On the structure of noncommutative mapping schemes

Maysam Maysami Sadr

arxiv: 1907.09969 · v1 · pith:NU6WB2WCnew · submitted 2019-07-23 · 🧮 math.AG · math.RA

On the structure of noncommutative mapping schemes

Maysam Maysami Sadr This is my paper

Pith reviewed 2026-05-24 17:07 UTC · model grok-4.3

classification 🧮 math.AG math.RA

keywords noncommutative schemesind-schemesquantum groupsHopf algebrasmapping schemesdual formalismalgebraic duality

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The pith

A dual functorial formalism defines ind-schemes of mappings between schemes, G-mappings for quantum groups, and homomorphisms between quantum groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a dual functorial formalism to consider three types of objects: the ind-scheme of mappings between two schemes, the ind-scheme of G-mappings between two G-schemes where G is a quantum group, and the ind-scheme of group homomorphisms between two quantum groups. Schemes are treated as dual to unital associative algebras and quantum groups as dual to Hopf algebras. This setup allows the coordinate algebras of these ind-schemes to be identified with tensor products or Hom-objects in the respective categories. A reader would care if this formalism provides a consistent way to extend mapping spaces to noncommutative settings.

Core claim

Three types of objects are considered in a dual functorial formalism: (i) ind-scheme of mappings between two schemes, (ii) ind-scheme of G-mappings between two G-schemes for a quantum group G, and (iii) ind-scheme of group homomorphisms between two quantum groups, where schemes and quantum groups are dual to unital associative algebras and Hopf algebras.

What carries the argument

The dual functorial formalism identifying coordinate algebras of the ind-schemes with tensor products or Hom-objects in categories of algebras and Hopf algebras.

Load-bearing premise

That the dual functorial formalism correctly identifies the coordinate algebras of the indicated ind-schemes with the appropriate tensor products or Hom-objects in the categories of algebras and Hopf algebras.

What would settle it

An explicit computation where the coordinate algebra of a mapping ind-scheme differs from the expected tensor product or Hom-object in the algebra category would falsify the identification.

read the original abstract

The following three types of objects are considered in a dual functorial formalism: (i) ind-scheme of mappings between two schemes, (ii) for a quantum group G, ind-scheme of G-mappings between two G-schemes, and (iii) ind-scheme of group homomorphisms between two quantum group. By schemes and quantum groups here we mean objects which are respectively dual to unital associative algebras and Hopf algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper announces a dual functorial setup linking mapping ind-schemes for schemes, G-schemes, and quantum group homomorphisms but contains no constructions, proofs, or checks.

read the letter

The core announcement is that three mapping-type ind-schemes can be treated together inside one dual functorial language, where ordinary schemes are dual to unital algebras and quantum groups to Hopf algebras. That simultaneous packaging is the only concrete claim visible. It does organize three related but usually separate objects under the same duality, which could save specialists a bit of bookkeeping if the identifications actually work out in examples. Beyond that, nothing is shown: no coordinate rings are written down, no functorial properties are derived, and no comparison with existing work on quantum homogeneous spaces or mapping schemes appears. The text reads as a definition rather than a result. Because the manuscript supplies no derivations or consistency checks, it is impossible to tell whether the dual identifications are tautological or whether they produce new structure. The central assumption—that the ind-scheme coordinate algebras match the expected tensor or Hom objects—remains untested on the page. This note is aimed at a narrow group already working inside noncommutative algebraic geometry who might want a compact language for these objects. For anyone outside that circle the paper adds nothing usable. It does not contain enough verifiable content to justify sending it to referees; the authors would need to supply explicit constructions and at least one nontrivial example before the claim can be evaluated.

Referee Report

1 major / 0 minor

Summary. The manuscript announces a dual functorial formalism in which three types of ind-schemes are considered: (i) the ind-scheme of mappings between two schemes, (ii) the ind-scheme of G-mappings between two G-schemes for a quantum group G, and (iii) the ind-scheme of group homomorphisms between two quantum groups, where schemes are dual to unital associative algebras and quantum groups are dual to Hopf algebras.

Significance. The announcement of such a formalism, if substantiated with explicit identifications of coordinate algebras via tensor products or Hom-objects, could provide a framework for studying mapping spaces in noncommutative algebraic geometry. As presented, however, the text contains only this definitional statement with no constructions, theorems, or verifications.

major comments (1)

[Abstract] Abstract, paragraph 1: the central claim that the three ind-schemes are considered inside the dual functorial formalism rests on the unstated assumption that coordinate algebras are correctly identified with the appropriate tensor products or Hom-objects in the categories of algebras and Hopf algebras; no derivation or explicit identification is supplied.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their report. Our manuscript is a brief announcement of a dual functorial formalism and does not contain detailed constructions. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 1: the central claim that the three ind-schemes are considered inside the dual functorial formalism rests on the unstated assumption that coordinate algebras are correctly identified with the appropriate tensor products or Hom-objects in the categories of algebras and Hopf algebras; no derivation or explicit identification is supplied.

