Ideal theory

In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930.

This article is about the mathematical theory. For the usage in political philosophy, see Ideal theory (politics).

The ideal theory in question had been based on elimination theory, but in line with David Hilbert's taste moved away from algorithmic methods. Gröbner basis theory has now reversed the trend, for computer algebra.

The importance of the idea of a module, more general than an ideal, probably led to the perception that ideal theory was too narrow a description. Valuation theory, too, was an important technical extension, and was used by Helmut Hasse and Oscar Zariski. Bourbaki used commutative algebra; sometimes local algebra is applied to the theory of local rings. Douglas Northcott's 1953 Cambridge Tract Ideal Theory (reissued 2004 under the same title) was one of the final appearances of the name.

Topology determined by an ideal

Let R be a ring and M an R-module. Then each ideal ${\displaystyle {\mathfrak {a}}}$ of R determines a topology on M called the ${\displaystyle {\mathfrak {a}}}$-adic topology such that a subset U of M is open if and only if for each x in U there exists a positive integer n such that

${\displaystyle x+{\mathfrak {a}}^{n}M\subset U.}$

With respect to this ${\displaystyle {\mathfrak {a}}}$-adic topology, ${\displaystyle \{x+{\mathfrak {a}}^{n}M\}_{n}}$ is a basis of neighbourhoods of ${\displaystyle x}$ and makes the module operations continuous; in particular, ${\displaystyle M}$ is a possibly non-Hausdorff topological group. Also, M is a Hausdorff topological space if and only if ${\textstyle \bigcap _{n>0}{\mathfrak {a}}^{n}M=0.}$ Moreover, when ${\displaystyle M}$ is Hausdorff, the topology is the same as the metric space topology given by defining the distance function: ${\displaystyle d(x,y)=2^{-n}}$ for ${\displaystyle x\neq y}$, where ${\displaystyle n}$ is an integer such that ${\displaystyle x-y\in {\mathfrak {a}}^{n}M-{\mathfrak {a}}^{n+1}M}$.

Given a submodule N of M, the ${\displaystyle {\mathfrak {a}}}$-closure of N in M is equal to ${\textstyle \bigcap _{n>0}(N+{\mathfrak {a}}^{n}M)}$, as shown easily.

Now, a priori, on a submodule N of M, there are two natural ${\displaystyle {\mathfrak {a}}}$-topologies: the subspace topology induced by the ${\displaystyle {\mathfrak {a}}}$-adic topology on M and the ${\displaystyle {\mathfrak {a}}}$-adic topology on N. However, when ${\displaystyle R}$ is Noetherian and ${\displaystyle M}$ is finite over it, those two topologies coincide as a consequence of the Artin–Rees lemma.

When ${\displaystyle M}$ is Hausdorff, ${\displaystyle M}$ can be completed as a metric space; the resulting space is denoted by ${\displaystyle {\widehat {M}}}$ and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to):

${\displaystyle {\widehat {M}}=\varprojlim M/{\mathfrak {a}}^{n}M}$

where the right-hand side is the completion of the module ${\displaystyle M}$ with respect to ${\displaystyle {\mathfrak {a}}}$.

Example: Let ${\displaystyle R=k[x_{1},\dots ,x_{n}]}$ be a polynomial ring over a field and ${\displaystyle {\mathfrak {a}}=(x_{1},\dots ,x_{n})}$ the maximal ideal. Then ${\displaystyle {\widehat {R}}=k[\![x_{1},\dots ,x_{n}]\!]}$ is a formal power series ring.

R is called a Zariski ring with respect to ${\displaystyle {\mathfrak {a}}}$ if every ideal in R is ${\displaystyle {\mathfrak {a}}}$-closed. There is a characterization:

R is a Zariski ring with respect to ${\displaystyle {\mathfrak {a}}}$ if and only if ${\displaystyle {\mathfrak {a}}}$ is contained in the Jacobson radical of R.

In particular a Noetherian local ring is a Zariski ring with respect to the maximal ideal.

System of parameters

A system of parameters for a local Noetherian ring of Krull dimension d with maximal ideal m is a set of elements x₁, ..., x_d that satisfies any of the following equivalent conditions:

1. m is a minimal prime over (x₁, ..., x_d).
2. The radical of (x₁, ..., x_d) is m.
3. Some power of m is contained in (x₁, ..., x_d).
4. (x₁, ..., x_d) is m-primary.

Every local Noetherian ring admits a system of parameters.

It is not possible for fewer than d elements to generate an ideal whose radical is m because then the dimension of R would be less than d.

If M is a k-dimensional module over a local ring, then x₁, ..., x_k is a system of parameters for M if the length of M / (x₁, ..., x_k)M is finite.

Reduction theory

The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extract detailed information about the behaviors of blow-ups.

Given ideals J ⊂ I in a ring R, the ideal J is said to be a reduction of I if there is some integer m > 0 such that ${\displaystyle JI^{m}=I^{m+1}}$.[1] For such ideals, immediately from the definition, the following hold:

• For any k, ${\displaystyle J^{k}I^{m}=J^{k-1}I^{m+1}=\cdots =I^{m+k}}$.
• J and I have the same radical and the same set of minimal prime ideals over them[2] (the converse is false).

If R is a Noetherian ring, then J is a reduction of I if and only if the Rees algebra R[It] is finite over R[Jt].[3] (This is the reason for the relation to a blow up.)

A closely related notion is that of analytic spread. By definition, the fiber cone ring of a Noetherian local ring (R, ${\displaystyle {\mathfrak {m}}}$) along an ideal I is

${\displaystyle {\mathcal {F}}_{I}(R)=R[It]\otimes _{R}\kappa ({\mathfrak {m}})\simeq \bigoplus _{n=0}^{\infty }I^{n}/{\mathfrak {m}}I^{n}}$.

The Krull dimension of ${\displaystyle {\mathcal {F}}_{I}(R)}$ is called the analytic spread of I. Given a reduction ${\displaystyle J\subset I}$, the minimum number of generators of J is at least the analytic spread of I.[4] Also, a partial converse holds for infinite fields: if ${\displaystyle R/{\mathfrak {m}}}$ is infinite and if the integer ${\displaystyle \ell }$ is the analytic spread of I, then each reduction of I contains a reduction generated by ${\displaystyle \ell }$ elements.[5]

Local cohomology in ideal theory

Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.

Let ${\displaystyle M}$ be a module over a ring ${\displaystyle R}$ and ${\displaystyle I}$ an ideal. Then ${\displaystyle M}$ determines the sheaf ${\displaystyle {\widetilde {M}}}$ on ${\displaystyle Y=\operatorname {Spec} (R)-V(I)}$ (the restriction to Y of the sheaf associated to M). Unwinding the definition, one sees:

${\displaystyle \Gamma _{I}(M):=\Gamma (Y,{\widetilde {M}})=\varinjlim \operatorname {Hom} (I^{n},M)}$.

Here, ${\displaystyle \Gamma _{I}(M)}$ is called the ideal transform of ${\displaystyle M}$ with respect to ${\displaystyle I}$.[6]

References

1. Huneke & Swanson 2006, Definition 1.2.1
2. Huneke & Swanson 2006, Lemma 8.1.10
3. Huneke & Swanson 2006, Theorem 8.2.1.
4. Huneke & Swanson 2006, Corollary 8.2.5.
5. Huneke & Swanson 2006, Proposition 8.3.7
6. Eisenbud 2005, Appendix 10B. harvnb error: no target: CITEREFEisenbud2005 (help)

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
• Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432