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Detecting flatness over smooth bases. (English) Zbl 1267.13021

In this very well-written paper the authors establish the following interesting theorem:
Let \(K\) be a field, \(R\) an essentially smooth \(K-\)algebra, \(A\) an algebra essentially of finite type over \(R\) and \(M\) a finite \(A-\)module. If \(M^{\otimes^d_R}\) is torsion-free over \(R\) for some \(d\geq \text{dim}R\), then \(M\) is flat over \(R\).
Recall that \(R\) is an essentially smooth \(K-\)algebra, if \(R\) is an algebra essentially of finite type over \(K\) and also \(R\otimes_KK'\) is regular for every field extension \(K\subseteq K'\). Some important points in this theorem are: \((1)\) the field \(K\) is arbitrary, it is not even assumed to be perfect; \((2)\) it holds for any finite \(A-\)module \(M\), not just \(A\). So that this theorem generalizes the analytic case \(K=\mathbb{C}\) proved by Adamus, Bierstone, and Milman [“Geometric Auslander criterion for flatness of an analytic mapping”, Am. J. Math., to appear] and the classical case by S. Lichtenbaum [Ill. J. Math. 10, 220–226 (1966; Zbl 0139.26601)] wherein \(R\) is regular local and \(M\) is a finitely generated \(R-\)module.
Among others, the rigidity of Tor is proven for the same kind of rings and modules as those in the above theorem. As well, in the particular case where \(R\) is regular of dimension at most \(2\), \(A\) is a Notherian \(R\)-algebra and \(M\) is a finite \(A\)-module, it is shown that the torsionfree-ness of \(M\otimes_RM\) implies the flatness of \(M\) as an \(R\)-module. This is an extension of a theorem of W. V. Vasconcelos [J. Pure Appl. Algebra 122, No.3, 313-321 (1997; Zbl 0885.13006)].

MSC:

13C12 Torsion modules and ideals in commutative rings
13C11 Injective and flat modules and ideals in commutative rings
14A15 Schemes and morphisms
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References:

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[12] K. Takahashi, H. Terakawa, K.-I. Kawasaki, and Y. Hinohara, A note on the new rigidity theorem for Koszul complexes, Far East J. Math. Sci. (FJMS) 20 (2006), no. 3, 269 – 281. · Zbl 1101.13038
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