Let S be an unramified regular local ring of mixed characteristic two and R the integral closure of S in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of S obtained by lifting to S the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of R and show that R admits a birational small Cohen-Macaulay module. It is noted that R is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$.

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具有混合特征的双二次扩展上的双理性小Cohen-Macaulay模的存在

设S为混合特征2的无分支规则局部环，R为S的商因数的二次封闭式中的S的整体闭环，该商数是通过邻接足够普通的方形自由元素的根获得的$F，G\in \mathrm{小号}$。让${\mathrm{小号}}^{\mathrm{2个}}$表示通过将Frobenius映射图上的图像提升至S而获得的S的子环。$\mathrm{小号}/\mathrm{2个}\mathrm{小号}$。当至少一项$F，G\in {\mathrm{小号}}^{\mathrm{2个}}$，我们刻画了R的Cohen-Macaulayness的特征，并证明R接受了双边小Cohen-Macaulay模块。请注意，如果出现以下情况，R不会自动成为Cohen-Macaulay$F，G\in {\mathrm{小号}}^{\mathrm{2个}}$ 或者如果 $F，G\notin {\mathrm{小号}}^{\mathrm{2个}}$。