1 Erratum to: manuscripta math. 160, 239–264 (2019) https://doi.org/10.1007/s00229-018-1056-6

An error in Proposition 5.11 was pointed out by professor Adrian Langer. In the statement of the proposition, the variety Z must be projective. Therefore, Proposition 5.11 should have been stated as follows:

Proposition 5.11

Let X, \(\Delta \), Z and f be as in Definition 5.1, and Y be a normal variety. Assume that \(f:X\rightarrow Z\) can be factored into projective morphisms \(g:X\rightarrow Y\) with \(g_*{{\mathcal {O}}}_X\cong {{\mathcal {O}}}_Y\) and \(h:Y\rightarrow Z\). Suppose that Z is projective.

  1. (1)

    If \((f,\Delta )\) is F-split, then so is h.

  2. (2)

    Assume that Y is smooth. If \((g,\Delta )\) and h are F-split, then so is \((f,\Delta )\).

  3. (3)

    The converse of (2) holds if \(K_Y\sim _{{{\mathbb {Z}}}_{(p)}}h^*K_Z\).

Furthermore, the proof of statements (2) and (3) of the proposition includes an unclear part and an unsuitable part, so the proof should be modified as follows:

Proof of Proposition 5.11

Let \(e>0\) be an integer. Now we have the following commutative diagram:

Here, \(\pi ^{(e)}:=(F_{Y/Z}^{(e)})_X\). We first show (1). The above diagram induces the commutative diagram of \({{\mathcal {O}}}_{Y_{Z^e}}\)-modules

where the left vertical morphism is an isomorphism because of the flatness of \((F_Z^e)_Y\). Since the lower horizontal morphism splits, so does the upper one.

Next, we show (2) and (3). As explained in Observation 5.4, if \((g,\Delta )\) is F-split, then there exists an effective \({{\mathbb {Z}}}_{(p)}\)-Weil divisor \(\Delta '\ge \Delta \) on X such that \(K_{X/Y}+\Delta '\) is \({{\mathbb {Z}}}_{(p)}\)-linearly trivial and that \((g,\Delta ')\) is also F-split. Therefore, when we prove (2), we may assume that \(\Delta \) is a \({{\mathbb {Z}}}_{(p)}\)-Weil divisor and that \((p^e-1)(K_{X/Y}+\Delta )\sim 0\) for every \(e>0\) divisible enough. In this case, \((p^e-1)(K_{X/Z}+\Delta ) \sim (p^e-1)g^* K_{Y/Z}\), so \({\mathcal {L}}^{(e)}_{(X/Z,\Delta )} \cong {g^{(e)}}^* {\mathcal {N}}_1^{(e)}\) for a line bundle \({\mathcal {N}}_1^{(e)}\) on \(Y^{e}\), and \( {\mathcal {L}}^{(e)}_{(X/Y,\Delta )} \cong {\mathcal {O}}_{X^e} \cong {g^{(e)}}^*{\mathcal {O}}_{Y^e}. \) By an argument similar to the above, when we show (3), we may suppose that for every \(e>0\) divisible enough, \({\mathcal {L}}^{(e)}_{(X/Y,\Delta )} \cong {g^{(e)}}^* {\mathcal {M}}_1^{(e)}\) for a line bundle \({\mathcal {M}}_1^{(e)}\) on \(Y^e\), and \( {\mathcal {L}}^{(e)}_{(X/Z,\Delta )} \cong {\mathcal {O}}_{X^e} \cong {g^{(e)}}^* {\mathcal {O}}_{Y^e}. \) In summary, since we now prove (2) and (3), we may assume that for every \(e>0\) divisible enough, \({{\mathcal {L}}}^{(e)}_{(X/Y,\Delta )} \cong {g^{(e)}}^*{\mathcal {M}}^{(e)}\) and \({{\mathcal {L}}}^{(e)}_{(X/Z,\Delta )} \cong {g^{(e)}}^*{\mathcal {N}}^{(e)}\) for line bundles \({\mathcal {M}}^{(e)}\) and \({\mathcal {N}}^{(e)}\) on \(Y^e\).

Let \(V\subseteq Y\) be an open subset such that g is flat at every point in \(X_V:=g^{-1}(V)\) and \({\mathrm {codim}}(Y\setminus V)\ge 2\). Let \(u:U\rightarrow X_V\) be the open immersion of the regular locus of \(X_V\). Set \(g':=g\circ u:U\rightarrow Y\). We then have \({g'}_*{{\mathcal {O}}}_U\cong {{\mathcal {O}}}_Y\) because of the assumptions. Therefore, for every line bundle \({\mathcal {N}}\) on Y, we see that

$$\begin{aligned} H^0\left( U, (g^* {\mathcal {N}})|_U \right) = H^0(U, {g'}^*{\mathcal {N}}) \cong H^0(Y, {g'}_*{g'}^*{\mathcal {N}}) \cong H^0(Y, {\mathcal {N}}) \cong H^0(X, g^*{\mathcal {N}}) \end{aligned}$$

by the projection formula. In addition, by the flatness of \(F_Z^e\), we get \({{g'}_{Z^e}}_*{{\mathcal {O}}}_{U_{Z^e}}\cong {{\mathcal {O}}}_{Y_{Z^e}}\), and so \( H^0\left( U_{Z^e}, {\mathcal {O}}_{U_{Z^e}}\right) \cong H^0\left( X_{Z^e}, {\mathcal {O}}_{X_{Z^e}}\right) \) by an argument similar to the above. Hence, we have the following commutative diagram:

