Equimultiplicity Theory of Strongly $F$-regular rings

TL;DR

This paper develops a unified framework for analyzing various $F$-invariants like Hilbert--Kunz multiplicity and $F$-signature over strongly $F$-regular rings, enhancing understanding of singularities.

Contribution

It introduces techniques that unify the study of $F$-invariants and their behavior under localization in strongly $F$-regular rings.

Findings

Unified approach to $F$-invariants

Insights into singularity measurements

Behavior of invariants under localization

Abstract

We explore the equimultiplicity theory of the $F$-invariants Hilbert--Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F$-regular rings. Techniques introduced in this article provide a unified approach to the study of these $F$-invariants under localization and as measurements of singularities.

Equations

$$I_e=\{r\in R\mid \phi (F_{\ast }^{e}r)\in \mathfrak{m},\forall \phi \in {\mathrm{Hom}}_R(F_{\ast }^{e}R,R)\}$$

$$I_e(I)=\{r\in R\mid \phi (F_{\ast }^{e}r)\in I,\forall \phi \in {\mathrm{Hom}}_R(F_{\ast }^{e}R,R)\}.$$

$$\underset{n_1,\dots ,n_h\to \infty }{\lim }\frac{1}{n_1\cdots n_h}s(I,(x_1^{n_1},\dots ,x_h^{n_h}))=\sum _{P}e(x_1,\dots ,x_h;R/P)s(I{R}_P),$$

$${\mathrm{e}}_{\mathrm{HK}}(I)=\underset{e\to \infty }{\lim }\frac{\lambda (R/I^{[p^e]})}{p^{e\dim R}}.$$

$${\mathrm{e}}_{\mathrm{HK}}(R)=\underset{e\to \infty }{\lim }\frac{\lambda (R/{\mathfrak{m}}^{[p^e]})}{p^{e\dim (R)}}=\underset{e\to \infty }{\lim }\frac{\mu (F_{\ast }^{e}R)}{\mathrm{rank}(F_{\ast }^{e}R)}.$$

$$I_e(\mathfrak{a})=\{r\in R\mid \phi (F_{\ast }^{e}r)\in \mathfrak{a},\forall \phi \in {\mathrm{Hom}}_R(F_{\ast }^{e}R,R)\}.$$

$$\lambda \left(\frac{I_e(J,x)}{I_e(J)}\right)\ge \lambda \left(\frac{I_e(J)+(x^{p^e})}{I_e(J)}\right)=\lambda (R/(I_e(J):x^{p^e}))\ge \lambda (R/I_e(\mathfrak{m})).$$

$$s(\mathfrak{a})=\underset{e\to \infty }{\lim }\frac{{\mathrm{e}}_{\mathrm{HK}}(I_e(\mathfrak{a}))}{p^{e\dim (R)}}.$$

$$\underset{e\to \infty }{\lim }\frac{1}{p^{e\dim R}}\left|\lambda (R/I_e(\mathfrak{a}))-{\mathrm{e}}_{\mathrm{HK}}(I_e(\mathfrak{a}))\right|=0,$$

$$\underset{e\to \infty }{\mathrm{liminf}}\frac{{\mathrm{rank}}_M(F_{\ast }^{e}R)}{\mathrm{rank}(F_{\ast }^{e}R)}>0.$$

$$\underset{e\to \infty }{\mathrm{liminf}}\frac{{\mathrm{rank}}_M(F_{\ast }^{e}R)}{\mathrm{rank}(F_{\ast }^{e}R)}\ge \underset{e\to \infty }{\mathrm{liminf}}\frac{a_{e-e_0}(R)}{\mathrm{rank}(F_{\ast }^{e}R)}=\frac{s(R)}{\mathrm{rank}(F_{\ast }^{e_0}R)}>0.$$

$$F_{\ast }^{e}R\cong R^{\oplus a_e(R)}\oplus M_{e_0}^{\oplus {\mathrm{rank}}_{M_{e_0}}(F_{\ast }^{e}R)}\oplus N_e.$$

$$s(R_P)=\underset{e\to \infty }{\lim }\frac{a_e(R_P)}{\mathrm{rank}(F_{\ast }^{e}R)}$$

$$F_{\ast }{I}_e(\mathfrak{m})={\mathfrak{m}}^{\oplus a_e(R)}\oplus M_e={\mathfrak{m}}^{\oplus a_e(R)}\oplus \left(\{\eta \in M_e\mid \phi (\eta )\in P,\forall \phi \in {\mathrm{Hom}}_R(M_e,R)\}+\mathfrak{m}{M}_e\right).$$

$$M_e=\{\eta \in M_e\mid \phi (\eta )\in P,\forall \phi \in {\mathrm{Hom}}_R(M_e,R)\},$$

$$(I_e(P),x_1^{p^e},\dots ,x_i^{p^e}):x_{i+1}^{p^e}=(I_e(P),x_1^{p^e},\dots ,x_i^{p^e}).$$

$$I_e((P,x_1,\dots ,x_i):x_{i+1})=I_e(P,x_1,\dots ,x_i)=(I_e(P),x_1^{p^e},\dots ,x_i^{p^e}).$$

$$I_e((P,x_1,\dots ,x_i):x_{i+1})$$

$$I_e(R)=\frac{P^{[2^e]}{:}_S(xy-zw{)}^{2^e-1}}{(xy-zw)}.$$

$${\mathrm{e}}_{\mathrm{HK}}(R)=\underset{e\to \infty }{\lim }\frac{1}{\mathrm{rank}(F_{\ast }^{e}R)}(a_e(R)+b_e\mu (M_{e_0})+\mu (N_e))$$

$${\mathrm{e}}_{\mathrm{HK}}(R_P)=\underset{e\to \infty }{\lim }\frac{1}{\mathrm{rank}(F_{\ast }^{e}R)}(a_e(R)+b_e\mu ((M_{e_0}{)}_P)+\mu ((N_e{)}_P)).$$

$${\mathrm{e}}_{\mathrm{HK}}(R_P)$$

$$\le \underset{e\to \infty }{\lim }\frac{1}{\mathrm{rank}(F_{\ast }^{e}R)}\left(a_e(R)+b_e\mu (M_{e_0})+\mu (N_e)\right)-\underset{e\to \infty }{\mathrm{liminf}}\frac{b_e}{\mathrm{rank}(F_{\ast }^{e}R)}$$

$$={\mathrm{e}}_{\mathrm{HK}}(R)-\underset{e\to \infty }{\mathrm{liminf}}\frac{b_e}{\mathrm{rank}(F_{\ast }^{e}R)},$$

$${\mu }_R(F_{\ast }^{e}R/P{F}_{\ast }^{e}R)={\mu }_R(F_{\ast }^{e}R)={\mu }_{R_P}(F_{\ast }^e{R}_P)={\mu }_{R_P}(F_{\ast }^e{R}_P/P{F}_{\ast }^e{R}_P)\le {\mu }_R(F_{\ast }^{e}R/P{F}_{\ast }^{e}R).$$

$${\mu }_R(F_{\ast }^{e}R)={\mu }_R(F_{\ast }^{e}R/P{F}_{\ast }^{e}R)={\mu }_{R_P}(F_{\ast }^e{R}_P/P{F}_{\ast }^e{R}_P)={\mu }_{R_P}(F_{\ast }^e{R}_P).$$

$${\beta }_i^F(R)=\underset{e\to \infty }{\lim }\frac{\mu ({\Omega }_i^e(R))}{\mathrm{rank}(F_{\ast }^{e}R)}$$

$${\chi }_i^F(R)=\underset{e\to \infty }{\lim }\sum _{j=0}^i(-1{)}^j\frac{\mu ({\Omega }_{i-j}^e(R))}{\mathrm{rank}(F_{\ast }^{e}R)}=\sum _{j=0}^i(-1{)}^j{\beta }_{i-j}^F(R).$$

$${\mathrm{rank}}_R({\Omega }_i^e(R))={\chi }_{i-1}^e(R)+(-1{)}^i\mathrm{rank}(F_{\ast }^{e}R).$$

$$0\to {\Omega }_i^e(R)\to R^{\oplus {\beta }_{i-1}^e(R)}\to \cdots \to R^{\oplus \mu (F_{\ast }^{e}R)}\to F_{\ast }^{e}R\to 0.$$

$${\chi }_i^e(R)\ge {\chi }_{i-1}^e(R)+(-1{)}^i\mathrm{rank}(F_{\ast }^{e}R)-{\chi }_{i-1}^e(R)=(-1{)}^i\mathrm{rank}(F_{\ast }^{e}R)$$

Full text

Polstra was supported in part by NSF Postdoctoral Research Fellowship DMS $\#1703856$.

