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Flatness testing and torsion freeness of analytic tensor powers. (English) Zbl 1084.32005
Let \(f\colon X \to Y\) be a morphism of germs of complex analytic spaces where \(X\) is reduced of pure dimension and \(Y\) is smooth of dimension \(n\).
The text gives four conditions each one sufficient for the following characterisation of flatness to hold: The morphism \(f\) is flat if and only if the \(n\)-th analytic tensor power \(\mathcal O_X \hat \otimes_{\mathcal O_Y} \cdots \hat\otimes_{\mathcal O_Y} \mathcal O_X\) is torsion-free as an \(\mathcal O_Y\)-module. The four conditions given are as follows:
(1) \(n < 3\).
(2) \(f\colon X \to Y\) is a Nash morphism of Nash germs.
(3) The singular locus of \(X\) is mapped into a proper analytic subgerm of \(Y\).
(4) The local ring \(\mathcal O_X\) is Cohen-Macauley.
By (3), the result can be seen as a generalisation to the analytic domain of Auslander’s result [M. Auslander, Ill. J. Math. 5, 631–647 (1961; Zbl 0104.26202)] saying that a module \(M\) finite over an unramified regular local ring \(R\) of dimension \(n > 0\) is free (which is equivalent to torsion-freeness in the finite case) over \(R\) if and only if its \(n\)-th tensor power \(M^{\otimes_R^n}\) is torsion-free as an \(R\)-module. The theorem proven in this article can also be seen as a step towards the general conjecture [J. Adamus, J. Pure Appl. Algebra 193, No. 1–3, 1–9 (2004; Zbl 1054.32004)] saying that a local analytic algebra \(A\) over a local analytic algebra \(R\), which is regular of dimension \(n\), is \(R\)-flat if and only if its \(n\)-th analytic tensor power is torsion-free as an \(R\)-module.
Let \(X^n\) be the \(n\)-fold fibre product of \(X\) over \(Y\) and \(f^n\colon X^n \to Y\) the induced morphism. The hard implication of the theorem is to prove that torsion-freeness of the \(n\)-th analytic power follows from the flatness assumption under either conditions (1) to (4).
In order to handle cases (1) and (2), the author makes use of the concept of algebraic vertical components (a component of \(X\) is algebraic vertical if an arbitrarily small representative is mapped by \(f\) into a proper analytic subset) by using the following equivalence: The morphism \(f^n\) has no algebraic vertical components if and only if \(\mathcal O_{X^n}\) is a torsion-free \(\mathcal O_Y\)-module. In the first two cases, it is shown that restricting \(f^n\) to any geometric vertical component \(W\) (a component is geometric vertical if an arbitrarily small representative is mapped into a nowhere dense subset of a neighbourhood of the basepoint) of \(X^n\) yields a Gabrielov regular morphism, which in turn can be used to deduce that \(W\) is actually algebraic vertical. Thus \(\mathcal O_{X^n}\) is a torsion-free \(O_Y\)-module if and only if \(f^n\) has no geometric vertical component, and this is equivalent to \(f\) being flat by a result of Galligo-Kwieciński [A. Galligo and M. Kwieciński, J. Algebra 232, No. 1, 48–63 (2000; Zbl 1016.14001)].
The case (3) is handled using techniques of Galligo-Kwieciński and the following equivalence, which holds whenever \(X\) is of pure dimension and \(Y\) reduced and irreducible of dimension \(n\): The morphism \(f\) is open if and only if the reduced \(n\)-th analytic tensor power of \(\mathcal O_X\) over \(\mathcal O_Y\) is torsion-free ans an \(\mathcal O_Y\)-module.
The final case (4) is proven by observing that flatness in this case is the same as openness [G. Fischer, Complex analytic geometry (Lecture Notes in Mathematics 538, Springer-Verlag, Berlin-Heidelberg-New York) (1976; Zbl 0343.32002)].
The article ends with a discussion on Gabrielov regularity of fibre products. The author gives a criterion when a fibre power of an analytic morphism is Gabrielov regular.
MSC:
32B05 Analytic algebras and generalizations, preparation theorems
13J07 Analytical algebras and rings
32C15 Complex spaces
14P20 Nash functions and manifolds
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References:
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