Note that we are assuming that \({\mathcal {L}}^{(e)}_{(X/Z,\Delta )} \cong {g^{(e)}}^*{\mathcal {N}}\) for a line bundle \({\mathcal {N}}\) on \(Y^e\). Since the splitting of \(\phi ^{(e)}_{(X/Z,\Delta )}\) is clearly equivalent to the surjectivity of \(H^0\left( X_{Z^e},\phi _{(X/Z,\Delta )}^{(e)}\right) \), we see that the F-splitting of \((f,\Delta )\) and that of \((f|_U:U\rightarrow Z,\Delta |_U)\) are equivalent. By an argument similar to the above, we find that the F-splitting of \((g,\Delta )\) and that of \((g|_U,\Delta |_U)\) are also equivalent.

Assume that we can choose \(V=Y\) and \(U=X\), i.e. X and Y are regular and g is flat. Let \(e>0\) be an integer. By the flatness of g, we have the following commutative diagram:

This implies that

$$\begin{aligned} {\mathcal {H}}om\left( {\pi ^{(e)}}^{\sharp },{{\mathcal {O}}}_{X_{Z^e}}\right) \cong {{g}_{Z^e}}^*{\mathcal {H}}om\left( {F_{Y/Z}^{(e)}}^{\sharp },{{\mathcal {O}}}_{V_{Z^e}}\right) ={{g}_{Z^e}}^*\phi _{Y/Z}^{(e)}. \end{aligned}$$

Applying the functor \({\mathcal {H}}om(\underline{\quad },{{\mathcal {O}}}_{X_{Z^e}})\) and the Grothendieck duality to the natural morphism

$$\begin{aligned} {{\mathcal {O}}}_{X_{Z^e}} \xrightarrow {{\pi ^{(e)}}^{\sharp }} {\pi ^{(e)}}_*{{\mathcal {O}}}_{X_{Y^e}} \rightarrow {F_{X/Z}^{(e)}}_*{{\mathcal {O}}}_{X^e}(\lceil (p^e-1)\Delta \rceil ), \end{aligned}$$

we obtain the morphism

$$\begin{aligned} \phi ^{(e)}_{(X/Z,\Delta )}:{F_{X/Z}^{(e)}}_*{{\mathcal {L}}}^{(e)}_{(X/Z,\Delta )} \xrightarrow {{\pi ^{(e)}}_*\phi ^{(e)}_{(X/Y,\Delta )} \otimes {\omega }_{\pi ^{(e)}}} {g_{Z^e}}^*{F_{Y/Z}^{(e)}}_*{{\mathcal {L}}}^{(e)}_{Y/Z} \xrightarrow {{g_{Z^e}}^*\phi ^{(e)}_{Y/Z}} {{\mathcal {O}}}_{X_{Z^e}}. \end{aligned}$$

Note that

$$\begin{aligned} {\omega }_{\pi ^{(e)}} \cong {\omega }_{X_{Y^e}}\otimes {\pi ^{(e)}}^*{\omega }_{X_{Z^e}} \cong {g_{Z^e}}^*{\omega }_{Y^e/Z^e}^{1-p^e} \quad and \quad {g_{Z^e}}^*{F_{Y/Z}^{(e)}}_*{{\mathcal {L}}}^{(e)}_{Y/Z} \cong {\pi ^{(e)}}_*{g_{Y^e}}^*{{\mathcal {L}}}^{(e)}_{Y/Z}. \end{aligned}$$

We now prove the assertion. If \((g,\Delta )\) is F-split and h is F-split, then both of \(\phi ^{(e)}_{(X/Y,\Delta )}\) and \(\phi ^{(e)}_{Y/Z}\) split for every \(e>0\) divisible enough. Therefore, \(\phi ^{(e)}_{(X/Z,\Delta )}\) also splits, i.e. \((f,\Delta )\) is F-split. Conversely, suppose that \((f,\Delta )\) is F-split and that \((p^e-1)K_{Y/Z}\sim 0\) for an \(e>0\). Then, \({{\mathcal {L}}}^{(e)}_{Y/Z}\cong {{\mathcal {O}}}_{Y_{Z^e}}\) and \({\omega }_{\pi ^{(e)}}\cong {{\mathcal {O}}}_{X_{Y^e}}\). Fix an \(e>0\) divisible enough. Since \(H^0\left( X_{Z^e},\phi _{(X/Z,\Delta )}^{(e)}\right) \) is surjective, \(H^0\left( X_{Z^e},{\pi ^{(e)}}_*\phi _{(X/Y,\Delta )}^{(e)}\right) \) is a nonzero morphism, and hence so is \(H^0\left( X_{Y^e},\phi ^{(e)}_{(X/Y,\Delta )}\right) \). This morphism is surjective because of \(H^0(X_{Y^e},{{\mathcal {O}}}_{X_{Y^e}})\cong H^0(Y^e,{{\mathcal {O}}}_{Y^e})\cong k\). Thus, \(\phi ^{(e)}_{(X/Y,\Delta )}\) splits, and so \((g,\Delta )\) is F-split. Note that the F-splitting of h follows directly from (1). \(\square \)