1. Introduction

The most intrinsic feature of a ring $R$ of prime characteristic $p>0$ is the Frobenius endomorphism given by taking the $p$-powers, $x\mapsto x^{p}$. Let $F^{e}_{*}R$ be the $R$-module obtained by restricting scalars along the $e$th Frobenius endomorphism. For sake of simplicity assume that $(R,\mathfrak{m},k)$ is local and $F$-finite, meaning $R$ is a local ring and $F^{e}_{*}R$ is a finitely generated $R$-module for each $e\in\mathbb{N}$. At the root of prime characteristic commutative algebra and algebraic geometry is Kunz’s fundamental result characterizing flatness of the Frobenius endomorphism.

Theorem 1.1** ([Kun69]).**

Let $(R,\mathfrak{m},k)$ be an $F$-finite local ring of prime characteristic $p>0$. Then $R$ is regular if and only if $F^{e}_{*}R$ is a free $R$-module for some (equivalently, all) $e\in\mathbb{N}$.

Motivated by Kunz’s theorem, it is natural to study non-regular prime characteristic rings by studying algebraic, geometric, and homological properties of the family of $R$-modules $\{F^{e}_{*}R\}_{e\in\mathbb{N}}$ which distinguish $R$ from a regular local ring. We consider the following measurements:

1. (1)

$\mu(F^{e}_{*}R)$, the minimal number of generators of $F^{e}_{*}R$ as an $R$-module; 2. (2)

$a_{e}(R)$, the largest rank of a free summand of $F^{e}_{*}R$; 3. (3)

$\beta_{i}^{e}(R):=\dim_{k}(\mbox{Tor}_{i}^{R}(k,F^{e}_{*}R))$, the $i$th Betti number of $F^{e}_{*}R$; 4. (4)

$\chi_{i}^{e}(R):=\sum_{j=0}^{i}(-1)^{j}\beta^{e}_{i-j}(R)$.

The asymptotic ratio of the above numbers, as compared with the rank of $F^{e}_{*}R$, produces several interesting and important numerical invariants unique to rings of prime characteristic.

1. (1)

Hilbert–Kunz multiplicity: $\displaystyle\operatorname{e_{HK}}(R)=\lim_{e\to\infty}\mu(F^{e}_{*}R)/\operatorname{rank}(F^{e}_{*}R)$, [Mon83]. 2. (2)

$F$-signature: $\displaystyle\operatorname{s}(R)=\lim_{e\to\infty}a_{e}(R)/\operatorname{rank}(F^{e}_{*}R)$, [HL02, SVdB97, Tuc12]. 3. (3)

The $i$th Frobenius Betti number: $\displaystyle\beta_{i}^{F}(R)=\lim_{e\to\infty}\beta_{i}^{e}(R)/\operatorname{rank}(F^{e}_{*}R)$, [AL08]. 4. (4)

The $i$th Frobenius Euler characteristic: $\displaystyle\chi_{i}^{F}(R)=\lim_{e\to\infty}\chi_{i}^{e}(R)/\operatorname{rank}(F^{e}_{*}R)$, [DPY18].

This article concerns the equimultiplicity theory of the above numerical invariants, a topic initiated by the second author in [Smi19]. Specifically, we are interested in understanding when the above measurements are unchanged under localization. Our main result in this direction is the following:

Theorem A**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring of prime characteristic $p>0$ and let $P\in\operatorname{Spec(R)}$. Let $\{\nu_{e}(R)\}_{e\in\mathbb{N}}$ be one of the following sequences of numbers:

• •

$\{\mu(F^{e}_{*}R)\}_{e\in\mathbb{N}}$

• •

$\{a_{e}(R)\}_{e\in\mathbb{N}}$

• •

$\{\beta_{i}^{e}(R)\}_{e\in\mathbb{N}}$

• •

$\{\chi_{i}^{e}(R)\}_{e\in\mathbb{N}}$

Let $\displaystyle\nu(R)=\lim_{e\to\infty}\nu_{e}(R)/\operatorname{rank}(F^{e}_{*}R)$. Then the following are equivalent:

1. (1)

$\nu(R)=\nu(R_{P})$; 2. (2)

For each $e\in\mathbb{N}$, $\nu_{e}(R)=\nu_{e}(R_{P})$.

In the scenario that $\{\nu_{e}(R)\}_{e\in\mathbb{N}}$ is the sequence of numbers $\{\mu(F^{e}_{*}R)\}_{e\in\mathbb{N}}$ then Theorem A is a significant improvement of [Smi19, Corollary 5.18], where the same theorem was proven under the additional assumption that $R/P$ is a regular local ring.

It has been known for some time that Hilbert–Kunz multiplicity, $F$-signature, and Frobenius Betti numbers serve as measurements of singularities, see [WY00, HY02], [HL02, AL03], and [AL08] respectively. Frobenius Euler characteristic was developed in [DPY18] as a tool to prove that the functions $\beta_{i}^{F}\colon\operatorname{Spec(R)}\to\mathbb{R}$ sending $P\mapsto\beta_{i}^{F}(R_{P})$ are upper semi-continuous and it was unclear from those techniques whether or not Frobenius Euler characteristic could be used to detect regular rings. Prior to this article, only the first Frobenius Euler characteristic was proven to serve as a measurement of singularity, see [Li08, Main Theorem (iv)]. In the present article, we show Frobenius Euler characteristic does indeed serve as a measurement of singularities under the strongly $F$-regular hypothesis.

Theorem B**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring of prime characteristic $p>0$. Then the following are equivalent:

1. (1)

$R$ is a regular local ring; 2. (2)

$\chi_{i}^{e}(R)=(-1)^{i}\operatorname{rank}(F^{e}_{*}R)$ for every $e\in\mathbb{N}$; 3. (3)

$\chi_{i}^{e}(R)=(-1)^{i}\operatorname{rank}(F^{e}_{*}R)$ for some $e\in\mathbb{N}_{\geq 1}$; 4. (4)

$\chi_{i}^{F}(R)=(-1)^{i}$.

We conjecture that Theorem B can be proven under weaker hypotheses. It seems likely that, similar to Hilbert–Kunz multiplicity and Frobenius Betti numbers, one would only need to assume the completion of the local ring $R$ at the maximal ideal has no low-dimensional components in order to know that $\chi_{i}^{F}(R)=(-1)^{i}$ implies $R$ is regular.

The $F$-signature of a local ring $(R,\mathfrak{m},k)$ can be studied through splitting ideals, a notion originating in [AE06]. For each $e\in\mathbb{N}$ the $e$th splitting ideal is

[TABLE]

and the $F$-signature of $R$ is realized as the limit $\operatorname{s}(R)=\lim_{e\to\infty}\operatorname{\lambda}(R/I_{e})/p^{e\dim(R)}$. To better understand the behavior of $F$-signature under localizations we consider relative splitting ideals: for each ideal $I\subseteq R$ and $e\in\mathbb{N}$ let

[TABLE]

Observe that $I_{e}(\mathfrak{m})=I_{e}$. If $I\subseteq R$ is an $\mathfrak{m}$-primary ideal then we can define an $F$-signature relative to the ideal $I$ as $\operatorname{s}(I)=\lim_{e\to\infty}\operatorname{\lambda}(R/I_{e}(I))/p^{e\dim(R)}$, a limit we will observe exists. Not only do relative splitting ideals allow us to understand the behavior of $F$-signature in the context of Theorem B, we prove the following associativity type formula for $F$-signature which is of independent interest. The corresponding formula for Hilbert–Samuel multiplicity is a classic and very useful result of Lech ([Lec57]), the version for Hilbert–Kunz multiplicity can be found in [Smi19, Proposition 5.4] and was extensively used therein.

