On generalized deformation problems. (English) Zbl 1533.13016
Let \((R,\mathfrak{m})\) be a Noetherian local ring, \(I\) an ideal of \(R\) such that \(pd(R/I)<\infty\). If \(R/I\) satisfies a property \(\mathcal{P}\), does it follow that \(R\) satisfies \(\mathcal{P}\)? The author is calling this the generalized deformation problem. First it is shown that the problem is solved affirmatively in the positive prime characteristic case, for F-injectivity in the Cohen-Macaulay case and for F-rationality for excellent rings. In the general case, the problem is solved affirmatively for the properties \((S_k), (R_k)\ \text{ and }\ (S_{k+1})\) and \((R_k)\) under additional assumptions, as well as for the property of being a domain.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)
MSC:
| 13D10 | Deformations and infinitesimal methods in commutative ring theory |
| 13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14B07 | Deformations of singularities |
Keywords:
generalized deformation problem; finite projective dimension; Frobenius closure; tight closure; \(F\)-injective; \(F\)-rational; Serre’s conditionsReferences:
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