Greeting. As a part of my project i'm trying to understand the following paragraph of This article;
Here is what i understood so far: Let $X$ be a scheme and $L$ a locally free sheaf of rank $g$ over $X$. Let $T_L$ be the functor from $(Sch/X)^{opp}$ to $Sets$ sending an $X$ scheme $S$ with structural morphism $f$ to $Isom_X(O_S^g,f^*L)$. This functor is representable by a scheme $\underline{T_L}$, moreover $GL_g$ acts on the right by $\omega\gamma=\omega \circ \gamma$. Let $\pi:T_L\rightarrow X$ the structural morphism. $Hom_X({T_L},{T_L})$ is naturally isomorphic to $Isom_X(O_{T_L},\pi^*L)$, taking the identity element we obtain a universal trivialization $\omega:O_{T_L}\simeq \pi^*L$.
What remains for me to understand is the following: How does the writer view $\pi_*O_{T_L}$ as a sheaf of sections on the form $f(\omega)$? What does $f$ stands for? I think $f$ is a function from $Isom_X(O_{T_L},\pi^*L)$ to $X$ but i can't see the relation with the sheaf $\pi_*O_{T_L}$. Thanks in advance!
Let $u_X:X\rightarrow S$ be a scheme with $V:=Spec(R)\subseteq S$ an open subscheme with $u_X^{-1}(V)=U:=Spec(A)$, and let $\mathcal{L}_X$ be a locally trivial $\mathcal{O}_X$-module of rank $g$ where $\mathcal{L}_X(U)\cong L$. Let $u:S:=Spec(A)\rightarrow T:=Spec(R)$ be the induced map and let $L:=A\{e_1,..,e_g\}$ be the free $A$-module of rank $g$. Let $\mathcal{L}$ be the sheafification of $L$. It seems in this situation there is an isomorphism $T_{\mathcal{L}}\cong Spec(B)$ where $B:=A[x_{ij},t]/(det(x_{ij})t-1)$, where $i,j=1,..,g$. Hence $A[x_{ij}]$ is the coordinate ring of the scheme $Mat_g(A)$ of square matrices of rank $g$ with coefficients in $A$, and $B$ is the coordinate ring of the group scheme $GL(g,A)$ of invertible $g\times g$-matrices $M:=(a_{ij})$ with $a_{ij}\in A$ and $det(M)\in A^*$ a unit. There is a canonical right action of $GL(g,A)$ on $B$ and a canonical structure morphism $\pi:T_{\mathcal{L}}\rightarrow Spec(A)$.
In this situation it follows the global sections of the sheaf $\pi_*\mathcal{O}_{T_\mathcal{L}}$ is the ring $A[x_{ij},t]/(det(x_{ij})t-1)$ viewed as an $A$-algebra, and there is an inclusion of sheaves
I1. $\pi_*\mathcal{O}_{T_\mathcal{L}}[\kappa^*]\subseteq \pi_*\mathcal{O}_{T_\mathcal{L}}$ is a subsheaf
I have not read the paper, but it seems the subsheaf $\pi_*\mathcal{O}_{T_\mathcal{L}}[\kappa^*]$ is defined as the set of matrices $m$ in $B$ that are "semi-invariant" under the right action of the group $U$ of upper triangular matrices in $GL(g,A)$. Given $b\in U$ it follows $mb=\kappa^*(b)m$. You must figure out how $\kappa^*$ acts on elements $b\in U$. It seems the group $T$ acts on $U$.