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Correction To: Frobenius and homological dimensions of complexes

  • Taran FunkEmail author
  • Thomas Marley
Correction
  • 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

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 [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. \(\square \)

Notes

Acknowledgements

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

Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.University of Nebraska-LincolnLincolnUSA

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