Authors: The manuscript states the three types of ind-schemes as objects considered inside the dual functorial formalism, where schemes are dual to unital associative algebras and quantum groups to Hopf algebras. This relies on the standard categorical duality without supplying new derivations or explicit tensor-product identifications, as the text is limited to the definitional announcement. revision: no

standing simulated objections not resolved

The manuscript contains only the definitional statement with no constructions, theorems, or verifications.

Circularity Check

0 steps flagged

No significant circularity; purely definitional setup

full rationale

The paper announces consideration of three types of ind-schemes in a dual functorial formalism and defines schemes/quantum groups via duality to unital associative algebras and Hopf algebras. No equations, derivations, predictions, fitted parameters, or load-bearing self-citations are present. The text consists solely of this definitional announcement without any claimed theorem, computation, or reduction that could exhibit circularity by construction. The framework is self-contained as stated.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; the ledger is therefore limited to the explicit dualities stated there.

axioms (2)

domain assumption Schemes are dual to unital associative algebras
Stated explicitly in the abstract as the meaning of 'schemes'.
domain assumption Quantum groups are dual to Hopf algebras
Stated explicitly in the abstract as the meaning of 'quantum groups'.

pith-pipeline@v0.9.0 · 5581 in / 1263 out tokens · 19909 ms · 2026-05-24T17:07:00.238092+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 4 internal anchors

[1]

Artin, B

M. Artin, B. Mazur, Etale homotopy , Lecture Notes in Mathematics 100, Springer–Verlag, 1969

work page 1969
[2]

Cherenaeck, L

P. Cherenaeck, L. Guerra, Spaces of morphisms between algebraic spaces , In the mathematical heritage of CF Gauss, World Scientiﬁc (1991), 100–118

work page 1991
[3]

Gersten, Homotopy theory of rings , J

S.M. Gersten, Homotopy theory of rings , J. Algebra, 19 no. 3 (1971), 396–415

work page 1971
[4]

Kambayashi, Pro-aﬃne algebras, ind-aﬃne groups and the Jacobian proble m, J

T. Kambayashi, Pro-aﬃne algebras, ind-aﬃne groups and the Jacobian proble m, J. Algebra, 185 no. 2 (1996), 481–501

work page 1996
[5]

Kambayashi, M

T. Kambayashi, M. Miyanishi, On two recent views of the Jacobian conjecture , Contemp. Math., 369 (2005), 113–138

work page 2005
[6]

Kontsevich, A.L

M. Kontsevich, A.L. Rosenberg, Noncommutative smooth spaces, In the Gelfand mathematical seminars, 1996-1999, pp. 85–108, Birkhuser Boston, 2000. (arXiv:math/9 812158 [math.AG])

work page 1996
[7]

Milne, Basic theory of aﬃne group schemes , Available online: www.jmilne.org, 2012

J.S. Milne, Basic theory of aﬃne group schemes , Available online: www.jmilne.org, 2012

work page 2012
[8]

Quantum Functor $Mor$

M.M. Sadr, Quantum functor Mor , Math. Pannonica, 21 no. 1 (2010), 77–88. (arXiv:0807.5124 [math.OA])

work page internal anchor Pith review Pith/arXiv arXiv 2010
[9]

A Kind of Compact Quantum Semigroups

M.M. Sadr, A kind of compact quantum semigroups , Int. J. Math. Math. Sci., 2012 (2012), Article ID 725270, 10 pages. (arXiv:0808.2740 [math.OA])

work page internal anchor Pith review Pith/arXiv arXiv 2012
[10]

On the quantum groups and semigroups of maps between noncommutative spaces

M.M. Sadr, On the quantum groups and semigroups of maps between noncomm utative spaces, Czechoslo- vak Math. J., 67 no. 1 (2017): 97–121. (arXiv:1506.06518 [math.QA])

work page internal anchor Pith review Pith/arXiv arXiv 2017
[11]

Shafarevich, On some inﬁnite-dimensional groups , Rend

I.R. Shafarevich, On some inﬁnite-dimensional groups , Rend. Mat. Appl., 25 (1966), 208-212

work page 1966
[12]

Shafarevich, On some inﬁnite-dimensional groupsII , Math

I.R. Shafarevich, On some inﬁnite-dimensional groupsII , Math. USSRIzvestija, 18 (1982), 185-194

work page 1982
[13]

