# When is the product of two ideals equal to their intersection?

Consider a ring $$A$$ and an affine scheme $$X=\operatorname{Spec}A$$ . Given two ideals $$I$$ and $$J$$ and their associated subschemes $$V(I)$$ and $$V(J)$$, we know that the intersection $$I\cap J$$ corresponds to the union $$V(I\cap J)=V(I)\cup V(J)$$. But a product $$IJ$$ gives a new subscheme $$V(IJ)$$ which has same support as the union but can be bigger in an infinitesimal sense. For example if $$I=J$$ you get a scheme $$V(I^2)$$ which is equal to "double" $$V(I)$$.

Vague Question : What is geometric interpretation of $$V(IJ)$$ in general?

Precise question: When is $$I\cap J=IJ$$?

Everybody knows the case $$I+J=A$$ but this is absolutely not necessary. For example if $$A$$ is a UFD and $$f,g$$ are relatively prime then $$(f)(g)=(f)\cap(g)$$, but in general $$(f)+(g)\neq A$$ (e.g. $$f=X, g=Y \in k[X, Y]$$).

Thank you very much.

• Feb 6, 2017 at 12:41

To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history:

When $$A$$ is regular, the Tor functor satisfies the following property:

(1) $$\text{Tor}_1^A(M,N) = 0$$ implies $$\text{Tor}_i^A(M,N) = 0$$ for $$i>0$$ for any two finitely generated modules.

(this is a theorem by Auslander in the geometric and unramified case and Lichtenbaum in the ramified case. (1) is called the rigidity of Tor).

It turns out that when $$A$$ is regular and local (so one can talk about depth), (1) implies

(2) $$\text{depth} (M) + \text{depth}(N) = \dim A + \text{depth} {M\otimes_AN}$$

This stunning formula looks exactly the same as the property of "proper intersection" in intersection theory, except that one uses depth instead of dimension. Note that if $$M=A/I, N=A/J$$ then $$M\otimes N = A/(I+J)$$, which represents the intersection of $$V(I)$$ and $$V(J)$$, so this is very geometric.

(3) Talking about intersection theory, by Serre formula for intersection multiplicity, as all the Tors vanish, one can compute the intersection multiplicity of $$V(I), V(J)$$ by counting the length at the minimal components (i.e. the naive way). So you will have a generalization of Bezout theorem.

Finally, if $$V(I)$$ and $$V(J)$$ only intersect at isolated closed points, (2) implies (1) locally on the support of the intersection, so

(4) If $$V(I) \cap V(J)= \{m_1, \cdots, m_n \}$$ then $$I\cap J = IJ$$ if and only if $$A/I, A/J$$ are locally Cohen-Macaulay at the points $$m_i$$s.

You can find the last statement in Serre's Local Algebra book, V.6, Theorem 4, p 110 of the English version.

PS: Also, David did not mention his own interesting contribution, here.

• Nice exhaustive answer, so let me ask a stupid reference. I do not want to prove that if $f,g$ have no common factor in a UFD then $(f)\cap(g)=(f\cdot g)$ in a paper I am writing, but I found no explicit reference: do you know any? Adapting Serre's criterion seems a bit overkilling, for a UFD...Thanks. Aug 27, 2013 at 10:54
• Filippo: Any thing in $(f)\cap (g)$ would be of the form $fx=gy$. By writing both sides a product of irreducibles one concludes that $g$ divides $x$... Aug 28, 2013 at 0:47
• ...hmm, I guess I agree and that NO reference is by far the best option. I was probably a bit puzzled when I asked, sorry. Aug 28, 2013 at 4:38
• In (4), I guess there is a missing "proper intersection" hypothesis, dim(A/I) + dim(A/J) = dim(A) (as in the Serre Local Algebra reference). Feb 18, 2022 at 4:45
• @VictorWang: the depth formula forces this hypothesis, as dim is at least depth for each module and dim A=depth A. Feb 25, 2022 at 17:47

Answer to the precise question: When $\mathrm{Tor}^1(A/I, A/J)=0$.

