The Theorem and its proof can be found here.
Specifically, i am stuck at the fourth paragraph of the proof. Let me give some context: Let $I$ be a graded ideal over a polynomial ring $S=K[x_1,\dots,x_n]$ over a field $K$ and consider a monomial order $>$ such that $x_1>x_2>\cdots>x_n$. Let $I$ be a graded ideal and let $\operatorname{gin}_<(I)$ be its generic initial ideal. Let $\alpha$ be an elementary upper triangular matrix such that $\alpha \left( \operatorname{gin}_<(I)_d\right) \neq \operatorname{gin}_<(I)_d$, where the subscript $d$ denotes component of degree $d$. Let $t$ be the $K$-dimension of $\operatorname{gin}_<(I)$ and let $w_1,\dots,w_t$ be a basis of monomials, and $w=w_1 \wedge \cdots \wedge w_t$. Finally, define $\alpha(w) = \alpha(w_1)\wedge \cdots \wedge \alpha(w_t)$. Then the claim that i can not see why is true, is the following:
Claim: There is an exterior monomial $u$ in the support of $\alpha(w)$, such that $u> w$.
Question: How do we see that?
