Correction To: Frobenius and homological dimensions of complexes

1 Correction To: Collectanea Mathematica https://doi.org/10.1007/s13348-019-00260-7

The proof of Theorem 3.2 in the paper contains an error (namely in the use of Lemma 3.1 when $T={}^{e}\!R$, which is only a faithful R-module when R is reduced). We give a new proof of this Theorem (slightly strengthened to streamline the proof) which avoids the use of Lemma 3.1.

Theorem 3.2

Let $(R, \mathfrak {m}, k)$ be a d-dimensional Cohen–Macaulay local ring of prime characteristic p and which is F-finite. Let $e\geqslant \log _p e(R)$ be an integer, M an R-complex, and $r=\max \{1,d\}$.

• (a) Suppose there exists an integer $t> \sup {\text {H}}^*(M)$ such that ${\text {Ext}}^i_R({}^{e}\!R, M)=0$ for $t\leqslant i\leqslant t+r-1$. Then M has finite injective dimension.

• (b) Suppose there exists an integer $t>\sup {\text {H}}_*(M)$ such that ${\text {Tor}}_i^R({}^{e}\!R, M)=0$ for $t\leqslant i\leqslant t+r-1$. Then M has finite flat dimension.

Proof

We first note that if (a) holds in the case ${\text {dim}}R=d$, then (b) also holds in the case ${\text {dim}}R=d$: For, suppose the hypotheses of (b) hold for a complex M. Then by Lemma 2.5(a), ${\text {Ext}}^i_R({}^{e}\!R, M^{{\text {v}}})\cong {\text {Tor}}_i^R({}^{e}\!R, M)^{{\text {v}}}=0$ for $t\leqslant i\leqslant t+r-1$. As $\sup {\text {H}}^*(M^{{\text {v}}})=\sup {\text {H}}_*(M)$, we have by (a) that ${\text {id}}_R M^{{\text {v}}}<\infty$. Hence, ${\text {fd}}_R M<\infty$ by Corollary 2.6(a).

Thus, it suffices to prove (a). As in the original proof, we may assume that M is a module concentrated in degree zero and ${\text {Ext}}^i_R({}^{e}\!R,M)=0$ for $i=1,\dots ,r$. We proceed by induction on d, with the case $d=0$ being established by Proposition 2.8. Suppose $d\geqslant 1$ (so $r=d$) and we assume both (a) and (b) hold for complexes over local rings of dimension less than d.

Let $\mathfrak {p}\ne \mathfrak {m}$ be a prime ideal of R. As R is F-finite, we have ${\text {Ext}}^i_{R_{\mathfrak {p}}}({}^{e}\!R_{\mathfrak {p}}, M_{\mathfrak {p}})=0$ for $1\leqslant i\leqslant d$. As $d\geqslant \max \{1, {\text {dim}}R_{\mathfrak {p}}\}$ and $e(R)\geqslant e(R_{\mathfrak {p}})$ (see [12]), we have ${\text {id}}_{R_\mathfrak {p}} M_{\mathfrak {p}}<\infty$ by the induction hypothesis. Hence, ${\text {id}}_{R_\mathfrak {p}} M_{\mathfrak {p}}\leqslant {\text {dim}}R_{\mathfrak {p}}\leqslant d-1$ by [4, Proposition 4.1 and Corollary 5.3]. It follows that $\mu _i(\mathfrak {p}, M)=0$ for all $i\geqslant d$ and all $\mathfrak {p}\ne \mathfrak {m}$.

For convenience, we let S denote the R-algebra ${}^{e}\!R$. Let J be a minimal injective resolution of M. We have by assumption that

$$\begin{aligned} {\text {Hom}}_R(S, J^0)\xrightarrow {\phi ^0} {\text {Hom}}_R(S, J^1)\rightarrow \cdots \rightarrow {\text {Hom}}_R(S, J^{d}) \xrightarrow {\phi ^{d}} {\text {Hom}}_R(S, J^{d+1}) \end{aligned}$$

(3.1)

is exact. Let L be the injective S-envelope of ${\text {coker}}{\phi ^{d}}$ and $\psi :{\text {Hom}}_R(S, J^{d+1})\rightarrow L$ the induced map. Hence,

$$\begin{aligned} 0\rightarrow {\text {Hom}}_R(S, J^0)\rightarrow \cdots \xrightarrow {\phi ^d} {\text {Hom}}_R(S, J^{d+1}) \xrightarrow {\psi } L \end{aligned}$$

is acyclic and in fact the start of an injective S-resolution of ${\text {Hom}}_R(S, M)$.

