I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help on a possible approach would be much appreciated. Thanks very much!
Given:
- The polynomial ring $A = \mathbb{R}[x, y]$. It can be graded according to total polynomial degree as $A = \oplus_{k\geq 0} A_k$, with the Hilbert function of the $i^{th}$ piece given by $HF_i(A)= i+1$.
- The $A$-module $M = \oplus_{i}[\alpha_i]A$ generated by formally independent elements $\alpha_i$ coming from some finite set of cardinality $m$.
- The submodule $M \supset N = \sum [\beta_i] A$, where each generator $[\beta_i] = [\alpha_j] - [\alpha_k]$ for some $j$ and $k$.
- A surjective map $f: \{[\alpha_i]\} \rightarrow \{[\gamma_j]\}$, where $\gamma_j$ are formally independent elements of another finite set of cardinality $p < m$. Assume that, $$ \forall [\beta_i] = [\alpha_j] - [\alpha_k], \quad f([\beta_i]) := f([\alpha_j]) - f([\alpha_k]) \neq 0. $$
- A map, $$ g : \sum_{i \in I} [\alpha_i]a_i \mapsto \alpha_{\max(I)} a_{\max(I)}\;, $$ where $A \ni a_{i} \neq 0$ for all $i \in I$, and $I \subseteq \{1, \dots, m\}$.
- The submodules $N_0$ and $N_1$, $$ N_0 = \sum f([\beta_i])A, \qquad N_1 = \sum f\circ g([\beta_i])A. $$
Claim: Assuming the natural grading on $N_0$ and $N_1$, the following inequality holds: $$ HF_i(N_0) \geq HF_i(N_1). $$
EDIT
A simplified/alternate version of the problem statement on has been posted on SE by posing the problem in terms of real vector spaces. If that helps on nailing down an approach, you can find it here: https://math.stackexchange.com/questions/2728748/vector-space-dimension-after-nonlinearly-mapping-spanning-vectors
