Hypersurface Singularities of Finite Cohen‐Macaulay Type
Hypersurface Singularities of Finite Cohen‐Macaulay Type
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DOI：
10.1112/plms/s3-58.2.258
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发表时间：
1989-03
影响因子：
1.8
通讯作者：
Ø. Solberg
中科院分区：
文献类型：
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作者：
Ø. Solberg
令A= k [[zl，...，是域k上的变量z中的形式幂级数环，且令f= f（zx，.，zr）是A中的非零非单位。在[19]中，Knorrer证明了：若R= A/（f）是C上单孤立超曲面奇点的局部环，则R上不可分解极大Cohen-Macaulay模的同构类只有1000个（R是有限Cohen-Macaulay型）.文[12]中证明了其匡威，因此文[12]和[19]中的结果给出了C上单超曲面奇点的模理论刻画。在[12]和[19]中，他们也找到了特征不等于2的代数闭域k上的所有孤立超曲面奇点f使得R= A/（f）是有限Cohen-Macaulay型的. Knorrer证明中的一个重要步骤是证明了A中孤立奇点f的局部环是有限Cohen-Macaulay型的当且仅当f + u2 in-< 4 [[w]]是.这是通过使用具有阶数为2的群的斜群环来完成的。由于环A和A的斜群环之间的良好连接是基于群的阶在A中可逆的假设，这个证明在特征2中不起作用。我们的主要结果是：A中f的局部环是有限Cohen-Macaulay型的当且仅当A [[x，y]]中f + xy的局部环是，在任意域k上的所有特征.这是由[19]在特征不同于2的代数闭域上得出的。我们将使用[14]中的矩阵分解和[19]中的矩阵分解，以及基于几乎可裂序列存在性的有限Cohen-Macaulay型的准则[9]。
Let A= k [[zl,..., zr]\be the formal power series ring in the variables z, over a field k, and let f= f (zx,..., zr) be a non-zero non-unit in A. Then we call R= A/{f) the local ring of/In [19] Knorrer proved that if R= A/(f) is the local ring of a simple isolated hypersurface singularity over C, then there are only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules over R (R is of finite Cohen-Macaulay type). The converse is proved in [12], so the results in [12] and [19] give a module-theoretical characterization of the simple hypersurface singularities over C. In [12] and [19] they also find all the isolated hypersurface singularities/such that R= A/(f) is of finite Cohen-Macaulay type over an algebraically closed field k of characteristic different from 2.An important step in Knorrer's proof is to show that the local ring of an isolated singularity f in A is of finite Cohen-Macaulay type if and only if/+ u2 in-< 4 [[w]] is. This is done by using skew group rings with a group of order 2. Since the nice connection between a ring A and the skew group ring of A by a group of order 2 is based on the assumption that the order of the group is invertible in A, this proof does not work in characteristic 2. Our main result is that the local ring of/in A is of finite Cohen-Macaulay type if and only if the local ring of/+ xy in A [[x, y]] is, over an arbitrary field k in all characteristics. This follows from [19] over an algebraically closed field of characteristic different from 2. We will use matrix factorizations from [14] as in [19] and a criterion for finite Cohen-Macaulay type based on existence of almost split sequences [9].