By David J. Benson, Srikanth Iyengar, Henning Krause

The seminar specializes in a up to date answer, through the authors, of an extended status challenge in regards to the good module classification (of now not inevitably finite dimensional representations) of a finite staff. The facts attracts on rules from commutative algebra, cohomology of teams, and strong homotopy thought. The unifying topic is a thought of help which gives a geometrical method for learning quite a few algebraic constructions. The prototype for this has been Daniel Quillen’s description of the algebraic sort akin to the cohomology ring of a finite crew, in response to which Jon Carlson brought help forms for modular representations. This has made it attainable to use equipment of algebraic geometry to procure illustration theoretic details. Their paintings has encouraged the advance of analogous theories in quite a few contexts, particularly modules over commutative whole intersection earrings and over cocommutative Hopf algebras. one of many threads during this improvement has been the category of thick or localizing subcategories of varied triangulated different types of representations. This tale all started with Mike Hopkins’ class of thick subcategories of the correct complexes over a commutative Noetherian ring, by way of a class of localizing subcategories of its complete derived class, because of Amnon Neeman. The authors were constructing an method of handle such category difficulties, according to a development of neighborhood cohomology functors and help for triangulated different types with ring of operators. The e-book serves as an creation to this circle of ideas.

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Prove that H ∗ (G, k) ∼ = k[x], a polynomial algebra over k on a indeterminate x of degree 1. (11) Let G = Z/p with p ≥ 3 and char k = p. Prove that H ∗ (G, k) is the tensor product of an exterior algebra and a polynomial algebra, that is, one has an isomorphism of k-algebras: H ∗ (G, k) ∼ = Λ(x)⊗k k[y], with |x| = 1 and |y| = 2. (12) The K¨ unneth isomorphism yields an isomorphism of graded k-algebras H ∗ (G × H, k) ∼ = H ∗ (G, k) ⊗k H ∗ (H, k) where the tensor product on the right is the graded tensor product: (a ⊗ b) · (c ⊗ d) = (−1)|b||c| ac ⊗ bd .

1 Brown representability The following result is known as Brown representability theorem and is due to Keller [40] and Neeman [46]; it is a variation of a classical theorem of Brown [19] from homotopy theory. 11 (Brown). Let T be a compactly generated triangulated category. For a functor H : Top → Ab the following are equivalent. (1) The functor H is cohomological and preserves set-indexed coproducts. (2) There exists an object X in T such that H ∼ = HomT (−, X). The proof is in some sense constructive; it shows that any object in T arises as the homotopy colimit of a sequence of morphisms φ0 φ1 φ2 X0 −→ X1 −→ X2 −→ · · · such that X0 and the cone of each φi is a coproduct of objects of the form Σn C, with n ∈ Z and C an object from the set of compact objects generating T.

Write ΩC for the kernel of the composite surjection P → C. Prove that there is an exact sequence of kG-modules 0 → ΩC → P ⊕ A → B → 0 . One gets a triangle ΩC → A → B → in StMod(kG) as A ∼ = P ⊕ A there. Similarly, if we embed B into an injective module I and form the dual construction, we obtain an exact sequence 0 → B → I ⊕ C → Ω−1 A → 0 . This gives a triangle B → C → Ω−1 A → in StMod(kG). (17) Let A be an abelian category with enough injectives; for example, the category of modules over some ring.