Quantum families of maps and quantum semigroups on finite quantum spaces

P.M. So/suppress ltan,Quantum families of maps and quantum semigroups on ﬁnite qua ntum spaces , J. Geom. Phys., 59 (2009), 354-368. (arXiv:math/0610922 [math.OA])

work page internal anchor Pith review Pith/arXiv arXiv 2009
[14]

Sweedler, Hopf algebras, Mathematics Lecture Note Series, W

M.E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, New York, 196 9

work page
[15]

Woronowicz, Compact matrix pseudogroups , Comm

S.L. Woronowicz, Compact matrix pseudogroups , Comm. in Math. Phy., 111 no. 4 (1987), 613–665. Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan 45137-66731, Iran E-mail address : sadr@iasbs.ac.ir

work page 1987

[1] [1]

Artin, B

M. Artin, B. Mazur, Etale homotopy , Lecture Notes in Mathematics 100, Springer–Verlag, 1969

work page 1969

[2] [2]

Cherenaeck, L

P. Cherenaeck, L. Guerra, Spaces of morphisms between algebraic spaces , In the mathematical heritage of CF Gauss, World Scientiﬁc (1991), 100–118

work page 1991

[3] [3]

Gersten, Homotopy theory of rings , J

S.M. Gersten, Homotopy theory of rings , J. Algebra, 19 no. 3 (1971), 396–415

work page 1971

[4] [4]

Kambayashi, Pro-aﬃne algebras, ind-aﬃne groups and the Jacobian proble m, J

T. Kambayashi, Pro-aﬃne algebras, ind-aﬃne groups and the Jacobian proble m, J. Algebra, 185 no. 2 (1996), 481–501

work page 1996

[5] [5]

Kambayashi, M

T. Kambayashi, M. Miyanishi, On two recent views of the Jacobian conjecture , Contemp. Math., 369 (2005), 113–138

work page 2005

[6] [6]

Kontsevich, A.L

M. Kontsevich, A.L. Rosenberg, Noncommutative smooth spaces, In the Gelfand mathematical seminars, 1996-1999, pp. 85–108, Birkhuser Boston, 2000. (arXiv:math/9 812158 [math.AG])

work page 1996

[7] [7]

Milne, Basic theory of aﬃne group schemes , Available online: www.jmilne.org, 2012

J.S. Milne, Basic theory of aﬃne group schemes , Available online: www.jmilne.org, 2012

work page 2012

[8] [8]

Quantum Functor $Mor$

M.M. Sadr, Quantum functor Mor , Math. Pannonica, 21 no. 1 (2010), 77–88. (arXiv:0807.5124 [math.OA])

work page internal anchor Pith review Pith/arXiv arXiv 2010

[9] [9]

A Kind of Compact Quantum Semigroups

M.M. Sadr, A kind of compact quantum semigroups , Int. J. Math. Math. Sci., 2012 (2012), Article ID 725270, 10 pages. (arXiv:0808.2740 [math.OA])

work page internal anchor Pith review Pith/arXiv arXiv 2012

[10] [10]

On the quantum groups and semigroups of maps between noncommutative spaces

M.M. Sadr, On the quantum groups and semigroups of maps between noncomm utative spaces, Czechoslo- vak Math. J., 67 no. 1 (2017): 97–121. (arXiv:1506.06518 [math.QA])

work page internal anchor Pith review Pith/arXiv arXiv 2017

[11] [11]

Shafarevich, On some inﬁnite-dimensional groups , Rend

I.R. Shafarevich, On some inﬁnite-dimensional groups , Rend. Mat. Appl., 25 (1966), 208-212

work page 1966

[12] [12]

Shafarevich, On some inﬁnite-dimensional groupsII , Math

I.R. Shafarevich, On some inﬁnite-dimensional groupsII , Math. USSRIzvestija, 18 (1982), 185-194

work page 1982

[13] [13]

Quantum families of maps and quantum semigroups on finite quantum spaces

P.M. So/suppress ltan,Quantum families of maps and quantum semigroups on ﬁnite qua ntum spaces , J. Geom. Phys., 59 (2009), 354-368. (arXiv:math/0610922 [math.OA])

work page internal anchor Pith review Pith/arXiv arXiv 2009

[14] [14]

Sweedler, Hopf algebras, Mathematics Lecture Note Series, W

M.E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, New York, 196 9

work page

[15] [15]

Woronowicz, Compact matrix pseudogroups , Comm

S.L. Woronowicz, Compact matrix pseudogroups , Comm. in Math. Phy., 111 no. 4 (1987), 613–665. Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan 45137-66731, Iran E-mail address : sadr@iasbs.ac.ir

work page 1987