Ideal quotient

Template:Refimprove Template:Distinguish In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set

<math>(I : J) = \{r \in R \mid rJ \subseteq I\}</math>

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because <math>KJ \subseteq I</math> if and only if <math>K \subseteq (I : J)</math>. The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).

(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.

PropertiesEdit

The ideal quotient satisfies the following properties:

<math>(I :J)=\mathrm{Ann}_R((J+I)/I)</math> as <math>R</math>-modules, where <math>\mathrm{Ann}_R(M)</math> denotes the annihilator of <math>M</math> as an <math>R</math>-module.
<math>J \subseteq I \Leftrightarrow (I : J) = R</math> (in particular, <math>(I : I) = (R : I) = (I : 0) = R</math>)
<math>(I : R) = I</math>
<math>(I : (JK)) = ((I : J) : K)</math>
<math>(I : (J + K)) = (I : J) \cap (I : K)</math>
<math>((I \cap J) : K) = (I : K) \cap (J : K)</math>
<math>(I : (r)) = \frac{1}{r}(I \cap (r))</math> (as long as R is an integral domain)

Calculating the quotientEdit

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f₁, f₂, f₃) and J = (g₁, g₂) are ideals in k[x₁, ..., x_n], then

<math>I : J = (I : (g_1)) \cap (I : (g_2)) = \left(\frac{1}{g_1}(I \cap (g_1))\right) \cap \left(\frac{1}{g_2}(I \cap (g_2))\right)</math>

Then elimination theory can be used to calculate the intersection of I with (g₁) and (g₂):

Calculate a Gröbner basis for <math>tI+(1-t)(g_1)</math> with respect to lexicographic order. Then the basis functions which have no t in them generate <math>I \cap (g_1)</math>.

Geometric interpretationEdit

The ideal quotient corresponds to set difference in algebraic geometry.<ref>Template:Cite book, p.195</ref> More precisely,

If W is an affine variety (not necessarily irreducible) and V is a subset of the affine space (not necessarily a variety), then

<math>I(V) : I(W) = I(V \setminus W)</math>

where <math>I(\bullet)</math> denotes the taking of the ideal associated to a subset.

If I and J are ideals in k[x₁, ..., x_n], with k an algebraically closed field and I radical then

<math>Z(I : J) = \mathrm{cl}(Z(I) \setminus Z(J))</math>

where <math>\mathrm{cl}(\bullet)</math> denotes the Zariski closure, and <math>Z(\bullet)</math> denotes the taking of the variety defined by an ideal. If I is not radical, then the same property holds if we saturate the ideal J:

<math>Z(I : J^{\infty}) = \mathrm{cl}(Z(I) \setminus Z(J))</math>

where <math>(I : J^\infty )= \cup_{n \geq 1} (I:J^n)</math>.

ExamplesEdit

In <math>\mathbb{Z}</math> we have <math>((6):(2)) = (3)</math>.
In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal <math>I</math> of an integral domain <math>R</math> is given by the ideal quotient <math>((1):I) = I^{-1}</math>.
One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let <math>I = (xyz), J = (xy)</math> in <math>\mathbb{C}[x,y,z]</math> be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in <math>\mathbb{A}^3_\mathbb{C}</math>. Then, the ideal quotient <math>(I:J) = (z)</math> is the ideal of the z-plane in <math>\mathbb{A}^3_\mathbb{C}</math>. This shows how the ideal quotient can be used to "delete" irreducible subschemes.
A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient <math>((x^4y^3):(x^2y^2)) = (x^2y)</math>, showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal <math>I \subset R[x_0,\ldots,x_n]</math> the saturation of <math>I</math> is defined as the ideal quotient <math>(I: \mathfrak{m}^\infty) = \cup_{i \geq 1} (I:\mathfrak{m}^i)</math> where <math>\mathfrak{m} = (x_0,\ldots,x_n) \subset R[x_0,\ldots, x_n]</math>. It is a theorem that the set of saturated ideals of <math>R[x_0,\ldots, x_n]</math> contained in <math>\mathfrak{m}</math> is in bijection with the set of projective subschemes in <math>\mathbb{P}^n_R</math>.<ref>Template:Cite book</ref> This shows us that <math>(x^4 + y^4 + z^4)\mathfrak{m}^k</math> defines the same projective curve as <math>(x^4 + y^4 + z^4)</math> in <math>\mathbb{P}^2_\mathbb{C}</math>.

NotesEdit

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