An \(A_n\)-algebra \(A= (A,m_1, m_2, \ldots , m_n)\) is a special kind of \(A_\infty \)-algebra satisfying the \(A_\infty \)-relations involving just the \(m_i\) listed. We consider obstructions to extending an \(A_{n-1}\) algebra to an \(A_n\)-algebra. We enhance the known techniques by extending the Bousfield–Kan spectral sequence to apply to the homotopy groups of the space of minimal (i.e. \(m_1=0)\)\(A_\infty \)-algebra structures on a given graded projective module. We also consider the Bousfield–Kan spectral sequence for the moduli space of \(A_\infty \)-algebras. We compute up to the \(E_2\) terms and differentials \(d_2\) of these spectral sequences in terms of Hochschild cohomology.

中文翻译：

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增强的$$ A _ {\ infty} $$A∞-阻塞理论

一个\（A_N \） -代数\（A =（A，M_1，M_2，\ ldots，m_n）\）是一种特殊的\（A_ \ infty \） -代数满足\（A_ \ infty \） -仅涉及列出的\（m_i \）的关系。我们考虑了将\（A_ {n-1} \）代数扩展为\（A_n \）-代数的障碍。我们通过扩展Bousfield–Kan谱序列以应用到给定渐变射影模块上最小（即\（m_1 = 0）\）\（A_ \ infty \）-代数结构空间的同伦群来增强已知技术 。我们还考虑了\（A_ \ infty \）-代数的模空间的Bousfield–Kan谱序列。我们计算到\（E_2 \）项和这些光谱序列的微分\（d_2 \），以Hochschild同源性表示。