Theorem C**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local domain of prime characteristic $p>0$. Suppose that $I\subseteq R$ is an ideal such that $R/I$ is Cohen-Macaulay of dimension $h$ and $\underline{x}=x_{1},\ldots,x_{h}$ a parameter sequence on $R/I$. Then

[TABLE]

where the sum is taken over all prime ideals $P\supseteq I$ such that $\dim(R/I)=\dim(R/P)$.

Section 2 contains background results and basic properties of splitting ideals relative to an ideal. The proofs of Theorem A and Theorem B can be found in Section 3. We also use Section 3 to further explore the behavior of splitting ideals. For example, see Theorem 3.9 for a proof that $\operatorname{depth}(R/P)=\operatorname{depth}(R/I_{e}(P))$ whenever $P$ is a prime ideal of a strongly $F$-regular local ring satisfying $\operatorname{s}(R)=\operatorname{s}(R_{P})$. Section 4 is devoted to proving Theorem C.

Acknowledgements

The authors thank Alessandro De Stefani for valuable feedback on a preliminary draft of this article.

2. Preliminary Results

2.1. Hilbert–Kunz multiplicity

Monsky’s introduction of Hilbert–Kunz multiplicity is a continuation of Kunz’s work on prime characteristic rings in [Kun69, Kun76].

Definition 2.1**.**

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p>0$ and $I$ be an $\mathfrak{m}$-primary ideal. Denote ${I}^{[p^{e}]}=(x^{p^{e}}\mid x\in I)$. Then the Hilbert–Kunz multiplicity of $I$ is

[TABLE]

The Hilbert–Kunz multiplicity of a local ring $(R,\mathfrak{m},k)$ is the Hilbert–Kunz multiplicity of the maximal ideal $\mathfrak{m}$ and is denoted by $\operatorname{e_{HK}}(R)$. If $R$ is an $F$-finite domain then $\operatorname{rank}(F^{e}_{*}R)=p^{e\dim R}[k:k^{p^{e}}]$ by [Kun76, Proposition 2.3] and therefore

[TABLE]

Hence the definition of Hilbert–Kunz multiplicity presented in the introduction agrees with Definition 2.1.

2.2. $F$-signature and splitting ideals

Below is a definition, due to Tucker, which is a natural generalization of the splitting ideals and presents a natural extension of $F$-signature.

Definition 2.2**.**

Let $R$ be an $F$-finite ring and $\mathfrak{a}$ be an ideal. The $e$th splitting ideal of $\mathfrak{a}$ is defined as

[TABLE]

We record the following basic properties concerning splitting ideals, many of which mimic the behavior of the standard splitting ideals $I_{e}=I_{e}(\mathfrak{m})$.

Lemma 2.3**.**

Suppose $(R,\mathfrak{m})$ is an $F$-finite local ring of prime characteristic $p>0$ and Krull dimension $d$. Let $\mathfrak{a}\subset R$ be an ideal. Then the sequence of ideals $\{I_{e}(\mathfrak{a})\}$ satisfies the following properties:

1. (1)

$I_{e}(\mathfrak{a})$* is an ideal;* 2. (2)

$\mathfrak{a}^{[p^{e}]}\subseteq I_{e}(\mathfrak{a})$; 3. (3)

$I_{e}(\mathfrak{a})^{[p]}\subseteq I_{e+1}(\mathfrak{a})$; 4. (4)

$\varphi(F^{e_{0}}_{*}I_{e+e_{0}}(\mathfrak{a}))\subseteq I_{e}(\mathfrak{a})$* for every $e,e_{0}\in\mathbb{N}$ and $\varphi\in\operatorname{Hom}_{R}(F^{e_{0}}_{*}R,R)$;* 5. (5)

If $\mathfrak{a}$ is $\mathfrak{m}$-primary then the limit $\operatorname{s}(\mathfrak{a}):=\lim\limits_{e\to\infty}\frac{\operatorname{\lambda}(R/I_{e}(\mathfrak{a}))}{p^{ed}}$ exists and $\operatorname{\lambda}(R/I_{e}(\mathfrak{a}))=\operatorname{s}(\mathfrak{a})p^{ed}+O(p^{e(d-1)})$. The value $\operatorname{s}(\mathfrak{a})$ is referred to as the $F$-signature of $\mathfrak{a}$; 6. (6)

If $W$ is a multiplicative set then $I_{e}(\mathfrak{a})R_{W}=I_{e}(\mathfrak{a}R_{W})$; 7. (7)

$I_{e}(\mathfrak{a}:J)=I_{e}(\mathfrak{a}):J^{[p^{e}]}$* for all ideals $J$;* 8. (8)

If $P$ is a prime ideal then $I_{e}(P)$ is $P$-primary; 9. (9)

If $x\in R$ is regular element of $R/\mathfrak{a}$ then $x$ is regular element of $R/I_{e}(\mathfrak{a})$ for every $e\in\mathbb{N}$; 10. (10)

If $R$ is a regular local ring then $I_{e}(\mathfrak{a})=\mathfrak{a}^{[p^{e}]}$ for every $e\in\mathbb{N}$; 11. (11)

If $\mathfrak{b}\subseteq R$ is an ideal and $\mathfrak{a}\subseteq\mathfrak{b}$ then $I_{e}(\mathfrak{a})\subseteq I_{e}(\mathfrak{b})$;

Proof.

The proofs of (1)-(4) are straightforward and are left to the reader. Statement (5) then follows by [PT18, Corollary 4.5]. To prove (6) it is enough to observe $\operatorname{Hom}_{R}(F^{e}_{*}R,R)_{W}\cong\operatorname{Hom}_{R_{W}}(F^{e}_{*}R_{W},R_{W})$. For (7) we note that $a\in I_{e}(\mathfrak{a}:J)$ if and only if for all $\varphi\in\operatorname{Hom}_{R}(F^{e}_{*}R,R)$ we have $\varphi(F^{e}_{*}a)\in(\mathfrak{a}:J)$, or equivalently, $\varphi(F^{e}_{*}J^{[p^{e}]}a)=J\varphi(F^{e}_{*}a)\subseteq\mathfrak{a}$. Statements (8) and (9) easily follow from (7). Observation (10) follows from Theorem 1.1; if $F^{e}_{*}R$ is free then it is then easy to see that $F^{e}_{*}I_{e}(\mathfrak{a})=\mathfrak{a}F^{e}_{*}R$ from which it follows that $I_{e}(\mathfrak{a})=\mathfrak{a}^{[p^{e}]}$. Property (11) is trivial. ∎

Corollary 2.4**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and $F$-pure local ring of prime characteristic $p>0$. If $J\subsetneq I$ are ideals then $I_{e}(J)\subsetneq I_{e}(I)$. Moreover, if $R$ is strongly $F$-regular and $J,I$ are $\mathfrak{m}$-primary, then $\operatorname{s}(J)>\operatorname{s}(I)$.

Proof.

Without loss of generality we may assume $I=(J,x)$ and $x\not\in J$. Then $I_{e}(J):x^{p^{e}}=I_{e}(J:x)\subseteq I_{e}(\mathfrak{m})\neq R$, so $I_{e}(J)\subsetneq I_{e}(J)+(x^{p^{e}})\subseteq I_{e}(J,x).$

For the second part, observe that

[TABLE]

Therefore $\operatorname{s}(J)-\operatorname{s}((J,x))\geq\operatorname{s}(R)>0$. ∎

Similar to the usual $F$-signature, the $F$-signature of an $\mathfrak{m}$-primary ideal $\mathfrak{a}$ is realized as the limit of normalized Hilbert–Kunz multiplicities of the ideals $I_{e}(\mathfrak{a})$.

