Resolutions of symmetric ideals via stratifications of derived categories

2024/07/22 by Ganapathy, Karthik
#13A50 #13D02 #13D45 #18G80 #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2407.16071

Abstract

We propose a method to unify various stability results about symmetric ideals in polynomial rings by stratifying related derived categories. We execute this idea for chains of GL_n-equivariant modules over an infinite field k of positive characteristic. We prove the Le--Nagel--Nguyen--Römer conjectures for such sequences and obtain stability patterns in their resolutions as corollaries of our main result, which is a semiorthogonal decomposition for the bounded derived category of GL_∞-equivariant modules over S = k[x₁, x₂, …, x_n, …]. Our method relies on finite generation results for certain local cohomology modules. We also outline approaches (i) to investigate Koszul duality for S-modules taking the Frobenius homomorphism (of GL_∞) into account, and (ii) to recover and extend Murai's results about free resolutions of symmetric monomial ideals.

Related

• GL-algebras in positive characteristic II: the polynomial ring
• Equivariant Free Resolutions of Sequences of Symmetric Module
• General symmetric ideals
• The Herzog-Takayama resolution over a skew polynomial ring
• The Eliahou-Kervaire resolution over a skew polynomial ring