Let $k$ be a field and let $A$ be a finite type $k$-algebra that is a (global) complete intersection, i.e. there exists some presentation
$$ A = k[x_1,\dots,x_n]/I$$
such that $I$ has a generating set of cardinality $n - \dim_{Krull} A$.
Let now $A = k[X_1,\dots,X_N]/J$ be an arbitrary presentation of $A$.
(A) Can one conclude $J$ has a generating set of cardinality $N - \dim_{Krull} A$?
I assume the answer is no in general, see below. If this is true, does it become yes if $A$ is a graded ring?
(B) If the answer to (A) is no in general but we assume that $k[x_1,\dots,x_n]$ and $k[X_1,\dots,X_N]$ are positively graded (although perhaps the $x_i$'s and $X_j$'s have varying degrees), and that $I$ and $J$ are graded ideals, then does the answer to (A) become yes?
I assume the answer to (A) is no in general because the Stacks Project, Lemma 10.133.4, asserts a result that is close to (A) but localized: given a prime $\mathfrak{p}$ of $A$, then there exists $x\in A\setminus\mathfrak{p}$ such that $A_x$ is a global complete intersection if and only if $J_\mathfrak{P}$ in $k[X_1,\dots,X_N]_\mathfrak{P}$ can be generated by the right number of elements, where $\mathfrak{P}$ is the pullback of $\mathfrak{p}$ in $k[X_1,\dots,X_N]$. If $A$ is a global complete intersection, the first condition is met for every $\mathfrak{p}$, thus so is the second; but on the other hand this is also true if $A$ is a local, but not global, complete intersection (by which I mean there is a collection of $y_i$'s generating the unit ideal in $A$ such that each $A_{y_i}$ is a complete intersection). So a yes for (A) does not appear to follow from this lemma.
On the other hand, for a local $k$-algebra essentially of finite type, the analogous claim to (A) is true: see Stacks Project lemmas 10.133.6 and 10.133.7. And what is true in the local case is also often true in the graded-over-a-field case. Hence question (B).
Looking forward to your thoughts!