Artinian algebras and differential forms.

*(English)*Zbl 0931.13019A longstanding conjecture (BC) of R. Berger [Math. Z. 81, 326-354 (1963; Zbl 0113.26302)] is that a reduced curve over a field of characteristic zero is regular if and only if its sheaf of Kähler differentials is torsionfree. G. Cortiñas, S. C. Geller and C. A. Weibel [Math. Z. 228, No. 3, 569-588 (1998; Zbl 0907.13014)] formulated the Artinian Berger conjecture (ABC). This conjecture is stated as follows:

Let \(B\) be a commutative algebra over a perfect field \(k\). Call \(B\) a tame principal ideal algebra if it is isomorphic to a direct product of algebras of the form \(K[X]/(X^n)\), where \(K\) is a finite extension field of \(k\), and if \(k\) is of finite characteristic \(p\), the degree of \(K\) over \(k\) is prime to \(p\). For a finite dimensional \(k\)-algebra \(A\), let \(\tau(A)\) be the intersection of the kernels of the induced maps \(\varphi^*:\Omega^1_{A/k}\to\Omega^1_{B/k}\), ranging over all \(k\)-algebra morphisms from \(A\) to all tame principal ideal algebras \(B\). The (ABC) conjecture is that \(A\) is a tame principal ideal algebra if and only if \(\tau(A)=0\). – In characteristic zero, (ABC) implies (BC).

In the present paper, it is shown that (ABC) holds for certain graded algebras called standard graded algebras and for zero-dimensional Gorenstein graded algebras. A standard graded algebra is a positively graded algebra \(A=A_0\oplus A_1\oplus\cdots A_n\) such that \(M=A_1\oplus\cdots A_n\) is a maximal ideal and is generated by elements of degree one. That (ABC) holds for these implies (BC) for certain algebras, both graded and ungradable. That (ABC) holds for Gorenstein algebras has no implications about (BC) for Gorenstein rings of dimension one.

The results have some implications about when a finite dimensional algebra can be embedded in a principal ideal algebra. The proofs use an Artinian version of valuation theory called truncated valuation theory.

Let \(B\) be a commutative algebra over a perfect field \(k\). Call \(B\) a tame principal ideal algebra if it is isomorphic to a direct product of algebras of the form \(K[X]/(X^n)\), where \(K\) is a finite extension field of \(k\), and if \(k\) is of finite characteristic \(p\), the degree of \(K\) over \(k\) is prime to \(p\). For a finite dimensional \(k\)-algebra \(A\), let \(\tau(A)\) be the intersection of the kernels of the induced maps \(\varphi^*:\Omega^1_{A/k}\to\Omega^1_{B/k}\), ranging over all \(k\)-algebra morphisms from \(A\) to all tame principal ideal algebras \(B\). The (ABC) conjecture is that \(A\) is a tame principal ideal algebra if and only if \(\tau(A)=0\). – In characteristic zero, (ABC) implies (BC).

In the present paper, it is shown that (ABC) holds for certain graded algebras called standard graded algebras and for zero-dimensional Gorenstein graded algebras. A standard graded algebra is a positively graded algebra \(A=A_0\oplus A_1\oplus\cdots A_n\) such that \(M=A_1\oplus\cdots A_n\) is a maximal ideal and is generated by elements of degree one. That (ABC) holds for these implies (BC) for certain algebras, both graded and ungradable. That (ABC) holds for Gorenstein algebras has no implications about (BC) for Gorenstein rings of dimension one.

The results have some implications about when a finite dimensional algebra can be embedded in a principal ideal algebra. The proofs use an Artinian version of valuation theory called truncated valuation theory.

Reviewer: W.H.Gustafson (Lubbock)

##### MSC:

13N10 | Commutative rings of differential operators and their modules |

13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |

13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |

13A02 | Graded rings |

##### Keywords:

principal ideal algebra; Gorenstein algebra; truncated valuation theory; regular reduced curve; sheaf of Kähler differentials; Artinian Berger conjecture; standard graded algebras
PDF
BibTeX
XML
Cite

\textit{G. Cortiñas} and \textit{F. Krongold}, Commun. Algebra 27, No. 4, 1711--1716 (1999; Zbl 0931.13019)

Full Text:
DOI

##### References:

[1] | Berger R., Math. Zeit 81 pp 326– (1963) · Zbl 0113.26302 · doi:10.1007/BF01111579 |

[2] | Cortiñas G., Math. Zeit 81 (1963) |

[3] | Grauert H., Math. Annalen 153 pp 263– (1964) |

[4] | Saito K., Invent. Math 14 pp 123– (1971) · Zbl 0224.32011 · doi:10.1007/BF01405360 |

[5] | Zariski O., Grad. Texts in Math. 29 (1960) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.