2021/05/20 by Alexandru Chirvăsitu, Chirvasitu, Alexandru, Tao Hong +1
Mathematics · #08B20 #16S10 #16W50 #17B01 #17B70 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2105.09731
openalex publication_date 2021/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a diagram ⋯ → F3 → F2→ F1 of algebraic systems, where Fn denotes the free object on n generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the Fi are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the Fi are free Lie algebras then the limit in the category of graded Lie algebras is again free.