Theorem 2.5**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and reduced local ring of prime characteristic $p>0$ and $\mathfrak{a}$ an $\mathfrak{m}$-primary ideal. Then

[TABLE]

Proof.

The assertion is equivalent to saying that

[TABLE]

which is the content of [Tuc12, Corollary 3.7]. ∎

3. Equimultiplicity of $F$-invariants

We are interested in understanding when the invariants Hilbert–Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristics are unchanged under localization. Work of the second author in [Smi19] began this study for Hilbert–Kunz multiplicity where the following was proven:

Theorem 3.1** ([Smi19, Corollary 5.16]).**

Let $(R,\mathfrak{m})$ be an excellent weakly $F$-regular local ring of prime characteristic $p>0$ and $P\subset R$ a prime ideal such that $R/P$ is a regular local ring. Then the following are equivalent:

1. (1)

$\operatorname{e_{HK}}(R)=\operatorname{e_{HK}}(R_{P})$, 2. (2)

for each $e\in\mathbb{N}$, $\operatorname{\lambda}(R/\mathfrak{m}^{[p^{e}]})/p^{e\dim(R)}=\operatorname{\lambda}(R_{P}/P^{[p^{e}]}R_{P})/p^{e0pt(P)}$.

The techniques surrounding Theorem 3.1 involve a careful and challenging analysis of the behavior of the ideals $\{P^{[p^{e}]}\}$ and $\{(P,\underline{x})^{[p^{e}]}\}$ where $\underline{x}$ is a regular system of parameters modulo $P$. Using elementary techniques, we recover the above theorem without assuming $R/P$ is a regular local ring, but we do replace the assumption of weakly $F$-regular with the conjecturally equivalent assumption that $R$ is strongly $F$-regular. Our techniques stem from a novel, yet simple, observation that if a module $M$ is a direct summand of $F^{e_{0}}_{*}R$ for some $e_{0}\in\mathbb{N}$ and $R$ is strongly $F$-regular, then asymptotically there will be many direct summands of $F^{e}_{*}R$ isomorphic to $M$ as $e\to\infty$. To make this precise, we begin with some notation.

Notation 3.2*.*

Let $R$ be a ring and $N\subseteq M$ be finitely generated $R$-modules. Let $\operatorname{rank}_{N}(M)$ denote the maximal number of $N$-summands appearing in all possible direct sum decompositions of $M$.

The following lemma should be compared with [SVdB97, Proposition 3.3.1].

Lemma 3.3**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring. Suppose $M$ is a finitely generated $R$-module such that $\operatorname{rank}_{M}(F_{*}^{e_{0}}R)>0$ for some $e_{0}\in\mathbb{N}$. Then

[TABLE]

Proof.

Suppose that $F^{e_{0}}_{*}R\cong M\oplus N$. For each $e\in\mathbb{N}$ write $F^{e}_{*}R\cong R^{\oplus a_{e}(R)}\oplus M_{e}$. Then $F^{e+e_{0}}_{*}R\cong F^{e_{0}}_{*}R^{\oplus a_{e}(R)}\oplus F^{e_{0}}_{*}M_{e}$ and it follows that $M^{\oplus a_{e}(R)}$ is a direct summand of $F^{e+e_{0}}_{*}R$. In particular, $\operatorname{rank}_{M}(F_{*}^{e+e_{0}}R)\geq a_{e}(R)$ and therefore

[TABLE]

∎

3.1. $F$-signature and splitting ideals

We are prepared to present a proof of Theorem A for $F$-signature. But first:

Remark 3.4*.*

To make full use of Lemma 3.3 in the following theorem we remind the reader that the maximal rank of a free summand of a finitely generated module $M$ over a local ring $R$ is invariant of a choice of a direct sum decomposition. This is because $R$ is a direct summand of $M$ if and only if $\widehat{R}$ is a direct summand of $\widehat{M}$ and a complete local ring satisfies the Krull–Schmidt condition.

Theorem 3.5**.**

Let $(R,\mathfrak{m})$ be a strongly $F$-regular and $F$-finite local ring. Suppose $P\subset R$ is a prime ideal. Then $\operatorname{s}(R)=\operatorname{s}(R_{P})$ if and only if $a_{e}(R)=a_{e}(R_{P})$ for every $e\in\mathbb{N}$.

Proof.

If $a_{e}(R)=a_{e}(R_{P})$ for every $e\in\mathbb{N}$ then it is trivial that $\operatorname{s}(R)=\operatorname{s}(R_{P})$ since $\operatorname{rank}_{R}(F^{e}_{*}R)=\operatorname{rank}_{R_{P}}(F^{e}_{*}R_{P})$.

Suppose that $a_{e_{0}}(R_{P})>a_{e_{0}}(R)$ and write $F^{e_{0}}_{*}R\cong R^{\oplus a_{e_{0}}}\oplus M_{e_{0}}$. Then $(M_{e_{0}})_{P}$ has a free $R_{P}$-summand. For each $e\in\mathbb{N}$ by Remark 3.4 we may write

[TABLE]

Localizing at the prime $P$ we see that $a_{e}(R_{P})\geq a_{e}(R)+\operatorname{rank}_{M_{e_{0}}}(F_{*}^{e}R)$ and

[TABLE]

Therefore $\operatorname{s}(R_{P})>\operatorname{s}(R)$ by Lemma 3.3. ∎

The following theorem states that the splitting ideals of $R$ and that of a localization $R_{P}$ can be effectively compared whenever the Frobenius splitting numbers of $R$ and $R_{P}$ agree.

Theorem 3.6**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local ring of prime characteristic $p>0$, $P$ be a prime ideal. Then the following are equivalent:

1. (1)

$a_{e}(R)=a_{e}(R_{P})$, 2. (2)

$I_{e}((P,I))=I_{e}(P)+I^{[p^{e}]}$* for all ideals $I$,* 3. (3)

$I_{e}(\mathfrak{m})=I_{e}(P)+\mathfrak{m}^{[p^{e}]}$.

Proof.

Write $F^{e}_{*}R\cong R^{\oplus a_{e}(R)}\oplus M_{e}$. By definition, $F^{e}_{*}I_{e}(P)=P^{\oplus a_{e}(R)}\oplus\{\eta\in M_{e}\mid\varphi(\eta)\in P,\,\forall\varphi\in\operatorname{Hom}_{R}(M_{e},R)\}$. Hence, $a_{e}(R)=a_{e}(R_{P})$ if and only if $\operatorname{Hom}_{R}(M_{e},R)=\operatorname{Hom}_{R}(M_{e},P)$. It follows then that $\operatorname{Hom}_{R}(M_{e},R)=\operatorname{Hom}_{R}(M_{e},P+I)$, so $F^{e}_{*}(I_{e}(P)+I^{[p^{e}]})=(P,I)^{\oplus a_{e}(R)}\oplus(M_{e}+IM_{e})=(P,I)^{\oplus a_{e}(R)}\oplus M_{e}=F^{e}_{*}I_{e}(P+I)$. Thus (1) implies (2).

Since (2) trivially implies (3) it is left to show that the last condition implies the first. Suppose that $I_{e}(\mathfrak{m})=(I_{e}(P),\mathfrak{m}^{[p^{e}]})$. Then

[TABLE]

By Nakayama’s lemma we then get that

[TABLE]

i.e., $\operatorname{Hom}_{R}(M_{e},R)=\operatorname{Hom}_{R}(M_{e},P)$ and therefore $a_{e}(R)=a_{e}(R_{P})$. ∎

Theorem 3.5 and Theorem 3.6 imply the following:

Corollary 3.7**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring of prime characteristic $p>0$ and $P$ be a prime ideal. Then $\operatorname{s}(R)=\operatorname{s}(R_{P})$ if and only if $I_{e}(\mathfrak{m})=I_{e}(P)+\mathfrak{m}^{[p^{e}]}$ for every $e\in\mathbb{N}$.

