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I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation. Specifically, let be $X$ a reduced and irreducible complete curve over a field $L_0$ and set $\bar{X}= L \otimes X$, where $L$ is an extension of $L_0$. Mumford considers a torsion-free coherent $O_\bar{X}$-module $F$ on $\bar{X}$ and a maximal flag of coherent sub $O_\bar{X}$-modules $F_n$ \begin{gather*} F=F_0 \supset F_{-1} \supset \dotsb \supset F_{-t}=F(-P), \end{gather*} where $P=L \otimes P_0$, $P_0$ a regular closed point of $X$.

Then, both Mumford and Goss define $F_n$ for all $n$ by requiring \begin{gather*} F_{j+t}= F_j(P). \end{gather*} After that, both mention that, from general properties of coherent cohomology one sees that $\chi (F_n) = n$ and $h^0(F_n) \leq h^0(F_{n+1})$ and $h^1(F_n) \geq h^1(F_{n+1})$. Why $\chi(F_n)=n$ and the inequalities hold?

I'm a very novice at this theme, so if anyone of you knows where I can study more of this topic (and preliminaries), it would be very helpful!

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    $\begingroup$ You may want to include some details about what $F_n$ looks like, in order to make this more self contained. $\endgroup$
    – Donu Arapura
    Commented Jun 8 at 14:14
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    $\begingroup$ I added more information. Thanks for you advice. $\endgroup$
    – MChocko
    Commented Jun 8 at 17:57
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    $\begingroup$ Is $F$ assumed to have Euler characteristic zero? Otherwise this all seems wrong. There should be an exact sequence $0\to F_n \to F_{n+1} \to O_P \to 0$, and taking the long exact sequence in cohomology will show that the Euler characteristic goes up by 1 when n does and we have the inequalities on cohomology. $\endgroup$
    – Jack Huizenga
    Commented Jun 9 at 0:27
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    $\begingroup$ On page 184, Proposition 6.2.3 the hypothesis that $\chi(F)=0$ is explicitly established. The proposition concludes that $h^0(F)=h^1(F)=0$. $\endgroup$
    – F Zaldivar
    Commented Jun 9 at 18:07
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    $\begingroup$ @MChocko, Mumford dam.brown.edu/people/mumford/alg_geom/papers/… on its several Data A and Data B bijections, starting on page 116 also has the hypothesis that $\chi(F)=0$. $\endgroup$
    – F Zaldivar
    Commented Jun 9 at 18:45

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