contraction in is .

Let be a multiplicatively closed subset of , and be the minimal primary decomposition of in . Let , and suppose is numbered so that meets , but not . Then:

Moreover, these are minimal primary decompositions.

The set of prime ideals belonging to is said to be isolated if it satisfies the following condition: if is a prime ideal belonging to , and for some , then .

Second uniqueness theorem: Let be a minimal primary decomposition of ideal , and let be an isolated set of prime ideals of . Then, is independent of the decomposition. In particular, isolated primary components are uniquely determined by .

Integral dependence

Let be a ring, and a subring of . Then, element of is said to be integral over if is the root of the polynomial with coefficients in , that is, if satisfies:

Let . If a rational number is integral over , where have no common factor, we have:

the being rational integers. Hence, divides , , and .

The following are equivalent:

1. is integral over .
2. is a finitely generated -module.
3. is contained in a subring of , such that is a finitely generated -module.
4. There exists a faithful -module which is finitely generated as an -module.

To prove (i) (ii), we have:

for all . Hence, by induction, all positive powers of lie in the -module generated by . Hence, lie in the -module generated by . Hence, is generated by .

C is termed nonsingular if for , so that has a well-defined tangent at every point .

is an example of something that is algebraically closed, but not integrally closed. It can be perturbed as to avoid the unlucky accident of a "missing zero" over .

The going-up theorem

The going-down theorem

Valuations

Chain conditions

Noetherian rings

In a Noetherian ring , every ideal has a primary decomposition.

Artin rings

Discrete valuation rings

is a discrete valuation of , a surjective map so that:

By convention, .

Valuation ring of a discrete valuation is given by .

For valuation ring :

The Nullstellansatz

Variety

Tautologically, ; is a variety.

The Nullstellansatz states that:

1. If then
2. , the radical of the ideal

A variety is irreducible if it cannot be expressed as the union of two proper subvarieties:

; is a prime ideal.

The following reverse-inclusions are obvious:

Zariski topology

Zariski topology is a topology where the only closed sets are the algebraic ones, the zeros of polynomials. The Zariski topology on a variety is Noetherian.

corresponds exactly to the Zariski topology on , :

where .

Localization

is a local ring has a unique maximal ideal at .

, a localization of at p = .

For a general construction, let be a multiplicative set in , a prime ideal so that . Then , where is an equivalence relation.

is an exact functor in that, if and , then and .

Then, for ideal in , , and for any ideal of A, .

These three statements are equivalent:

1. is local if it has a unique maximal ideal .
2. is the unique maximal ideal.
3. If is a maximal ideal and , then is unit.

Support and annihilator

Support of M is defined as . Here , the module of fractions. Assassin . Annihilator of M over A is defined as .

Example: If , then . If , then annihilator .

1. If is generated by a single element , such that , then .
2. If and , then .
3. If and , then .
4. If , then .

In the disjoint union , is termed as the stalk of over .