# Correction To: Frobenius and homological dimensions of complexes

- 20 Downloads

## 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

*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 \(\mathfrak {p}\ne \mathfrak {m}\).

*S*denote the

*R*-algebra \({}^{e}\!R\). Let

*J*be a minimal injective resolution of

*M*. We have by assumption that

*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,

*S*-resolution of \({\text {Hom}}_R(S, M)\).

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

*J*, which is a minimal injective resolution of

*M*:

*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

*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

## Notes

### Acknowledgements

We thank Olgur Celikbas and Yongwei Yao for bringing this error to our attention.