Primary decomposition

The primary decomposition is a term from commutative algebra . In a primary decomposition, sub-modules are represented as the average of primary sub- modules. Existence and uniqueness can be proven under certain conditions. The primary decomposition of an ideal is a generalization of the decomposition of a number into its prime numbers . On the other hand, the primary decomposition is the algebraic basis for decomposing an algebraic variety into its irreducible components.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

definition

If a sub-module of a module is above a ring , a primary decomposition of is a representation of as an average: ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle U}$

${\displaystyle U=P_{1}\cap \dots \cap P_{n}\quad (n\geq 1)}$

of -primary sub-modules . (They are prime ideals of the ring .) ${\displaystyle p_{i}}$ ${\displaystyle P_{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle R}$

The primary decomposition is called reduced if the following applies:

1. For is ${\displaystyle i\neq j}$ ${\displaystyle p_{i}\neq p_{j}}$
2. ${\displaystyle \bigcap _{j\neq i}P_{j}\nsubseteq P_{i}}$

In the case of a reduced primary decomposition, they are also referred to as primary components . ${\displaystyle P_{i}}$

existence

If a finitely generated module is over a Noetherian ring , then every real sub-module of has a decomposition into irreducible sub- modules due to Noetherian induction . Since irreducible sub-modules of finitely generated modules over a Noetherian ring are already primary, the decomposition into irreducible sub-modules is already a primary decomposition. If one now replaces all components that are primary to the same prime ideal with their intersection, which is itself primary, and omits all components that are not required, one obtains a reduced primary decomposition. In particular, every ideal has as a sub-module of a decomposition into primary ideals. ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle I}$ ${\displaystyle R}$

Uniqueness

Is a sub-module of a module over a Noetherian ring and ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle U=P_{1}\cap \dots \cap \ P_{n}}$

a reduced primary decomposition into -primary sub-modules, so is ${\displaystyle p_{i}}$

${\displaystyle \{p_{1},\dots ,p_{n}\}=Ass(M/U)}$

${\displaystyle Ass(M/U)}$is the set of associated prime ideals of . In particular, the set of prime ideals occurring with a reduced primary decomposition is clearly defined. ${\displaystyle M/U}$

If there is a minimal element of the set , then is equal . The minimal elements of corresponding primary components are by and defined uniquely. ${\displaystyle p_{i}}$ ${\displaystyle \{p_{1},\dots ,p_{n}\}}$ ${\displaystyle P_{i}}$ ${\displaystyle R/p_{i}\cdot U}$ ${\displaystyle Ass(M/U)}$ ${\displaystyle M}$ ${\displaystyle U}$

If a primary component does not belong to a minimal element of , then an embedded primary component is called. These are not necessarily unique (see below). ${\displaystyle P_{i}}$ ${\displaystyle Ass(M/U)}$ ${\displaystyle P_{i}}$

Lasker-Noether theorem

The statements about the existence and uniqueness of the primary decomposition in Noether's rings are also called Lasker-Noether's theorem. It reads

Every ideal of a Noetherian ring permits a reduced primary decomposition . The Primradikale of are uniquely determined; it is precisely the prime ideals of form , with all elements running through. ${\displaystyle I\subset R}$ ${\displaystyle R}$ ${\displaystyle I=P_{1}\cap \ldots \cap P_{n}}$ ${\displaystyle p_{i}}$ ${\displaystyle P_{i}}$ ${\displaystyle I:(x):=\{r\in R|\ rx\in I\}}$ ${\displaystyle x}$ ${\displaystyle R}$

This theorem was first proven by Emanuel Lasker , who is best known as the world chess champion , for polynomial rings over a body. Emmy Noether then recognized that the arguments can be traced back to the ascending chain condition and therefore apply more generally to Noetherian rings. That explains the naming of this sentence. The generalization to finitely generated modules over a Noetherian ring is then routine. ${\displaystyle K[X_{1},\ldots ,X_{n}]}$

sentences

Is a multiplicatively closed subset of a ring and ${\displaystyle S}$ ${\displaystyle R}$

${\displaystyle U=\bigcap _{1\leq i\leq n}P_{i}}$

a reduced primary decomposition of a sub-module with -primary sub-modules of , so is ${\displaystyle U\subset M}$ ${\displaystyle p_{i}}$ ${\displaystyle P_{i}}$ ${\displaystyle M}$

${\displaystyle U_{S}=\bigcap _{p_{i}\cap S=\emptyset }(P_{i})_{S}}$

a reduced primary representation of . ${\displaystyle U_{S}}$

Examples

In whole numbers

For example, is in whole numbers

${\displaystyle k=p_{1}^{n_{1}}\cdot \dots \cdot p_{n}^{n_{i}}}$

with prime numbers , the primary decomposition is the main ideal generated by ${\displaystyle p_{i}}$ ${\displaystyle k}$

${\displaystyle (k)=(p_{1}^{n_{1}})\cap \dots \cap (p_{n}^{n_{i}})}$.

In a coordinate ring

If there is a body , then it has the ideal ${\displaystyle K}$

${\displaystyle I:=(x^{2},xy)\subset K[x,y]}$

the primary decompositions:

${\displaystyle I=(x)\cap \ (x,y)^{2}=(x)\cap (x^{2},y)}$

${\displaystyle (x,y)^{2}}$is primary as the power of a maximal ideal ; in the ring every zero divisor is nilpotent , so the ideal is also primary. Both and are -primary. This example shows that the primary decomposition itself is not unique, but the associated prime ideals are. ${\displaystyle K[x,y]/(x^{2},y)}$ ${\displaystyle (x^{2},y)}$ ${\displaystyle (x,y)^{2}}$ ${\displaystyle (x^{2},y)}$ ${\displaystyle (x,y)}$

literature

• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah, Macdonald: Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9

Individual evidence

1. ↑ Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, sentence C.32., P. 235
2. ↑ Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, sentence C.30., P. 235
3. ↑ Ernst Kunz: Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, Korollar C.28., P. 234
4. ^ O. Zariski, P. Samuel: Commutative Algebra I , Springer-Verlag (1975), ISBN 3-540-90089-6