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Vol. 52 No. 1 (2026)
Serdica Mathematical Journal

Articles

Fundamental comparison, base-change, and descent theorems in the \(K\)-theory of non-commutative \(n\)-ary \(\Gamma\)-semirings

https://doi.org/10.55630/serdica.2026.85-108

Published 13-04-2026

  • Chandrasekhar Gokavarapu
    • ,
  • Dasari Madhusudhana Rao

Chandrasekhar Gokavarapu
Acharya Nagarjuna University and Government College (Autonomous)

Dasari Madhusudhana Rao
Acharya Nagarjuna University and Government College for Women (Autonomous)

Abstract

We develop a comparison, base--change, and descent framework for the algebraic \(K\)-theory of non-commutative \(n\)-ary \(\Gamma\)-semirings. Working in the Quillen-exact (and Waldhausen) setting of bi--finite, slot--sensitive \(\Gamma\)-modules and perfect complexes, we construct functorial maps on \(K\)-theory induced by extension and restriction of scalars under explicit \(\Gamma\)-flatness hypotheses in the relevant positional slots. We prove derived Morita invariance (via tilting bi--module complexes), establish Beck–Chevalley type base-change for cartesian squares, and deduce a projection formula compatible with the multiplicative structure coming from positional tensor products. Passing to the non--commutative \(\Gamma\)-spectrum  \(\operatorname{Spec}^{\mathrm{nc}}_\Gamma(T)\), we show locality for perfect objects and derive Zariski hyperdescent for \(\mathbb{K}(\mathbf{Perf})\), together with excision and localization sequences for closed immersions and fpqc descent for \(\Gamma\)-flat covers. Finally, we interpret \(K_\Gamma(X)\) geometrically as the $K$--theory of the stable \(\infty\)-category of \(\Gamma\)-perfect complexes, describe its universal property in \(\Gamma\)-linear non-commutative motives, and record compatibility with cyclotomic and Chern-type trace maps.

2020 Mathematics Subject Classification:

19D10, 19E08, 16Y60, 18E30, 55U35

Keywords

non-commutative \(n\)-ary \(\Gamma\)-semiring, positional tensor product, exact category, Waldhausen category, Quillen \(Q\)-construction, \(S_\bullet\)-construction, perfect \(\Gamma\)-complex, derived Morita invariance, base-change, projection formula, Zariski descent, fpqc descent, excision, localization, non-commutative motives, cyclotomic trace

Full text

Author Details

Chandrasekhar Gokavarapu

Department of Mathematics
Acharya Nagarjuna University
Guntur, Andhra Pradesh, India, PIN: 522510
and
Government College (Autonomous)
Rajahmundry, Andhra Pradesh, India, PIN: 533105
e-mail: chandrasekhargokavarapu@gmail.com

Dasari Madhusudhana Rao

Department of Mathematics
Acharya Nagarjuna University
Guntur, Andhra Pradesh, India, PIN: 522510
and
Government College for Women (Autonomous)
Guntur, Andhra Pradesh, India, PIN: 522001
e-mail: dmrmaths@gmail.com

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