The techniques surrounding Theorem 3.5 provide a novel proof that the $F$-signature of a local ring is $1$ if and only if $R$ is a regular local ring.

Theorem 3.8** ([HL02, Corollary 16]).**

Let $(R,\mathfrak{m})$ be an $F$-finite local ring of prime characteristic $p>0$. Then $\operatorname{s}(R)=1$ if and only if $R$ is a regular local ring.

Proof.

Having positive $F$-signature implies $R$ is strongly $F$-regular.111The converse also holds, see [AL03, Main Theorem]. Hence, $R$ is a domain, so $R_{(0)}$ is a regular ring and, therefore, $\operatorname{s}(R_{(0)})=\operatorname{s}(R)$. Invoking Theorem 3.5 we then have that $a_{e}(R)=a_{e}(R_{(0)})=\operatorname{rank}(F^{e}_{*}R)$. Therefore $F^{e}_{*}R$ is a free $R$-module and $R$ is a regular local ring by Theorem 1.1. ∎

The advantage of the proof of Theorem 3.8 is that it directly uses Kunz’s Theorem while the proof of [HL02, Corollary 16] invokes the fact that $R$ must be regular if $\operatorname{e_{HK}}(R)=1$ ([WY00, HY02]). We may also adapt our approach to give a somewhat novel proof that Hilbert–Kunz multiplicity of a formally unmixed local ring is $1$ if and only if $R$ is a regular local ring, see Theorem 3.11 below.

Theorem 3.9**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring of prime characteristic $p>0$. Suppose that $P\in\operatorname{Spec(R)}$, $\operatorname{s}(R)=\operatorname{s}(R_{P})$, and $\underline{x}=x_{1},\ldots,x_{h}$ is a sequence of elements in $R$. Then the following are equivalent:

1. (1)

$\underline{x}$* is a regular sequence on $R/P$;* 2. (2)

$\underline{x}$* is a regular sequence on $R/I_{e}(P)$ for each $e\in\mathbb{N}$;* 3. (3)

$\underline{x}$* is a regular sequence on $R/I_{e}(P)$ for some $e\in\mathbb{N}$.*

In particular, $\operatorname{depth}(R/P)=\operatorname{depth}(R/I_{e}(P))$ for all $e\in\mathbb{N}$.

Proof.

Let $x_{1},\ldots,x_{h}$ be a regular sequence on $R/P$. To show that $x_{1},\ldots,x_{h}$ is a regular sequence on $R/I_{e}(P)$ it is equivalent to check that for every $0\leq i\leq h-1$

[TABLE]

By Theorem 3.5 and Theorem 3.6 we have that $(I_{e}(P),x^{p^{e}}_{1},\ldots,x_{i}^{p^{e}})=I_{e}(P,x_{1},\ldots,x_{i})$ and by (7) of Lemma 2.3 we have that $(I_{e}(P,x_{1},\ldots,x_{i})):x_{i+1}^{p^{e}}=I_{e}((P,x_{1},\ldots,x_{i}):x_{i+1}).$ But $x_{1},\ldots,x_{h}$ is a regular sequence on $R/P$ and therefore by a second application of Theorem 3.5 and Theorem 3.6 we see that

[TABLE]

Now suppose that for some $e\in\mathbb{N}$ that $x_{1},\ldots,x_{h}$ is a regular sequence on $R/I_{e}(P)$. Then for each $0\leq i\leq h-1$ we have by (7) of Lemma 2.3, Theorem 3.5, and Theorem 3.6 that

[TABLE]

By Corollary 2.4 we must have that $(P,x_{1},\ldots,x_{i}):x_{i+1}=(P,x_{1},\ldots,x_{i})$ for each $0\leq i\leq h-1$ and therefore $x_{1},\ldots,x_{h}$ is indeed a regular sequence on $R/P$. ∎

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring of prime characteristic $p>0$. Observe by (9) of Lemma 2.3 that $\operatorname{depth}(R/I_{e}(P))\geq 1$ for every $e\in\mathbb{N}$ and $P\in\operatorname{Spec(R)}\setminus\{\mathfrak{m}\}$. However, it does not follow that $\operatorname{depth}(R/I_{e}(P))=\operatorname{depth}(R/P)$ if we do not assume $\operatorname{s}(R)=\operatorname{s}(R_{P})$.

Example 3.10*.*

Consider the regular local ring $S$ of prime characteristic $2$ obtained by localizing $\mathbb{F}_{2}[x,y,z,w]$ at the maximal ideal $(x,y,z,w)$ and let $R=S/(xy-zw)$. Then $R$ is a strongly $F$-regular isolated singularity. Consider the height $1$ prime ideal $P=(x,z)$. By the techniques surrounding Fedder’s criterion [Fed83], c.f. [Gla96, Theorem 2.3], for each $e\in\mathbb{N}$ we have that

[TABLE]

Observe that $R/P$ is a regular local ring of dimension $2$, yet one can check that $I_{1}(R)=(xz,x^{2},z^{2})$ and $R/I_{1}(R)$ has depth $1$.

3.2. Hilbert-Kunz multiplicity

Now, we prove Theorem A for Hilbert–Kunz multiplicity.

Theorem 3.11**.**

Let $(R,\mathfrak{m})$ be a strongly $F$-regular and $F$-finite local ring of dimension $d$ and $P\in\operatorname{Spec(R)}$. Then the following are equivalent:

1. (1)

$\operatorname{e_{HK}}(R)=\operatorname{e_{HK}}(R_{P})$; 2. (2)

$\operatorname{\lambda}(R/\mathfrak{m}^{[p^{e}]})/p^{ed}=\operatorname{\lambda}(R_{P}/P^{[p^{e}]})/p^{e0pt(P)}$* for every $e\in\mathbb{N}$;* 3. (3)

$\mu(F^{e}_{*}R)=\mu(F^{e}_{*}R_{P})$* for every $e\in\mathbb{N}$;* 4. (4)

$F^{e}_{*}R/PF^{e}_{*}R$* is a free $R/P$-module for every $e\in\mathbb{N}$.*

Proof.

Conditions (2) and (3) are equivalent by [Kun76, Proposition 2.3] and clearly conditions (2) and (3) imply (1). To show that condition (1) implies condition (3) suppose that $\mu(F^{e_{0}}_{*}R)>\mu(F^{e_{0}}_{*}R_{P})$. If we write $F^{e_{0}}_{*}R\cong R^{\oplus a_{e_{0}}(R)}\oplus M_{e_{0}}$ then $\mu(M_{e_{0}})>\mu((M_{e_{0}})_{P})$. Let $b_{e}=\operatorname{rank}_{M_{e_{0}}}(F^{e}_{*}R)$. By Remark 3.4 we may write $F^{e}_{*}R\cong R^{\oplus a_{e}}\oplus(M_{e_{0}})^{\oplus b_{e}}\oplus N_{e}$ and it follows that

[TABLE]

and

[TABLE]

Therefore

[TABLE]

a value strictly less than $\operatorname{e_{HK}}(R)$ by Lemma 3.3.

Now suppose that $\operatorname{e_{HK}}(R)=\operatorname{e_{HK}}(R_{P})$. To show that $F^{e}_{*}R/PF^{e}_{*}R$ is a free $R/P$-module observe first that by Nakayama’s Lemma, $\mu_{R_{P}}(F^{e}_{*}R_{P})=\mu_{R_{P}}(F^{e}_{*}R_{P}/PF^{e}_{*}R_{P})$. Therefore

[TABLE]

Therefore, as an $R/P$-module, we have that $F^{e}_{*}R/PF^{e}_{*}R$ is generated by $\operatorname{rank}_{R/P}(F^{e}_{*}R/PF^{e}_{*}R)$ elements and must be free.

