张量复形：从更高张量构造的多线性自由分解
Tensor complexes: Multilinear free resolutions constructed from higher tensors
摘要：
交换环上的自由模的最基本的复形是Koszul复形，它由向量构造（即，1-张量），以及从矩阵（即，α 2-张量）。本文的主题是一个多线性模拟这些复杂的，我们从一个任意的更高的张量。我们的结构提供了详细的最小自由分解的新的例子，以及各种各样的复合物，包括一个统一的观点：的Eisenbud-Schreyer纯决议，和Gelfand-Kapranov-Zelevinsky和Weyman计算超行列式的复合物。此外，我们提供的应用研究的纯决议和Boij-Soederberg理论，包括建设无限多个新的家庭的纯决议和第一次明确描述的微分Eisenbud-Schreyer纯决议。
英文摘要：
The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon-Northcott and the Buchsbaum-Rim complexes, which are constructed from a matrix (i.e., a 2-tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety of complexes including: the Eagon-Northcott, Buchsbaum-Rim and similar complexes, the Eisenbud-Schreyer pure resolutions, and the complexes used by Gelfand-Kapranov-Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij-Soederberg theory, including the construction of infinitely many new families of pure resolutions and the first explicit description of the differentials of the Eisenbud-Schreyer pure resolutions.
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