We give a criterion for the complete reducibility of modules of finite length satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$ elements such that the difference between the skew-symmetric opposite action of the left action and the right action on these elements are Laurent polynomials in the variable. If $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ for every $\mathbb{Z}$-graded $V$-bimodule $M$, then every left $V$-module of finite-length satisfying a composability condition is completely reducible. In particular, since a lower-bounded $\mathbb{Z}$-graded vertex algebra $V$ is a special meromorphic open-string vertex algebra and left $V$-modules are in fact what has been called generalized $V$-modules with lower-bounded weights (or lower-bounded generalized $V$-modules), this result provides a cohomological criterion for the complete reducibility of lower-bounded generalized modules of finite length for such a vertex algebra. We conjecture that the converse of the main theorem above is also true. We also prove that when a grading-restricted vertex algebra $V$ contains a subalgebra satisfying some familiar conditions, the composability condition for grading-restricted generalized $V$-modules always holds and we need $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ only for every $\mathbb{Z}$-graded $V$-bimodule $M$ generated by a grading-restricted subspace in our complete reducibility theorem.

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一个（亚纯开弦）顶点代数的第一个上同调、推导和约简

我们使用代数的第一上同调给出了满足亚纯开弦顶点代数 $V$ 的可组合性条件的有限长度模的完全可约性的标准。对于 $V$-bimodule $M$，令 $\hat{H}^{1}_{\infty}(V, M)$ 是 $V$ 的第一个上同调，系数在 $M$。令 $\hat{Z}^{1}_{\infty}(V, M)$ 是 $\hat{H}^{1}_{\infty}(V, M)$ 的子空间规范同构为从权重$1$元素的右顶点算子的零模获得的导数空间，使得这些元素上的左动作和右动作的偏斜对称相反动作之间的差异是变量中的洛朗多项式。如果 $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ 对于每个 $\mathbb{Z} $-graded $V$-bimodule $M$, 那么每个满足可组合性条件的有限长度的左 $V$-模都是完全可约的。特别是，由于下界 $\mathbb{Z}$-graded 顶点代数 $V$ 是一个特殊的亚纯开弦顶点代数，而左 $V$-modules 实际上就是所谓的广义 $V$-具有下界权重的模块（或下界广义$V$-modules），该结果为这种顶点代数的有限长度的下界广义模块的完全可约性提供了上同调标准。我们推测上述主定理的逆命题也成立。我们还证明，当分级限制顶点代数 $V$ 包含满足一些熟悉条件的子代数时，