# Can we control the number of homogeneous generators of a f.g. homogeneous ideal?

Let $$G$$ be an abelian group and $$R$$ be a $$G$$-graded ring.

Question $$1$$:

Is there a map $$\phi:\mathbb{N}\rightarrow\mathbb{N}$$ such that for every $$n\in \mathbb{N}$$ and any homogeneous ideal $$I$$ of $$R$$ generated by $$n$$ elements, $$I$$ can be generated by $$\phi(n)$$ homogeneous elements ?

If we denote by $$\mu_R(I)$$ the minimal number of homogeneous generators of $$I$$, this is equivalent to the following:

Is for every $$n\in \mathbb{N}$$, $$Sup_I \ \ \mu_R(I) < \infty$$, where $$I$$ runs over the set of homogeneous ideal of $$R$$ generated by $$n$$ elements ?

Question $$2$$:

We consider the following assertions.

$$(1)$$ There is a map $$\psi:\mathbb{N}\rightarrow\mathbb{N}$$ such that for every $$n\in\mathbb{N}$$, and any homogeneous ideal $$I$$ of $$R$$ generated by $$n$$ homogeneous elements and any homogeneous element $$a\in R$$, $$(I:a)$$ can be generated by $$\psi(n)$$ homogeneous elements.

$$(2)$$ There is a map $$\phi:\mathbb{N}\rightarrow\mathbb{N}$$ such that for every $$n\in\mathbb{N}$$, and any homogeneous ideal $$I$$ of $$R$$ generated by $$n$$ homogeneous elements and any homogeneous element $$a\in R$$, $$(I:a)$$ can be generated by $$\phi(n)$$ elements.

Obviously, $$(1)\Rightarrow(2)$$. But I have been unable to determine if the converse is true in general.

Thank you very much.

No, not in general. For instance, consider the $$\mathbb{Z}$$-graded ring $$R$$ with $$R_n=\mathbb{Z}/(n)$$ for each $$n$$ where all products of homogenenous elements of nonzero degree are $$0$$. Given any pairwise coprime integers $$n_1,\dots,n_k$$, consider the element $$x$$ which is $$1$$ in degrees $$n_1,\dots,n_k$$ and $$0$$ in all other degrees. Then $$x$$ generates a homogeneous ideal, since each homogeneous part of $$x$$ can be written as $$mx$$ for some $$m\in\mathbb{Z}$$ (choose $$m$$ which is $$1$$ mod $$n_i$$ and $$0$$ mod $$n_j$$ for all $$j\neq i$$). But clearly $$(x)$$ cannot be generated by fewer than $$k$$ homogeneous elements. So, there are principal homogeneous ideals in $$R$$ which require arbitrarily large numbers of homogeneous generators.