Can the resultant ideal $\mathrm{Res}(f, g)$ of two homogeneous polynomials be defined in terms of the projective vanishing locus $V_+(f, g)$?

Let $A$ be a commutative ring and $f, g \in A[S, T]$ be two homogeneous polynomials in two variables of homogeneous degrees $d$, resp. $e$. Their resultant $\newcommand{\Res}{\mathrm{Res}}\Res(f, g)$ is defined to be the determinant of the linear map of free modules of rank $d + e$ $$ (f, g) \colon A[S, T]_{e - 1} \oplus A[S, T]_{d - 1} \to A[S, T]_{d + e - 1}, \qquad (H, K) \mapsto f H + g K. $$ with respect to the monomial bases on the various $A$-modules of homogeneous polynomials.

It is a result of classical elimination theory that if $\newcommand{\PP}{\mathbb{P}}\newcommand{\Spec}{\mathrm{Spec}} \pi \colon \PP^1_A \to \Spec(A)$ denotes the relative projective line over $A$, then the closed subset $V(\Res(f, g)) \subset \Spec(A)$ is the set-theoretic image of the projective vanishing locus $X := V_+(f, g) \subset \PP^1_A$. In particular, it follows that the radical $\sqrt{\langle \Res(f, g)\rangle}$ is an invariant of the closed sub-scheme $X = V_+(f, g) \subset \PP^1_A$.

I'm wondering whether in fact the ideal $\langle \mathrm{Res}(f, g)\rangle$ (or, equivalently, the scheme $V(\Res(f, g)) \subset \Spec(A)$) is an invariant of $X = V_+(f, g)$.

This is true if I assume additionally that either $f$ or $g$ are monic with respect to either of the variables $S$ or $T$. Indeed, say $f$ is monic with respect to $T$. In that case $V_+(f)$ is completely contained in the affine patch $D_+(S) \subset \PP^1_A$ and $X$ is in fact isomorphic to the finite $A$-scheme $\Spec(A[T] / \langle f(1, T), g(1, T)\rangle)$. In this case, the ideal $\langle \Res(f, g)\rangle$ is isomorphic to the $0$-th fitting ideal of the finite $A$-module $\Gamma(X, \mathscr{O}_{X}) \cong A[T] / \langle f(1, T), g(1, T)\rangle$, which is an invariant description.

A natural guess would therefore be that $\langle \Res(f, g)\rangle = \mathrm{Fitt}_0(\Gamma(X, \mathscr{O}_{X}))$ in general. Note that $\Gamma(X, \mathscr{O}_{X}) = \Gamma(\Spec(A), \pi_* \mathscr{O}_{X})$ is a finite $A$-module when $A$ is Noetherian by Grothendieck's coherent push-forward theorem, but I'm not actually sure if this is true without a Noetherian assumption (which I would like to avoid). The Fitting ideal is only defined for finite modules.

Let me also add that I can show that $\langle\Res(f, g)\rangle$ depends only on the homogeneous ideal $\langle f, g\rangle$ of $A[S, T]$, but that the scheme $V_+(f, g)$ only determines the saturation $(\langle f, g\rangle : \langle S, T\rangle^\infty)$, so this is not enough.