Homotopy Invariant Commutative Algebra over Fields

Abstract

This article grew out of lectures given in the programme 'Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM (Barcelona) in Spring 2015. They give some basic homotopy invariant definitions in commutative algebra and illustrate their interest by giving a number of examples.

The algebraic chromatic splitting conjecture for Noetherian ring spectra

“…Recall that this means that there is an (abstract) equivalence Γ p R ≃ Σ ν I p . For more details on Gorenstein ring spectra, see [Gre16] or [BHV18]. For the following, we let M ∨ = Hom Rp (M, R p ) denote the dual of M in Mod Rp .…”

Local duality for structured ring spectra

“…The goal of this paper is to investigate a specific class of examples in detail, applying our methods in particular to modular representation theory. Moreover, we consider the relation between local and global duality for structured ring spectra, thereby providing a different perspective on the Gorenstein condition previously studied in depth by Greenlees [Gre16]. The main examples of ∞-categories of interest to us in this paper are coming from the modular representation theory of a finite group G over a field k of characteristic p dividing the order of G. There are two natural stable ∞-categories associated to the group algebra kG: The derived category D kG = D Mod kG and the stable module category StMod kG .…”

“…Generalizing the notion of Gorenstein for discrete commutative rings, we call a structured ring spectrum R absolute Gorenstein whenever it satisfies Gorenstein duality for all homogeneous prime ideals p in π * R, which means that the local cohomology π * Γ p R of R at p is given by the injective hull of the residue field (π * R)/p. This notion is closely related to Greenlees' definition of Gorenstein duality, see [DGI06,Gre16], but does not make reference to a residue ring spectrum k lifting (π * R)/p. Furthermore, recall that the forgetful functor f * : Mod S → Mod R corresponding to a map of ring spectra f : R → S has both a left and a right adjoint, given by induction f * and coinduction f !…”

The representation-ring-graded local cohomology spectral sequence for BPℝ⟨3⟩

“…There is a well known complex orientable spectrum BP 3 at the prime 2 (with coefficient ring Z (2) [v 1 , v 2 ]), and the spectrum BP R 3 is a version that takes into account the action of the group Q of order 2 by complex conjugation. The subscript ⋆ refers to grading over the real representation ring RO(Q) = {x + yσ | x, y ∈ Z}, where σ is the sign representation on R. The ideal J is generated by elements v 1 , v 2 , and H * J denotes local cohomology in the sense of Grothendieck.We will introduce the statement and indicate its interest by giving a quasi-historical account (see [10] for more context). We will make the assertions more precise in the process.…”

“…We will introduce the statement and indicate its interest by giving a quasi-historical account (see [10] for more context). We will make the assertions more precise in the process.…”