4. Asymptotic test ideals [0394]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

4. Asymptotic test ideals

We recall definitions and some basic properties from the theory of generalized and asymptotic test ideals developed in [BMS08, Sect. 2] and [Mus13, Sect. 3]. We refer to [ST12] for a more comprehensive overview of the theory of test ideals. Let $X$ be a smooth variety over a perfect field $k$ of characteristic $p>0$ . Let $F\colon X\to X$ denote the Frobenius morphism which is induced by the $p$ -th power ring morphism on affine subsets. Write

\omega_{X/k}=\det\Omega_{X/K}={\mathcal{O}}_{X}(K_{X/k})

for some canonical divisor $K_{X/k}$ on $X$ .

Let ${\mathfrak{a}}$ be an ideal in ${\mathcal{O}}_{X}$ and $e\in\mathbb{N}_{>0}$ . There is a unique ideal ${\mathfrak{a}}^{[p^{e}]}$ in ${\mathcal{O}}_{X}$ such that for every open affine $U$ in $X$ the ideal ${\mathfrak{a}}^{[p^{e}]}(U)$ in ${\mathcal{O}}_{X}(U)$ is generated by $\{u^{p^{e}}\,|\,u\in{\mathfrak{a}}(U)\}$ . We have [BMS08, bottom of p. 44]

(4.1)

{\mathfrak{a}}(U)=\{a\in{\mathcal{O}}_{X}(U)\,|\,a^{p^{e}}\in{\mathfrak{a}}^{[p^{e}]}(U)\}

We recall the following facts from [Mus13, p. 540]: There is a canonical trace map ${\rm Tr}\colon F_{*}(\omega_{X/k})\to\omega_{X/k}$ whose construction can be based on the Cartier isomorphism [Kat70, Thm. (7.2) and Eq. (7.2.3)]. Mustaţă gives an explicit description of the trace map [Mus13, top of p. 540]. Given $e\in\mathbb{N}_{>0}$ there is an iterated trace map ${\rm Tr}^{e}\colon F^{e}_{*}(\omega_{X/k})\to\omega_{X/k}$ . For an ideal $\mathfrak{a}$ in ${\mathcal{O}}_{X}$ there exists a unique ideal $\mathfrak{a}^{[1/p^{e}]}$ in ${\mathcal{O}}_{X}$ with

(4.2)

{\rm Tr}^{e}(F^{e}_{*}(\mathfrak{a}\cdot\omega_{X/k}))=\mathfrak{a}^{[1/p^{e}]}\cdot\omega_{X/k}.

This definition of $\mathfrak{a}^{[1/p^{e}]}$ is compatible with [BMS08, Def. 2.2]. Hence we have

(4.3)

\bigl({\mathfrak{a}}^{[p^{e}]}\bigr)^{[1/p^{e}]}={\mathfrak{a}}\subseteq\bigl({\mathfrak{a}}^{[1/p^{e}]}\bigr)^{[p^{e}]}

by [BMS08, Lemma 2.4(iv)].

Definition 4.1.

[BMS08, Def. 2.9] Given an ideal ${\mathfrak{a}}$ in ${\mathcal{O}}_{X}$ and $\lambda\in\mathbb{R}_{\geq 0}$ one defines the test ideal of ${\mathfrak{a}}$ of exponent $\lambda$ to be

\tau({\mathfrak{a}}^{\lambda}):=\bigcup_{e\in\mathbb{N}_{>0}}\Bigl({\mathfrak{a}}^{\lceil\lambda p^{e}\rceil}\Bigr)^{[1/p^{e}]}

where given $r\in\mathbb{R}$ we write $\lceil r\rceil$ for the smallest integer $\geq r$ .

Remark 4.2.

(i)

Observe that we have $\tau({\mathfrak{a}}^{\lambda})=({\mathfrak{a}}^{\lceil\lambda p^{e}\rceil})^{[1/p^{e}]}$ for large $e\in\mathbb{N}$ as $X$ is noetherian. The equality

(4.4) $\tau(({\mathfrak{a}}^{m})^{\lambda})=\tau({\mathfrak{a}}^{\lambda m})$

for $m\in\mathbb{N}$ shows that the notation in Definition 4.1 is compatible with taking powers of ideals [BMS08, Cor. 2.15]. We have $\tau({\mathfrak{a}}^{\lambda})\subseteq\tau({\mathfrak{b}}^{\lambda})$ for ideals ${\mathfrak{a}}\subseteq{\mathfrak{b}}$ in ${\mathcal{O}}_{X}$ [BMS08, Prop. 2.11(i)].
(ii)

Choose $e$ such that $\tau({\mathfrak{a}})=({\mathfrak{a}}^{[p^{e}]})^{[1/p^{e}]}$ . For any ideal ${\mathfrak{b}}$ in ${\mathcal{O}}_{X}$ such that ${\mathfrak{a}}^{[p^{e}]}\subseteq{\mathfrak{b}}^{[p^{e}]}$ we get ${\mathfrak{a}}\subseteq{\mathfrak{b}}$ from (4.1). Hence (4.3) implies

