Ideal quotient

Properties

The ideal quotient satisfies the following properties:

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

Calculating the quotient

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I=(f_1,f_2,f_3) and J=(g_1,g_2) are ideals in \mathbb k[x_1,\ldots,x_n], then :(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)

Then elimination theory can be used to calculate the intersection of I with (g_1) and (g_2): :I \cap (g_1) = tI + (1-t) (g_1) \cap \mathbb k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t) (g_2) \cap \mathbb k[x_1, \dots, x_n]

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

Geometric interpretation

The ideal quotient corresponds to set difference in algebraic geometry. 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

:where I(\bullet) denotes the taking of the ideal associated to a subset.

• If I and J are ideals in \mathbb k[x_1,\ldots,x_n], with \mathbb k an algebraically closed field and I radical then

:where \mathrm{cl}(\bullet) denotes the Zariski closure, and Z(\bullet) 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:

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

Examples

• In \mathbb{Z} we have ((6):(2)) = (3).
• In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal I of an integral domain R is given by the ideal quotient ((1):I) = I^{-1}.
• One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let I = (xyz), J = (xy) in \mathbb{C}[x,y,z] be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in \mathbb{A}^3_\mathbb{C}. Then, the ideal quotient (I:J) = (z) is the ideal of the z-plane in \mathbb{A}^3_\mathbb{C}. 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 ((x^4y^3):(x^2y^2)) = (x^2y), 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 I \subset R[x_0,\ldots,x_n] the saturation of I is defined as the ideal quotient (I: \mathfrak{m}^\infty) = \cup_{i \geq 1} (I:\mathfrak{m}^i) where \mathfrak{m} = (x_0,\ldots,x_n) \subset R[x_0,\ldots, x_n]. It is a theorem that the set of saturated ideals of R[x_0,\ldots, x_n] contained in \mathfrak{m} is in bijection with the set of projective subschemes in \mathbb{P}^n_R. This shows us that (x^4 + y^4 + z^4)\mathfrak{m}^k defines the same projective curve as (x^4 + y^4 + z^4) in \mathbb{P}^2_\mathbb{C}.

Notes

References

References

1. (1997). "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra". Springer.
2. Greuel, Gert-Martin. (2008). "A Singular Introduction to Commutative Algebra". Springer-Verlag.