By Claire Amiot (auth.), Aslak Bakke Buan, Idun Reiten, Øyvind Solberg (eds.)
This booklet positive aspects survey and learn papers from The Abel Symposium 2011: Algebras, quivers and representations, held in Balestrand, Norway 2011. It examines a truly energetic study quarter that has had a starting to be impression and profound effect in lots of different components of arithmetic like, commutative algebra, algebraic geometry, algebraic teams and combinatorics. This quantity illustrates and extends such connections with algebraic geometry, cluster algebra idea, commutative algebra, dynamical platforms and triangulated different types. additionally, it comprises contributions on extra advancements in illustration idea of quivers and algebras.
Algebras, Quivers and Representations is focused at researchers and graduate scholars in algebra, illustration conception and triangulate categories.
Read Online or Download Algebras, Quivers and Representations: The Abel Symposium 2011 PDF
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Additional resources for Algebras, Quivers and Representations: The Abel Symposium 2011
1(2) shows that so is A ⊗ γ τ ⊗ A : A ⊗ τ C τ ⊗ A → A ⊗ τ A (BA)τ A ⊗ A. Since ε ACA = ε A(BA)A (A ⊗ γ τ ⊗ A) holds by Sect. 4(5) and ε A(BA)A is a quasiisomorphism by Sect. 4(4), we see that ε ACA is a quasi-isomorphism. (23) =⇒ (iv) =⇒ (v) =⇒ (vi) =⇒ (23) are proved by similar arguments. 2 For each augmented DG algebra A with A 0 = 0 or A −1 = 0 there is a canonical surjective quasi-isomorphism of DG algebras α A : ΩBA → A. For each coaugmented DG coalgebra C with C 1 or C 0 = 0 there is a canonical injective quasi-isomorphism of DG coalgebras γ C : C → BΩC.
2 Comparison Maps The units and counts of the adjunctions in Sect. 4 give maps ϕ AM A⊗ε C ⊗M ε ACM : A ⊗ τ C τ ⊗ M −−−−−−→ A ⊗ k ⊗ M = A ⊗ M −−−→ M C⊗ηA ⊗X ψ CX ηCAX : X −−−→ C ⊗ X = C ⊗ k ⊗ X −−−−−→ C τ ⊗ A ⊗ τ X ϕ NA N ⊗ε C ⊗A ε N CA : N ⊗ τ C τ ⊗ A −−−−−→ N ⊗ k ⊗ A = N ⊗ A −−−→ N Y ⊗ηA ⊗C ψYC ηY AC : Y −−−→ Y ⊗ C = Y ⊗ k ⊗ C −−−−−→ Y τ ⊗ A ⊗ τ C (19) (20) (21) (22) which—in the order displayed—are morphisms of left DG A-modules, left DG Ccomodules, right DG A-modules, and right DG C-comodules, respectively.
Claim Opposite cap products turn N ⊗ X into a left DG ΞCA The computations are parallel to those used for Sect. 3, with λ = λΞ Convolution algebras are functorial in both arguments: o (N ⊗X) . 7 Adjunction Given morphisms ϑ : X → C ⊗ M of left DG C-comodules and θ : A ⊗ X → M of left DG A-modules the composed maps ϕ AM A⊗ε C ⊗M A⊗ϑ ωCA (ϑ) : A ⊗ X −−−→ A ⊗ C ⊗ M −−−−−−→ A ⊗ M −−→ M C⊗ηA ⊗X ψ CX C⊗θ ωAC (θ ) : X −−→ C ⊗ X −−−−−→ C ⊗ A ⊗ X −−→ C ⊗ M are morphisms of left DG A-modules and of left DG C-comodules, respectively.