Conversely, if $F^{e}_{*}R/PF^{e}_{*}R$ is a free $R/P$-module for every $e\in\mathbb{N}$ then $\mu_{R}(F^{e}_{*}R/PF^{e}_{*}R)=\mu_{R_{P}}(F^{e}_{*}R_{P}/PF^{e}_{*}R_{P})$ and therefore

[TABLE]

∎

The following corollary is the analogue of Theorem 3.9 for Hilbert–Kunz multiplicity.

Corollary 3.12**.**

Let $(R,\mathfrak{m})$ be a strongly $F$-regular and $F$-finite local ring of prime characteristic $p>0$. Suppose that $P\in\operatorname{Spec(R)}$ and $\operatorname{e_{HK}}(R)=\operatorname{e_{HK}}(R_{P})$. Then for each sequence of elements $\underline{x}=x_{1},\ldots,x_{h}$ the following are equivalent:

1. (1)

$\underline{x}$* is a regular sequence on $R/P$;* 2. (2)

$\underline{x}$* is a regular sequence on $R/P^{[p^{e}]}$ for each $e\in\mathbb{N}$;* 3. (3)

$\underline{x}$* is a regular sequence on $R/P^{[p^{e}]}$ for some $e\in\mathbb{N}$.*

In particular, $\operatorname{depth}(R/P)=\operatorname{depth}(R/P^{[p^{e}]})$ for every $e\in\mathbb{N}$.

Proof.

For any finitely generated $R$-module $M$ a sequence of elements $\underline{x}$ is a regular sequence on $M$ if and only if $\underline{x}$ is a regular sequence on $F^{e}_{*}M$. The corollary is immediate by Theorem 3.11 since the modules $F^{e}_{*}(R/P^{[p^{e}]})\cong F^{e}_{*}R/PF^{e}_{*}R$ are free $R/P$-modules. ∎

Corollary 3.12 is an improvement of an observation that can be made from [Smi19, Proposition 3.1 and Corollary 5.19]: if $(R,\mathfrak{m})$ is weakly $F$-regular, $P\in\operatorname{Spec(R)}$ satisfies $\operatorname{e_{HK}}(R)=\operatorname{e_{HK}}(R_{P})$, and $R/P$ is regular then $R/P^{[p^{e}]}$ is Cohen-Macaulay for $e\in\mathbb{N}$.

We utilize Theorem 3.11 and results of [AE08] and provide a novel proof that the Hilbert–Kunz multiplicity of a local ring is $1$ if and only if the ring is regular. We recall that a ring is unmixed if it is equidimensional and has no embedded components.

Theorem 3.13** ([WY00]).**

Let $(R,\mathfrak{m})$ be a formally unmixed local $F$-finite ring of prime characteristic $p>0$. Then $\operatorname{e_{HK}}(R)=1$ if and only if $R$ is a regular local ring.

Proof.

The assumption on Hilbert–Kunz multiplicity implies that $R$ is strongly $F$-regular, see [AE08, Corollary 3.6]. By Theorem 3.11 applied to $P=(0)$, $\mu(F_{*}R)=\operatorname{rank}F_{*}R$, so $F_{*}R$ is a free $R$-module and $R$ is regular by Theorem 1.1. ∎

3.3. Frobenius Betti numbers and Frobenius Euler characteristic

We now turn our attention to the behavior of Frobenius Betti numbers and Frobenius Euler characteristics under localizations.

Definition 3.14**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local domain of prime characteristic $p>0$. For each $e\in\mathbb{N}$ let $\Omega_{i}^{e}(R)$ be the $i$th syzygy in the minimal free resolution of $F^{e}_{*}R$. The $i$th Frobenius Betti number of $R$ is

[TABLE]

and the $i$th Frobenius Euler characteristic of $R$ is

[TABLE]

We refer the reader to [Li08, AL08, DSHNnB17, DPY18] for basics on Frobenius Betti numbers and [DPY18] for basics on Frobenius Euler characteristic. Our study begins with a simple application of the Auslander–Buchsbaum formula.

Lemma 3.15**.**

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p>0$. The following are equivalent:

1. (1)

$R$* is a regular local ring;* 2. (2)

$F^{e}_{*}R$* has finite projective dimension as an $R$-module for every $e\geq 1$;* 3. (3)

$F^{e}_{*}R$* has finite projective dimension for some $e\in\mathbb{N}$.*

Proof.

It is easy to see that $\operatorname{depth}(R)=\operatorname{depth}(F^{e}_{*}R)$ for every $e\in\mathbb{N}$. Hence by the Auslander–Buchsbaum formula, if the projective dimension of $F^{e}_{*}R$ is finite then $F^{e}_{*}R$ is a free $R$-module and the lemma follows from Theorem 1.1. ∎

Lemma 3.16**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local domain of prime characteristic $p>0$. Then

[TABLE]

Moreover, if $R$ is not regular then $\beta_{i}^{e}(R)>\operatorname{rank}_{R}(\Omega_{i}^{e}(R))=\chi_{i-1}^{e}(R)+(-1)^{i}\operatorname{rank}(F^{e}_{*}R)$.

Proof.

Rank is additive on exact sequences and there are long exact sequences

[TABLE]

By Lemma 3.15, if $R$ is not regular, then $\Omega_{i}^{e}(R)$ is not free, hence $\beta_{i}^{e}(R)=\mu(\Omega_{i}^{e}(R))>\operatorname{rank}(\Omega_{i}^{e}(R))$. ∎

Lemma 3.17**.**

Let $(R,\mathfrak{m})$ be a local $F$-finite domain of prime characteristic $p$ and let $e,i\in\mathbb{N}$ with $e\geq 1$. Then $\chi^{e}_{i}(R)\geq(-1)^{i}\operatorname{rank}(F^{e}_{*}R)$ with equality if and only if $R$ is a regular local ring.

Proof.

For $i=0$ the lemma follows from Theorem 1.1. If $i\geq 1$ and $e\in\mathbb{N}$ then $\chi_{i}^{e}(R)=\beta_{i}^{e}(R)-\chi_{i-1}^{e}(R)$. Applying Lemma 3.16 we arrive at

[TABLE]

with equality if and only if $R$ is a regular local ring. ∎

Lemma 3.18**.**

Let $(R,\mathfrak{m})$ be a local $F$-finite domain and $P\in\operatorname{Spec(R)}$. Then $\beta_{i}^{e}(R)=\beta_{i}^{e}(R_{P})$ if and only if $\chi_{i}^{e}(R)=\chi_{i}^{e}(R_{P})$ and $\chi_{i-1}^{e}(R)=\chi_{i-1}^{e}(R_{P})$. In particular, if $\beta_{1}^{e}(R)=\beta_{1}^{e}(R_{P})$ then $\mu(F^{e}_{*}R)=\mu(F^{e}_{*}R_{P})$.

Proof.

Observe first that

[TABLE]

and

[TABLE]

Suppose that $\beta_{i}^{e}(R)=\beta_{i}^{e}(R_{P})$. The function $\chi_{i}^{e}\colon\operatorname{Spec(R)}\to\mathbb{R}$ is upper semicontinuous, [DPY18, Proposition 3.1], therefore $\chi_{i}^{e}(R)\geq\chi_{i}^{e}(R_{P}).$ If $\beta_{i}^{e}(R)=\beta_{i}^{e}(R_{P})$ then

[TABLE]

but the function $\chi_{i-1}^{e}\colon\operatorname{Spec(R)}\to\mathbb{R}$ is also upper semicontinuous, therefore equality must hold. ∎

Similar to Lemma 3.3, if $R$ is strongly $F$-regular and a module $M$ appears as a direct summand of $\Omega_{i}^{e_{0}}(R)$ for some $e_{0}\in\mathbb{N}$ then $M$ appears as a direct summand of $\Omega_{i}^{e}(R)$ asymptotically many times as $e\to\infty$.

