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The Balmer Spectrum of Stable Module Categories Over Finite Group Schemes

We give an introduction to the Balmer spectrum of a general tensor-triangulated category and a classification of the tensor-thick ideals of such a category. This framework relies on a notion of support for each object. We then summarize the work of Benson, Iyengar, and Krause regarding two equivalent notions of support, namely the triangulated support and the cohomological support. After this, we specialize to the case of the stable module category stmod(A(1)) and use the previous results to give an explicit classification of the tensor-thick ideals of this category. These tensor-thick ideals are shown to correspond to the vanishing of Margolis homologies. In the 1980s Hopkins and Smith proved the Thick Subcategory Theorem, which is a crucial component in the Periodicity Theorem in chromatic homotopy theory. The Thick Subcategory Theorem says that the thick subcategories of the stable homotopy category of finite p-local spectra are nested and correspond to the vanishing of Morava K-theories. Hopkins then made the important observation that their method of classification could be applied to a more algebraic context. To this end, he sketched a classification result for the thick subcategories of the derived category D(proj R) of perfect complexes over a commutative noetherian ring R. More precisely, he established a correspondence between specialization closed sets subsets of proj R and the thick subcategories in the derived category. This approach was generalized by Balmer and Thomason to arbitrary tensortriangulated (tt) categories. In his work, Balmer defined the “prime spectrum” of a tt-category. The Balmer spectrum consists of prime tt-ideals, which are thick subcategories that behave like prime ideals in a commutative ring. Balmer then formulated an analog of the support of a module. Using this notion of support, he established a bijection between radical tt-ideals in a tt-category and the supports of objects in the category. In this same direction, Benson, Carlson, and Rickard proved an analog of Hopkins’s theorem in the stable category stmod(kG) of finitely generated kG-modules in [3], where kG is a finite group algebra. Recently, Benson, Iyengar, Krause, and Pevtsova extended this classification to finite group schemes in [5]. In this project we were interested in applying these results to the specific case of stmod(A(1)). The paper is structured as follows. In section 1, we introduce the Balmer spectrum of a tensortriangulated category following [2]. We also state Balmer’s classification of tt-ideals in such a category. After describing this general setting, in section 2 we introduce the triangulated support and the cohomological support of a compactly generated triangulated category and establish an equivalence between them. In section 3 we use this equivalence to give an explicit classification of the tt-ideals in stmod(A(1)). More specifically, we show that the Balmer spectrum consists of two prime ideals (h10, h11, v), (h11, v, w) of ExtA(1)(F2,F2). We then show that the vanishing of Q0-Margolis homology corresponds to the prime (h11, v, w), while the vanishing of Q1-Margolis homology corresponds to the prime (h10, h11, v). We end the paper with a section containing some work toward computing ExtA(1)(F2,M). This is done via a spectral sequence formulated in terms of Q1-Margolis homology. 1 Balmer Spectrum and Support In this section we outline a framework that generalizes the Thick Subcategory Theorem of HopkinsSmith. This framework relies on a notion of support for objects in a given category. The original notion of support comes from commutative algebra, in which the support of an R-module M is defined to be the set of primes p ⊂ R such that Mp 6= 0. We will closely follow the exposition of [2]. Definition 1. An additive category in which all Hom-sets take values in the category of abelian groups. A shift functor Ω−1 on a category T is an automorphism T → T . A triangle (X,Y, Z, u, v, w) consists of the following data: X u −→ Y v −→ Z w −→ Ω−1(X) A triangulated category is an additive category T with a shift functor Ω−1 and a class of triangles characterized by certain properties. Given an arbitrary triangulated category T , a thick subcategory S is a full subcategory of T which is closed under finite direct sums and summands. We will call a subcategory S ⊂ T localizing if it is thick and closed under small coproducts. Similarly, we will call a subcategory colocalizing if it is thick and closed under small products. Definition 2. A tensor-triangulated category is a triple (K,⊗, 1) consisting of a triangulated category K, a symmetric monoidal product ⊗ : K × K → K which is exact in each variable. We will