(4.5) ${\mathfrak{a}}\subseteq\tau({\mathfrak{a}}).$

Let ${\mathfrak{a}}_{\bullet}$ be graded sequence of ideals in ${\mathcal{O}}_{X}$ , i.e. a family $({\mathfrak{a}}_{m})_{m\in\mathbb{N}_{>0}}$ of ideals in ${\mathcal{O}}_{X}$ such that ${\mathfrak{a}}_{m}\cdot{\mathfrak{a}}_{n}\subseteq{\mathfrak{a}}_{m+n}$ for all $m,n\in\mathbb{N}_{>0}$ and ${\mathfrak{a}}_{m}\neq(0)$ for some $m>0$ .

Definition 4.3.

[Mus13, p. 541] Choose $\lambda\in\mathbb{R}_{\geq 0}$ . Define the asymptotic test ideal of exponent $\lambda$ as

\tau({\mathfrak{a}}_{\bullet}^{\lambda}):=\bigcup_{m\in\mathbb{N}}\tau({\mathfrak{a}}_{m}^{\lambda/m}).

Remark 4.4.

(i)

We have $\tau({\mathfrak{a}}^{\lambda}_{\bullet})=\tau({\mathfrak{a}}_{m}^{\lambda/m})$ for suitable $m\in\mathbb{N}$ which are divisible enough [Mus13, p. 541].
(ii)

For all $m\in\mathbb{N}$ we have [Mus13, p. 541, l. 4]

(4.6) $\tau({\mathfrak{a}}_{m})\subseteq\tau({\mathfrak{a}}^{m}_{\bullet}).$
(iii)

For all $m\in\mathbb{N}$ we have the Subadditivity Property [Mus13, Prop. 3.1(ii)]

(4.7) $\tau({\mathfrak{a}}_{\bullet}^{m\lambda})\subseteq\tau({\mathfrak{a}}_{\bullet}^{\lambda})^{m}.$

Definition 4.5.

Let $D$ be a divisor on $X$ with $h^{0}(X,{\mathcal{O}}_{X}(mD))\neq 0$ for some $m>0$ . Define the asymptotic test ideal of exponent $\lambda\in\mathbb{R}_{\geq 0}$ associated with $X$ and $D$ as

\tau(\lambda\cdot\|D\|):=\tau({\mathfrak{a}}_{\bullet}^{\lambda})

where ${\mathfrak{a}}_{\bullet}$ denotes the graded sequence of base ideals for $D$ , i.e. ${\mathfrak{a}}_{m}$ is the image of the natural map

H^{0}(X,{\mathcal{O}}(mD))\otimes_{k}{\mathcal{O}}_{X}(-mD)\to{\mathcal{O}}_{X}.

If $D$ is a $\mathbb{Q}$ -divisor such that $h^{0}(X,{\mathcal{O}}_{X}(mD))\neq 0$ for some positive integer $m$ such that $mD$ is a usual divisor then we put $\tau(\lambda\cdot\|D\|):=\tau(\lambda/r\cdot\|rD\|)$ for some $r\in\mathbb{N}$ such that $rD$ has integral coefficients.

We finish with a slight generalization of Mustaţă’s uniform generation property [Mus13, Thm. 4.1]. Observe that in loc. cit. it is required that the variety $X$ is projective over the ground field $k$ .

Theorem 4.6.

Let $R$ be a $k$ -algebra of finite type over a perfect field $k$ of characteristic $p>0$ . Let $X$ be an integral scheme of dimension $n$ which is projective over the spectrum of $R$ and smooth over $k$ . Let $D$ , $E$ , and $H$ be divisors on $X$ and $\lambda\in\mathbb{Q}_{\geq 0}$ such that

(i)

${\mathcal{O}}_{X}(H)$ is an ample, globally generated line bundle,
(ii)

$h^{0}(X,{\mathcal{O}}_{X}(mD))>0$ for some $m>0$ , and
(iii)

the $\mathbb{Q}$ -divisor $E-\lambda D$ is nef.

Then the sheaf $\mathcal{O}_{X}(K_{X/k}+E+dH)\otimes_{\mathcal{O}_{X}}\tau(\lambda\cdot\|D\|)$ is globally generated for all $d\geq n+1$ .

Proof.

We literally follow Mustaţă’s proof with two modifications. The proof requires Mumford’s theorem on Castelnuovo-Mumford regularity for the projective scheme $X$ over $R$ which holds also in this more general setting [BS13, 20.4.13]. Furthermore we replace the use of Fujita’s vanishing theorem to the sheaves $\mathscr{F}_{j}:=\mathcal{O}_{\mathscr{X}}(K_{X/k}+T_{j})$ , $j=1,\dots,r$ and the ample divisor $(d-i)H$ by an application of Keeler’s generalization [Kee03, Thm. 1.5]. ∎