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Let be a graded ring and an integer. We introduce and study -strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be -strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever . Many properties of the -strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate...
We give a simple proof that critical values of any Artin -function attached to a representation with character are stable under twisting by a totally even character , up to the -th power of the Gauss sum related to and an element in the field generated by the values of and over . This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.
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