For odd $$N\ge 5$$ N ≥ 5 , we establish a short exact sequence about motivic double zeta values $$\zeta ^{\mathfrak {m}}(r,N-r)$$ ζ m ( r , N - r ) with $$r\ge 3$$ r ≥ 3 odd, $$N-r\ge 2$$ N - r ≥ 2 . From this we classify all the relations among depth-graded motivic double zeta values $$\zeta ^{\mathfrak {m}}(r,N-r)$$ ζ m ( r , N - r ) with $$r\ge 3$$ r ≥ 3 odd, $$N-r\ge 2$$ N - r ≥ 2 . As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns relations among multiple zeta values of odd weight.

中文翻译：

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奇数权重的动机双 zeta 值

对于奇数 $$N\ge 5$$ N ≥ 5 ，我们建立一个关于动机双 zeta 值的短精确序列 $$\zeta ^{\mathfrak {m}}(r,Nr)$$ ζ m ( r , N - r ) 与 $$r\ge 3$$ r ≥ 3 奇数，$$Nr\ge 2$$ N - r ≥ 2 。由此我们将深度分级的动机双 zeta 值 $$\zeta ^{\mathfrak {m}}(r,Nr)$$ ζ m ( r , N - r ) 与 $$r\ge 3 之间的所有关系分类$$ r ≥ 3 奇数，$$Nr\ge 2$$ N - r ≥ 2 。作为推论，我们证实了 Zagier 关于矩阵秩的猜想，该矩阵涉及多个奇数权重的 zeta 值之间的关系。