As in the original proof, we obtain that the map $\psi$ is injective.

Now consider the complex J, which is a minimal injective resolution of M:

$$\begin{aligned} 0\rightarrow J^0\xrightarrow {\partial ^0} J^1\rightarrow \cdots \rightarrow J^{d-1} \xrightarrow {\partial ^{d-1}} J^d\xrightarrow {\partial ^d}\cdots \end{aligned}$$

The proof will be complete upon proving:

Claim:$\partial ^{d-1}$ is surjective.

Proof: As $\psi$ is injective we have from (3.1) that $\phi ^d=0$, and thus $\phi ^{d-1}$ is surjective. Let $C={\text {coker}} \partial ^{d-1}$ and $(-)^{{\text {v}}}$ the Matlis dual functor (as defined in Corollary 2.6). Then

$$\begin{aligned} 0\rightarrow C^{{\text {v}}}\rightarrow (J^d)^{{\text {v}}}\rightarrow \cdots \rightarrow (J^0)^{{\text {v}}}\rightarrow M^{{\text {v}}}\rightarrow 0 \end{aligned}$$

is exact. Note that $(J^i)^{{\text {v}}}$ is a flat R-module for all i (e.g., Corollary 2.6(b)). As the set of associated primes of any flat R-module is contained in the set of associated primes of R, and as R is Cohen–Macaulay of dimension greater than zero, to show $C^{{\text {v}}}=0$ it suffices to show $(C^{{\text {v}}})_{\mathfrak {p}}=0$ for all $\mathfrak {p}\ne \mathfrak {m}$. So fix a prime $\mathfrak {p}\ne \mathfrak {m}$. As S is a finitely generated R-module, we have ${\text {Tor}}_i^R(S,M^{{\text {v}}})\cong {\text {Ext}}^i_R(S,M)^{{\text {v}}}=0$ for $i=1,\dots ,d$ by Lemma 2.5(b). This implies ${\text {Tor}}_i^{R_{\mathfrak {p}}}(S_{\mathfrak {p}}, (M^{{\text {v}}})_{\mathfrak {p}})=0$ for $i=1,\dots ,d$. As $R_{\mathfrak {p}}$ is an F-finite Cohen–Macaulay local ring of dimension less than d, and $p^e\geqslant e(R)\geqslant e(R_{\mathfrak {p}})$, we have that ${\text {fd}}_{R_{\mathfrak {p}}}(M^{{\text {v}}})_{\mathfrak {p}}<\infty$ by the induction hypothesis on part (b). In particular, by [4, Corollary 5.3], ${\text {fd}}_{R_{\mathfrak {p}}} (M^{{\text {v}}})_{\mathfrak {p}}\leqslant {\text {dim}}R_{\mathfrak {p}}\leqslant d-1$ and thus $(C^{{\text {v}}})_{\mathfrak {p}}$ is a flat $R_{\mathfrak {p}}$-module. Then by either [15, Corollary 3.5] or [6, Theorem 3.1], we have

$$\begin{aligned} 0\rightarrow S_{\mathfrak {p}} \otimes _{R_{\mathfrak {p}}} (C^{{\text {v}}})_{\mathfrak {p}}\rightarrow S_{\mathfrak {p}}\otimes _{R_{\mathfrak {p}}} ((J^{d})^{{\text {v}}})_{\mathfrak {p}}\rightarrow S_{\mathfrak {p}}\otimes _{R_{\mathfrak {p}}} ((J^{d-1})^{{\text {v}}})_{\mathfrak {p}} \end{aligned}$$

(3.3)

is exact. Now, since $\phi ^{d-1}={\text {Hom}}_R(S,\partial ^{d-1})$ is surjective, we have, using duality and Lemma 2.5(b), that

$$\begin{aligned} 0\rightarrow S\otimes _R (J^{d})^{{\text {v}}}\rightarrow S\otimes _R (J^{d-1})^{{\text {v}}} \end{aligned}$$

is exact. Localizing this exact sequence at $\mathfrak {p}$ and comparing with (3.3), we have $S_{\mathfrak {p}} \otimes _{R_{\mathfrak {p}}} (C^{{\text {v}}})_{\mathfrak {p}}=0$. However, tensoring with $S_{\mathfrak {p}}$ over $R_{\mathfrak {p}}$ is faithful (e.g., [13, Proposition 2.1(c)]) and hence $(C^{{\text {v}}})_{\mathfrak {p}}=0$. Hence, $C^{{\text {v}}}=0$, and thus $C=0$, which completes the proof of the Claim. $\square$