Lemma 3.19**.**

Let $(R,\mathfrak{m})$ be an $F$-finite and strongly $F$-regular local ring of prime characteristic $p>0$ and $M$ be a finitely generated $R$-module. If $\operatorname{rank}_{M}(\Omega_{i}^{e_{0}}(R))>0$ for some $e_{0}\in\mathbb{N}$ then

[TABLE]

Proof.

Suppose $M$ is a direct summand of $\Omega_{i}^{e_{0}}(R)$. Observe that $F^{e+e_{0}}_{*}R$ has a direct summand $F^{e_{0}}_{*}R^{\oplus a_{e}}$. It readily follows that $\Omega_{e+e_{0}}(R)$ has $\Omega_{e_{0}}(R)^{\oplus a_{e}(R)}$ as a direct summand and therefore $\operatorname{rank}_{M}(\Omega_{e+e_{0}})\geq a_{e}(R)$. In particular,

[TABLE]

∎

We are now prepared to prove Theorem A for Frobenius Betti numbers and Frobenius Euler characteristics. We first present a proof of Theorem A for Frobenius Betti numbers.

Theorem 3.20**.**

Let $(R,\mathfrak{m})$ be an $F$-finite strongly $F$-regular local ring of prime characteristic $p>0$ and $P\in\operatorname{Spec(R)}$. Then for each integer $i\geq 0$, $\beta_{i}^{F}(R)=\beta_{i}^{F}(R_{P})$ if and only if $\beta_{i}^{e}(R)=\beta_{i}^{e}(R_{P})$ for every $e\in\mathbb{N}$.

Proof.

Clearly if $\beta_{i}^{e}(R)=\beta_{i}^{e}(R_{P})$ for every integer $e$ then $\beta_{i}^{F}(R)=\beta_{i}^{F}(R_{P})$. Suppose there exists an integer $e_{0}$ such that $\mu(\Omega_{i}^{e_{0}}(R_{P}))<\mu(\Omega_{i}^{e_{0}}(R))$. For each $e\in\mathbb{N}$ let $b_{e}=\operatorname{rank}_{\Omega_{i}^{e_{0}}(R)}(\Omega^{e}_{i}(R))$. Then we can write $\Omega^{e}_{i}(R)\cong\Omega^{e_{0}}_{i}(R)^{\oplus b_{e}}\oplus M_{e}$. Localizing at $P$,

[TABLE]

where $F_{P}$ is a free $R_{P}$-module. It readily follows that

[TABLE]

Therefore $\beta_{i}^{F}(R_{P})\leq\beta_{i}^{F}(R)-\liminf_{e\to\infty}\frac{b_{e}}{\operatorname{rank}(F^{e}_{*}R)}$ which is strictly less then $\beta_{i}^{F}(R)$ by Lemma 3.19. ∎

Following the proof of Theorem 3.11 we recover [AL08] for strongly $F$-regular rings.

Corollary 3.21**.**

Let $(R,\mathfrak{m})$ be an $F$-finite strongly $F$-regular local ring of prime characteristic $p>0$. Then for each integer $i\geq 0$, $\beta_{i}^{F}(R)=0$ if and only if $R$ is a regular local ring.

Proof.

By Lemma 3.15, $\beta_{i}^{F}(R_{(0)})=0=\beta_{i}^{F}(R)$. Therefore $\beta_{i}^{e}(R)=\beta_{i}^{e}(R_{(0)})=0$ and the claim follows from Lemma 3.15. ∎

Finally, we complete our proof of Theorem A by establishing an equimultiplicity criterion for Frobenius Euler characteristic.

Theorem 3.22**.**

Let $(R,\mathfrak{m})$ be an $F$-finite strongly $F$-regular local ring of prime characteristic $p>0$ and $P\in\operatorname{Spec(R)}$. Then for each integer $i\geq 0$, $\chi_{i}^{F}(R)=\chi_{i}^{F}(R_{P})$ if and only if $\chi_{i}^{e}(R)=\chi_{i}^{e}(R_{P})$ for every $e\in\mathbb{N}$.

Proof.

Without loss of generality we may assume $R$ is not regular. By Lemma 3.16, $\chi_{i}^{e}(R)=\chi_{i}^{e}(R_{P})$ if and only if $\operatorname{rank}(\Omega^{e}_{i+1}(R))=\operatorname{rank}(\Omega^{e}_{i+1}(R_{P}))$ and $\chi_{i}^{F}(R)=\chi_{i}^{F}(R_{P})$ if and only if

[TABLE]

Suppose there exists an integer $e_{0}$ such that $\operatorname{rank}(\Omega_{i+1}^{e_{0}}(R))\not=\operatorname{rank}(\Omega_{i+1}^{e_{0}}(R_{P}))$. Therefore $\Omega_{i+1}^{e_{0}}(R)_{P}$ has a nonzero free summand. Let $b_{e}=\operatorname{rank}_{\Omega_{i+1}^{e_{0}}(R)}(\Omega_{i+1}^{e}(R))$, by Lemma 3.19 $\liminf\limits_{e\to\infty}\frac{b_{e}}{\operatorname{rank}(F^{e}_{*}R)}>0$. Then for each integer $e\in\mathbb{N}$ the $R_{P}$-module $\Omega_{i+1}^{e}(R)_{P}$ contains a free summand of rank $b_{e}$. In particular, we have that $\operatorname{rank}(\Omega_{i+1}^{e}(R_{P}))\leq\operatorname{rank}(\Omega_{i+1}^{e}(R)_{P})-b_{e}$. Therefore

[TABLE]

∎

As with $F$-signature, Hilbert–Kunz multiplicity, and Frobenius Betti numbers, we now know that Frobenius Euler characteristic can be used to detect regular rings, provided we know the ring being studied is strongly $F$-regular.

Theorem 3.23**.**

Let $(R,\mathfrak{m})$ be an $F$-finite strongly $F$-regular local ring of prime characteristic $p>0$. The following are equivalent:

1. (1)

$R$* is a regular local ring;* 2. (2)

$\chi^{e}_{i}(R)=(-1)^{i}\operatorname{rank}(F^{e}_{*}R)$* for every $e\in\mathbb{N}$;* 3. (3)

$\chi^{e}_{i}(R)=(-1)^{i}\operatorname{rank}(F^{e}_{*}R)$* for some $e\in\mathbb{N}$;* 4. (4)

$\chi_{i}^{F}(R)=(-1)^{i}$.

Proof.

The equivalence of $(1),(2)$, and $(3)$ is the content of Lemma 3.17 and $(4)$ is trivially implied by condition $(2)$. Now, an argument with the generic point as in Theorem 3.8 shows that $(4)$ implies $(2)$ by Theorem 3.22. ∎

4. An associativity formula for $F$-signature

Our proof of Theorem C begins with two technical lemmas.

Lemma 4.1**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local ring of prime characteristic $p>0$. Suppose that $I\subset R$ is an ideal such that $R/I$ is Cohen-Macaulay of dimension $h$ and $x_{1},x_{2},\ldots,x_{h}$ a parameter sequence on $R/I$. Then for all sequences of natural numbers $n_{1},n_{2},\ldots,n_{h}$ we have that

1. (1)

[TABLE]

and 2. (2)

[TABLE]

Proof.

We may pass to $I^{\prime}=I+(x_{2}^{n_{2}},\ldots,x_{h}^{n_{h}})$ and assume that $\dim R/I=1$.

