# Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring

Question 1. Let $$R$$ be a polynomial ring over a field $$k$$. Assume that $$R$$ is graded in the usual way (i.e., each variable has degree $$1$$). Let $$I$$ and $$J$$ be two ideals of $$R$$ such that $$I$$ is generated in degree $$\leq a$$ (that is, there is a set of generators of $$I$$ that have degrees $$\leq a$$) and $$J$$ is generated in degree $$\leq b$$. Is it true that the ideal $$I \cap J$$ is generated in degree $$\leq a+b$$ ?

When $$R$$ is a univariate polynomial ring, this is simply claiming that the lcm of two polynomials of degrees $$\leq a$$ and $$\leq b$$ is a polynomial of degree $$\leq a+b$$; of course this is correct. But this kind of logic does not generalize to more variables. (It is easy to see that Question 1 has a positive answer when $$I$$ and $$J$$ are monomial ideals.)

If true, Question 1 would give a new (inductive) proof for the following theorem (part of the Subspace arrangement theorem of Derksen and Sidman -- see Theorem 2.1 in their A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements, arXiv:math/0109035v1):

Theorem 2. Let $$R$$ be a polynomial ring over a field $$k$$. Assume that $$R$$ is graded in the usual way (i.e., each variable has degree $$1$$). Let $$I_1, I_2, \ldots, I_t$$ be $$t$$ homogeneous ideals of $$R$$ that are all generated in degree $$1$$ (that is, each $$I_i$$ has the form $$I_i = V_i R$$ for some $$k$$-vector subspace $$V_i$$ of the degree-$$1$$ homogeneous component of $$R$$). Then, the ideal $$I_1 \cap I_2 \cap \cdots \cap I_t$$ is generated in degree $$\leq t$$.

Some more questions suggest themselves:

Question 3. (If the answer to Question 1 is negative:) Does the answer to Question 1 become positive if we replace "ideal" by "homogeneous ideal"?

(This would still be enough to reprove Theorem 2.)

Truth be told, I'm less interested in the polynomial case than I am in the exterior algebra case:

Question 4. What if we replace the polynomial ring by an exterior algebra?

(You can assume $$k$$ has characteristic $$0$$ for simplicity.)

Finally, if all these questions have negative answers, here is what I am really looking for:

Question 5. Is there an elementary proof of the analogue of Theorem 2 for exterior algebras (which is part of Theorem 9 in Francesca Gandini, Degree bounds for invariant skew polynomials, arXiv:2108.01767v1)?

"Elementary" means no use of nontrivial commutative algebra for me (I'm actually fine with Schur functors, although I have a hunch that they too can be avoided).

I'm putting the invariant theory tag on this question because the ultimate use of Question 5 is in Francesca Gandini's proof of the analogue of the Noether bound for the exterior algebra, which I have asked about in Noether's bound for anticommutative invariant theory (diff. forms instead of polynomials)? and am now trying to understand...

• It seems to me that question 1 should be easy to reduce to question 3, simply by homogenizing. Am I mistaken? That is, if the answer to question 3 is "yes", the answer to question 1 should also be "yes". Aug 30, 2021 at 19:28
• @user44191: I know how homogenization acts on specific polynomials; not sure how it works on ideals, given that it depends on the degree in an awkward way. But the idea doesn't strike me as outlandish. Aug 30, 2021 at 21:42

Take $$I=(a^3,b^3)$$ and $$J=(ac^2-bd^2)$$. Then according to Macaulay2, $$I\cap J$$ has generators in degrees $$7,8,9$$, for instance $$a^3c^6-b^3d^6$$. So the answers to Question 3 and 1 are no.