# Correction To: Frobenius and homological dimensions of complexes

Correction

## 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 $$r=\max \{1,d\}$$.
1. (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.

2. (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 ), 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. $$\square$$ 