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1. Generalized Bockstein maps and Massey products

   https://doi.org/10.1017/fms.2022.103  

   Lam, Yeuk Hay; Liu, Yuan; Sharifi, Romyar; Wake, Preston; Wang, Jiuya (January 2023, Forum of Mathematics, Sigma)

   Abstract Given a profinite group G of finite p -cohomological dimension and a pro- p quotient H of G by a closed normal subgroup N , we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H . We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H , we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H . We apply our study to prove lower bounds on the p -ranks of class groups of certain nonabelian extensions of $$\mathbb {Q}$$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology. 

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2. On the Lie algebra structure of integrable derivations

   https://doi.org/10.1112/blms.12884  

   Briggs, Benjamin; Rubio_y_Degrassi, Lleonard (December 2023, Bulletin of the London Mathematical Society)

   Abstract Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group. 

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3. Maximal Tori in HH 1 and the Fundamental Group

   https://doi.org/10.1093/imrn/rnac026  

   Briggs, Benjamin; Rubio y Degrassi, Lleonard (March 2023, International Mathematics Research Notices)

   Abstract We investigate maximal tori in the Hochschild cohomology Lie algebra $${\operatorname {HH}}^1(A)$$ of a finite dimensional algebra $$A$$, and their connection with the fundamental groups associated to presentations of $$A$$. We prove that every maximal torus in $${\operatorname {HH}}^1(A)$$ arises as the dual of some fundamental group of $$A$$, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $$A$$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras. 

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4. Cup products on curves over finite fields

   Frauke M. Bleher and Ted Chinburg (January 2021, arXivorg)

   In this paper we compute cup products of elements of the first étale cohomology group of μℓ over a smooth projective geometrically irreducible curve C over a finite field k when ℓ is a prime and #k≡1 mod ℓ. Over the algebraic closure of k, such products are values of the Weil pairing on the ℓ-torsion of the Jacobian of C. Over k, such cup products are more subtle due to the fact that they naturally take values in Pic(C)⊗Zμ~ℓ rather than in the group μ~ℓ of ℓth roots of unity in k∗. 

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5. Cohomology of finite tensor categories: Duality and Drinfeld centers

   https://doi.org/10.1090/tran/8548  

   Negron, Cris; Plavnik, Julia (March 2022, Transactions of the American Mathematical Society)

   We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. 

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