Some perspectives on homotopy obstructions

Satya Mandal () and Bibekananda Mishra ()
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Satya Mandal: University of Kansas
Bibekananda Mishra: University of Kansas

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 1, 294-300

Abstract: Abstract For a projective A-module P, with $$n=rank(P)\ge 2$$ n = r a n k ( P ) ≥ 2 , the Homotopy obstruction sets $$\pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(P)\right)$$ π 0 L O ( P ) were defined, in [6], to detect whether P has a free direct summand or not. These sets have a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some further perspective on these sets $$\pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(P)\right)$$ π 0 L O ( P ) . In particular, under similar regularity and other conditions, we prove that if P, Q are two projective A-modules, with $$rank(P)=rank(Q)=d$$ r a n k ( P ) = r a n k ( Q ) = d and $$\det (P) \cong \det Q$$ det ( P ) ≅ det Q , then $$\pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(Q)\right) \cong \pi _0\left( {{\mathcal {L}}{\mathcal {O}}}(P)\right)$$ π 0 L O ( Q ) ≅ π 0 L O ( P ) .

Keywords: Projective modules; Chow groups (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00005-y

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