Selected Matches for: Items Authored by Michler, Ruth I.

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CMP 1 707 527 (99:17) 14Fxx (13Cxx)
Michler, Ruth I.(1-NTXS)
On the number of generators of the torsion module of differentials. (English. English summary)
Proc. Amer. Math. Soc., to appear.

{A review for this item is in process.}

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99c:13032 13D03 (19D55)
Michler, Ruth I.(1-NTXS)
Cyclic homology of affine hypersurfaces with isolated singularities. (English. English summary)
J. Pure Appl. Algebra 120 (1997), no. 3, 291--299.

Let $A$ be a reduced hypersurface algebra with isolated singularities over a field of characteristic zero. This paper is dedicated to the computation of the Hodge decomposition of the cyclic homology of such a hypersurface in terms of the de Rham cohomology. This extends results of M. Vigue-Poirrier [J. Pure Appl. Algebra 72 (1991), no. 1, 95--108; MR 92h:19004] to the non-graded case. This is applied to compute the Hodge decomposition of the cyclic homology for the plane nodal cubic.

Similar computations have appeared in work by R. Hubl [S. Bruderle and E. Kunz, Math. Ann. 299 (1994), no. 1, 57--76; MR 95b:13017 (Appendix, pp. 72--76)]. See also the work of the Buenos Aires Cyclic Homology Group [J. Pure Appl. Algebra 83 (1992), no. 3, 205--218; MR 94f:16020].

Reviewed by Joseph P. Brennan

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97b:13033 13N05
Michler, Ruth I.(3-QEN)
Torsion of differentials of affine quasi-homogeneous hypersurfaces. (English. English summary)
Rocky Mountain J. Math. 26 (1996), no. 1, 229--236.

The paper under review examines the torsion of the modules $\Omega\sp i\sb {A/k}$ of differential forms of a reduced hypersurface $A=k[X\sb 1,\cdots,X\sb N]/(F)=k[x\sb 1,\cdots,x\sb N]$ ($k$ a field with char$(k)=0$), defined by a quasi-homogeneous polynomial $F$, having an isolated singularity at the origin.

It is known that in this case only $\Omega\sp {N-1}\sb {A/k}$ and $\Omega\sp N\sb {A/k}$ have torsion, and that we have $\dim\sb k({\rm tors}(\Omega\sp {N-1}\sb {A/k}))=\dim\sb k({\rm tors}(\Omega\sp N\sb {A/k}))$. Assume that $X\sb i$ has weight $\lambda\sb i$ and that $\deg(F)=n$. The author proves that ${\rm tors}(\Omega\sp {N-1}\sb {A/k})$ is a cyclic $A$-module, generated by the differential form $$\omega\sb 0=\sum(-1)\sp {i+1}\frac{\lambda\sb i}{n}x\sb idx\sb 1\cdots\widehat{dx\sb i}\cdots dx\sb N,$$ and she proceeds to show that ${\rm tors}(\Omega\sp {N-1}\sb {A/k})\cong \Omega\sp N\sb {A/k}$. For the proof the author uses explicit calculations based on the description of $\Omega\sp cdot\sb {A/k}$ in terms of differential forms of $k[X\sb 1,\cdots,X\sb N]/k$ and $F$, and a Koszul complex argument.

Reviewed by Reinhold Hubl

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96k:13036 13N05 (14B05)
Michler, Ruth (3-QEN)
Torsion of differentials of hypersurfaces with isolated singularities. (English. English summary)
J. Pure Appl. Algebra 104 (1995), no. 1, 81--88.

Let $K$ be an algebraically closed field of characteristic $0, R=K[X\sb 1,\cdots,X\sb N], F\in R$ a reduced polynomial defining an affine hypersurface having only isolated singularities and $A=R/(F)$. It is known [cf. K. Lebelt, Math. Ann. 211 (1974), 183--197; MR 51 #8092]) that the torsion submodule $T(\Omega\sp i\sb {A/K})$ is zero for $i\not=$ $N-1,N$. The author proves that $T(\Omega\sp {N-1}\sb {A/K})$ is $A$-isomorphic to $I/J$, where $J=(\partial F/\partial X\sb 1,\cdots,\partial F/\partial X\sb n)R$ and $I=(J\colon F)$. She also shows that $\dim\sb KT(\Omega\sp {N-1}\sb {A/K})=\dim\sb K\Omega\sp N\sb {A/K}$. The paper contains two examples of reduced plane curves having a single isolated singularity at the origin: $F\colon X\sp 3Y\sp 2+Y\sp 5+X\sp 7=0$ for which $\partial F/\partial X,\partial F/\partial Y$ is an $R$-sequence but $T(\Omega\sp 1\sb {A/K})$ is not cyclic and $F\colon(c+X)\sp 2Y\sp 2/2+cX\sp 2/2+X\sp 3/3=0 (c\not=0)$ for which $\partial F/\partial X,\partial F/\partial Y$ is not an $R$-sequence.

Reviewed by Tiberiu Dumitrescu

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96a:19003 19D55
Michler, Ruth I.(1-NTXS)
Hodge-components of cyclic homology for affine quasi-homogeneous hypersurfaces. (English. English summary)
$K$-theory (Strasbourg, 1992).
Astérisque No. 226, (1994), 10, 321--333.

In this paper the author proves that the Hodge-components of Hochschild homology of a reduced affine hypersurface are given by torsion modules of Kahler differentials. Using results of T. Goodwillie, J.-L. Loday and U. Vetter, the author also proves a vanishing result for the Hodge-components (also called the $\lambda$-decomposition) of cyclic homology of affine hypersurfaces. The non-zero summands in this decomposition are computed in terms of the de Rham cohomology of the underlying algebra. Furthermore, in the case of a hypersurface defined by a quasihomogeneous polynomial, an explicit computation of the Hodge-components of cyclic homology is given. The reduced cyclic homology groups provide a topological invariant of algebras formed by the quotient of such polynomials.

\{For the entire collection see MR 95i:19001\}.

Reviewed by Gerald M. Lodder

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