Hilbert-Kunz function/Projective variety/Global sections and Frobenius pull-back/Section

Our approach to the computation of the Hilbert-Kunz multiplicity is by using the presenting sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{Y}(-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{Y}\longrightarrow 0$

and twists of its ${}e$-th Frobenius pull-backs, that is

$0\longrightarrow \operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{Y}(m-qd_{i}){\stackrel {f_{1}^{q},\ldots ,f_{n}^{q}}{\longrightarrow }}{\mathcal {O}}_{Y}(m)\longrightarrow 0$

(where ${}q=p^{e}$), and to relate the asymptotic behavior of

${}\operatorname {length} \,(R/I^{[q]})=\operatorname {dim} _{K}\,(R/I^{[q]})=\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(R/I^{[q]})_{m}\,$

to the asymptotic behavior of the global sections of the Frobenius pull-backs

${}(F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})(m)=\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\,.$

What we want to compute is just the cokernel of the complex of global sections of the above sequence, namely

$\operatorname {dim} _{K}\,(R/I^{[q]})_{m}=h^{0}(Y,{\mathcal {O}}_{Y}(m))-\sum _{i=1}^{n}h^{0}(Y,{\mathcal {O}}_{Y}(m-qd_{i}))+h^{0}(Y,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))\,.$

The summation over ${}m$ is finite (but the range depends on ${}q$), and the terms

${}h^{0}(Y,{\mathcal {O}}_{Y}(m))=\operatorname {dim} _{K}\,\Gamma (Y,{\mathcal {O}}_{Y}(m))=\operatorname {dim} _{K}\,R_{m}\,$

are easy to control, so we have to understand the behavior of the global syzygies

$H^{0}(Y,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))$

for all ${}q$ and ${}m$, at least asymptotically. This is a Frobenius-Riemann-Roch problem.