Proof: We have the exact sequence $$0 \to I \to A \to A/I \to 0$$ Tensoring with $A/J$, we get $$0 \to \mathrm{Tor}^1(A/I, A/J) \to I/(I \cdot J) \to A/J \to A/(I+J) \to 0.$$ The left hand term is $0$ because $A$ is flat as an $A$-module.

Now, what is the kernel of $I \mapsto A/J$? Clearly, it is $I \cap J$. So the kernel of $I/(I \cdot J) \to A/J$ is $(I \cap J)/(I \cdot J)$. We see that $I \cap J = I \cdot J$ if and only if $\mathrm{Tor}^1(A/I, A/J)=0$.

• The condition with Tor is looking more complicated than the question.
– user6976
Dec 13, 2010 at 14:22
• But it is more "geometric" since only $V(I)$ and $V(J)$ are involved. Dec 13, 2010 at 14:35
• I agree that the condition with Tor is more geometric --- it can be viewed as a kind of purity' of intersection (for instance, two smooth subvarieties of a smooth variety have this property if and only if their intersection has the expected dimension). Is there an accepted name for this condition? Dec 13, 2010 at 15:38
• @t3suji. Let V and W be closed integral subschemes of a nonsingular quasi-projective irreducible variety. Then, for any irreducible component Z of VcapW, it holds that codim Z <= codim V + codim W. (See Serre's Local Algebra.) We say that V and W intersect properly in Z if equality holds. A stronger condition is being in general position. If V and W are in general position all the higher Tor's vanish. The cycle [VcapW] associated to VcapW is then equal to the product cycle [V][W]. As far as I know, this is standard language in intersection theory for algebraic varieties. Dec 13, 2010 at 17:49
• @Ariyan. Thanks! (I somehow always forget the "proper intersection" terminology, that's why my use of "purity" in place of... "propriety" or "properness".) Dec 13, 2010 at 18:00

I apologize in advance for resurrecting such an old question but I absolutely could not resist the urge of sharing a precise characterization for $$I \cap J=IJ$$, that I read recently in a beautiful short paper, when $$I,J$$ are monomial ideals in a polynomial ring.

First some terminology: Let $$k$$ be a field and $$R=k[x_1,...,x_n]$$ . Every monomial ideal $$I$$ of $$R$$ has a unique minimal monomial set of generators, usually denoted by $$G(I)$$ . For a set of monomials $$T$$ in $$R$$, let $$\newcommand{\Supp}{\operatorname{Supp}}\Supp (T) :=\{i | x_i$$ divides $$m$$ for some $$m \in T \}$$ .

With this, we can state the characterization: Let $$I,J$$ be monomial ideals in $$k[x_1,...,x_n]$$ , then $$I \cap J=IJ$$ if and only if $$\Supp (G(I)) \cap \Supp (G(J))$$ is empty.

This is Theorem 2.2 in https://link.springer.com/article/10.1007/s12044-019-0509-5

A vague answer to the vague question:

When you want the union of $$V(I)$$ and $$V(J)$$ to behave well under deformations and to count with multiplicity', then you may prefer to use the ideal $$IJ$$ rather than $$I\cap J$$. Let me give an example:

Take $$V=V(x)$$, $$W=V(x-t)$$ and $$T=V(t)$$ denote $$V_0:=V\cap T=V(x,t)=W\cap T=:W_0$$. If you use intersection of ideals for the union of varieties you will get:

$$(V\cup W)\cap T=V(x^2,t)$$, and

$$(V_0\cup W_0)\cap T=V(x,t)$$.

While using product you will get:

$$(V\cup W)\cap T=V(x^2,t)=(V_0\cup W_0)\cap T$$.

• I am not totally convinced of this. If we consider the intersection of $X=V(y-x^2)$ with $Y_t=V(y-t)$ in the affine plane we have $X\cap Y_t$ consist of two points for every $t\neq 0$ so maybe we should expect a double point at the origin in the union $X\cup Y_0$ but the local ring there is reduced. However, I am changing from intersection to union thinking in this way so I am not sure if my intuition is correct. May 19, 2018 at 13:53