多项式环的上环的Serre维数与Euler类群
Serre dimension and Euler class group of overrings of polynomial rings
摘要：
设R是维数为d的交换诺特环，B=R[X_1，\ldots，X_m，Y_1 ^{ \pm 1}，\ldots，Y_n^{ \pm 1}]是R上的Laurent多项式环.若A=B[Y，f^{-1}]，对某个\in R[Y]，则证明了如下结果：（i）若f是一个首一多项式，则A的Serre维数是\leq d.在n=0的情况下，这个结果是由于Bhatwadekar，没有条件，f是一个首一多项式。(ii)由Bhatwadekar和Raja Sridharan定义的A的p阶欧拉类群E^p（A）对于p \geq max \{d+1，\dim A-p +3\}是平凡的。在m=n=0的情况下，这个结果是由于Mandal-Parker。
英文摘要：
Let R be a commutative Noetherian ring of dimension d and B=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ring over R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the following results: (i) If f is a monic polynomial, then Serre dimension of A is \leq d. In case n=0, this result is due to Bhatwadekar, without the condition that f is a monic polynomial. (ii) The p-th Euler class group E^p(A) of A, defined by Bhatwadekar and Raja Sridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, this result is due to Mandal-Parker.
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