纯自由分解的存在性
The Existence of Pure Free Resolutions
摘要：
令d1，…，dn是严格递增的整数序列。Boij和Söderberg [arXiv：math/0611081]证明了任意多项式环K[x_1，...，x_n]，其最小自由分解是纯类型（d1，.，dn），在这个意义上，它的第i个合是在度di中生成的。在本文中，我们证明了一个更强的陈述，在特征零：这样的模不仅存在，但可以采取GL（n）-等变。实际上，我们给出了两种不同的等变结构，并构造了外代数和Z/2-分次代数上的纯归结。利用Schur函子的组合学和Bott定理，构造了Grassmann簇上等变向量丛的直像。
英文摘要：
Let d1,...,dn be a strictly increasing sequence of integers. Boij and Söderberg [arXiv:math/0611081] have conjectured the existence of a graded module M of finite length over any polynomial ring K[x_1,..., x_n], whose minimal free resolution is pure of type (d1,...,dn), in the sense that its i-th syzygies are generated in degree di. In this paper we prove a stronger statement, in characteristic zero: Such modules not only exist, but can be taken to be GL(n)-equivariant. In fact, we give two different equivariant constructions, and we construct pure resolutions over exterior algebras and Z/2-graded algebras as well. The constructions use the combinatorics of Schur functors and Bott's Theorem on the direct images of equivariant vector bundles on Grassmann varieties.
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