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…
Best Answer
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.