We claim there exist short exact sequences

[TABLE]

Observe that if such short exact sequences exist then the first inequality is obvious since the length of the left piece of a short exact sequence is no more than the length of the middle term. The second inequality is equivalent to the inequality

[TABLE]

an inequality which follows from the first since we can filter $\operatorname{\lambda}(R/I_{e}(I+(x^{n})))$ as

[TABLE]

To show that the above short exact sequences exist, we first notice that

[TABLE]

Indeed, if $u\in I_{e}(I+(x^{n-1}))$ and $\varphi\in\operatorname{Hom}_{R}(F^{e}_{*}R,R)$ then

[TABLE]

Therefore there are right exact sequences

[TABLE]

To show injectivity of the first map observe that an element $u\in I_{e}(I+(x^{n-1}))$ satisfies $x^{p^{e}}u\in I_{e}(I+(x^{n+1}))$ if and only if $u\in I_{e}(I+(x^{n+1})):x^{p^{e}}$. By (7) of Lemma 2.3 we have that

[TABLE]

where the second equality follows by standard observations on parameter ideals in the Cohen-Macaulay ring $R/I$. ∎

The following technical lemma is very much in the spirit of [PT18, Theorem 4.3].

Lemma 4.2**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local domain of prime characteristic $p>0$ and of Krull dimension $d$. Suppose that $I\subset R$ is an ideal such that $R/I$ is Cohen-Macaulay of dimension $h$ and $\underline{x}=x_{1},\ldots,x_{h}$ a parameter sequence on $R/I$. Then there exists a constant $C\in\mathbb{R}$ such that for all $e,n_{1},n_{2},\ldots,n_{h}\in\mathbb{N}$

[TABLE]

Proof.

Denote by $\underline{N}$ a Cartesian product of natural numbers, $\underline{N}=(n_{1},n_{2},\ldots,n_{h})$, let $N=n_{1}n_{2}\cdots n_{h}$, and for each $\underline{N}$ let $\underline{x}^{\underline{N}}$ be the sequence of elements $x_{1}^{n_{1}},x_{2}^{n_{2}},\cdots,x_{h}^{n_{h}}.$ We are claiming there exists a constant $C$, depending only on $\operatorname{\lambda}(R/(I+(\underline{x})))$, such that for all $\underline{N}$

[TABLE]

We will first show that there exists a constant $C$ such that for all $\underline{N}$ and $e\in\mathbb{N}$

[TABLE]

The $R$-module $F_{*}R$ is finitely generated and torsion-free so there exists a short exact sequence

[TABLE]

where $T$ is a finitely generated torsion $R$-module. By (4) of Lemma 2.3

[TABLE]

and therefore there are right exact sequences

[TABLE]

where $T_{e}$ is the homomorphic image of $T/I_{e}(I+(\underline{x}^{\underline{N}}))T$. Therefore

[TABLE]

Suppose that $c\in R$ is a nonzero element which annihilates $T$. Because $(I+(\underline{x}^{N}))^{[p^{e}]}\subseteq I_{e}(I+(\underline{x}^{\underline{N}}))$ there exists a surjective map

[TABLE]

and we have that

[TABLE]

It is well known there exists $C\in\mathbb{R}$, depending only on the ring $R$, such that

[TABLE]

see [Pol18, Proposition 3.3] for example. Because $R/I$ is Cohen-Macaulay we know that

[TABLE]

If we divide the inequality in 4.1 by $\operatorname{rank}(F_{*}R)p^{(e+1)d}$ we obtain that

[TABLE]

for every $e\in\mathbb{N}$. The constant $\mu(T)C\operatorname{\lambda}(R/(I+\underline{x}))$ has no dependence on $e$ or $N$, so we replace $C$ by this constant and utilize [PT18, Lemma 3.5] to obtain that

[TABLE]

Obtaining inequalities of the form

[TABLE]

is almost identical to the above. Begin by examining a short exact sequence of the form

[TABLE]

where $T^{\prime}$ is a torsion $R$-module. By (3) of Lemma 2.3 we have that $\psi(I_{e}(I+(\underline{x}^{\underline{N}}))^{[p]})\subseteq I_{e+1}(I+(\underline{x}^{\underline{N}}))F_{*}R$ and so there are right exact sequences

[TABLE]

where $T^{\prime}_{e}$ is the homomorphic image of $T^{\prime}/I_{e+1}(I+(\underline{x}^{\underline{N}}))T^{\prime}$. The reader is now encouraged to follow the techniques above and the techniques of [PT18, Theorem 4.3] to obtain inequalities as described in 4.2. ∎

For the proof of the Theorem C we recall the following standard result: if $a_{m,n}$ is a bisequence such that

• •

$\lim\limits_{m,n\to\infty}a_{m,n}$ exists, and

• •

$\lim\limits_{n\to\infty}a_{m,n}$ exists for all $m$,

then $\lim\limits_{m,n\to\infty}a_{m,n}=\lim\limits_{m\to\infty}\lim\limits_{n\to\infty}a_{m,n}$.

Theorem 4.3**.**

Let $(R,\mathfrak{m})$ be an $F$-finite local ring of prime characteristic $p>0$ and of Krull dimension $d$. Suppose that $I\subseteq R$ is an ideal such that $R/I$ is Cohen-Macaulay of dimension $h$ and $\underline{x}=x_{1},\ldots,x_{h}$ a parameter sequence on $R/I$. Then

[TABLE]

where the sum is taken over all prime ideals $P\supseteq I$ such that $\dim(R/I)=\dim(R/P)$.

Proof.

Lemma 4.2 allows us to swap limits and identify

[TABLE]

Furthermore, by Lemma 4.1

[TABLE]

We prove the theorem by induction on $h$. Let us start with the case of $h=1$.

To prove the claim, let us introduce an auxiliary bisequence that will link the two sides of the formula together.

Claim 4.4**.**

For each pair of natural numbers $n,m\in\mathbb{N}$ let

[TABLE]

Then the bisequence $a_{n,m}$ satisfies the following properties:

1. (1)

$a_{n,0}=\operatorname{\lambda}(R/I_{e}(I+(x^{n})))$; 2. (2)

$\lim\limits_{m\to\infty}a_{n,m}=\operatorname{\lambda}(R/(I_{e}(I)+(x^{np^{e}})))$; 3. (3)

$a_{n,m}=a_{n+m,0}-a_{m,0}$.

Proof.

The first two properties are immediate from the definition.

For the third formula we first recall that if $J$ is an ideal and $x\notin J$ then $\operatorname{\lambda}(R/(J,x))=\operatorname{\lambda}(R/J)-\operatorname{\lambda}(R/J:x)$. Applying this to $J=I_{e}(I+(x^{n+m}))$ and $x^{np^{e}}$ we obtain by (7) of Lemma 2.3 that

[TABLE]

∎

Recall that for any bisequence $\sup_{n}\sup_{m}a_{n,m}=\sup_{n,m}a_{n,m}=\sup_{m}\sup_{n}a_{n,m}$. By definition, the sequence $a_{n,m}$ is increasing in $m$, so by the claim

[TABLE]

because $x$ is a regular element modulo $I_{e}(I)$ by (9) of Lemma 2.3. On the other hand, by Lemma 4.1 $a_{n,0}/n$ is an increasing function in $n\in\mathbb{N}$, so the claim also shows that

[TABLE]

Thus

[TABLE]

which proves the theorem in the case $h=1$ after passing to the limit as $e\to\infty$.

For $h\geq 2$ we may first consider the ideal $I^{\prime}=I+(x_{1}^{n},\ldots,x_{h-1}^{n})$ and get that

[TABLE]

where $Q$ varies through the prime ideals $P$ containing $I_{e}(I^{\prime})$ such that $\dim R_{Q}/PR_{Q}=\dim R_{Q}/I_{e}(I^{\prime})R_{Q}$. By induction,

[TABLE]

where $P$ varies through the prime ideals $P$ containing $I_{e}(I)$ such that $\dim R_{Q}/PR_{Q}=\dim R_{Q}/I_{e}(I)R_{Q}$. Thus

[TABLE]

The theorem follows by changing the order of summation and using the associativity formula for parameter ideals ([Lec57, Theorem 1]), see the proof of [Smi19, Theorem 4.9]. ∎