局部代数中的上同调跳跃轨迹与对偶性
Cohomological jump loci and duality in local algebra
AI总结：
本文针对局部dg代数的Koszul扩张上的dg模，引入了高阶支撑理论“上同调跳跃轨迹”，证明其在Grothendieck对偶下封闭，并得出有限完全交维数模的Betti阶数与复杂度在对偶下保持不变的结论。
摘要：
本文针对局部dg代数的Koszul扩张上的dg模，引入并研究了一种称为上同调跳跃轨迹的高阶支撑理论。这一设定的普适性适用于局部完全交环、外代数以及素特征域中某些群代数上的dg模。该族簇推广了上述各背景下已被充分研究的支撑簇。我们证明了上同调跳跃轨迹满足若干有趣的性质，包括在（Grothendieck）对偶下封闭。该支撑理论的主要应用是：在局部环上，对于具有有限完全交维数的有限生成模，其Betti阶数与复杂度的同调不变量在对偶下保持不变。
英文摘要：
In this article a higher order support theory, called the cohomological jump loci, is introduced and studied for dg modules over a Koszul extension of a local dg algebra. The generality of this setting applies to dg modules over local complete intersection rings, exterior algebras and certain group algebras in prime characteristic. This family of varieties generalizes the well-studied support varieties in each of these contexts. We show that cohomological jump loci satisfy several interesting properties, including being closed under (Grothendieck) duality. The main application of this support theory is that over a local ring the homological invariants of Betti degree and complexity are preserved under duality for finitely generated modules having finite complete intersection dimension.
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