Why the graded Betti numbers of ideal $I \subset k[x_1 , \cdots , x_n]$, are bounded by the graded Betti numbers of $\mathrm{gin}(I)$?

(Where $\mathrm{gin}(I)$, is the generic initial ideal of $I$ whit respect to a monomial order)

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This is about the question whether $\beta_{ij}(\mathrm{in}(I)) = \beta_{ij}(\mathrm{in}(g(I)))$. No, they need not be equal. E.g., take $I = (x^2, y^2)$ in $k[x,y]$. Then for a general $g$, $\mathrm{in}(g(I)) = (x^2,xy, y^3)$. Note that $\mathrm{in}(I) = I$ and that $\beta_{03}(I) = 0 \neq 1 = \beta_{03}(\mathrm{in}(g(I)))$.

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