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In this short note, we given a new proof of Mitchells theorem that $L_{Tleft(nright)} K(Z) cong 0$ for $n geq 2$. Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology for chromatically-localized algebraic K-theory.
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Let A be a connected graded algebra and let E denote its Ext-algebra. There is a natural A-infinity algebra structure on E, and we prove that this structure is mainly determined by the relations of A. In particular, the coefficients of the A-infinity products m_n restricted to the tensor powers of Ext^1 give the coefficients of the relations of A. We also relate the m_ns to Massey products.
We provide a more economical refined version of Evrards categorical cocylinder factorization of a functor [Ev1,2]. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred functor which satisfies the (dual) assumption of Quillens Theorem B.
Suppose k is a field of characteristic 2, and $n,mgeq 4$ powers of 2. Then the $A_infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_infty$-structure on $H^*(C_ntimes C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.
This paper contains a construction of generators and partial relations for $K_1$ of a simplicial Waldhausen category where cofiber sequences split up to weak equivalence. The main application of these generators and relations is to produce generators for $K_1$ of a (simplicial) assembler.
We strengthen some results in etale (and real etale) motivic stable homotopy theory, by eliminating finiteness hypotheses, additional localizations and/or extending to spectra from HZ-modules.
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