Riemannian Geometry on Associative Varieties

Arvid Siqveland

arxiv: 2604.10462 · v1 · submitted 2026-04-12 · 🧮 math.AG

Riemannian Geometry on Associative Varieties

Arvid Siqveland This is my paper

Pith reviewed 2026-05-10 16:22 UTC · model grok-4.3

classification 🧮 math.AG

keywords associative varietieslocal representing objectssimple modulesRiemannian geometryalgebraic geodesicsconnectionsassociative algebrasdifferential geometry

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The pith

Associative algebras support a generalization of algebraic varieties that admits Riemannian geometry with connections and algebraic geodesics.

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

The paper first shows that classical algebraic varieties can be defined over any field rather than only algebraically closed ones. It then constructs local representing objects for simple modules over associative algebras, using these to replace the usual localization at maximal ideals and thereby define associative varieties. Substituting smooth functions on Euclidean space for the polynomial ring in this construction permits the definition of Riemannian metrics, connections, and geodesic curves directly on these associative varieties. This introduces real differential geometry into the setting of associative algebras, extending classical geometry to a broader algebraic context.

Core claim

We prove that the classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields k. For associative algebras A, there exist local representing objects A_M for simple modules M. Replacing the localization in maximal ideals with the local representations in simple modules defines an associative generalization of varieties. Replacing R[n] with C^∞(R^n) allows us to define Riemannian geometry on associative varieties, including connections and algebraic geodesic curves.

What carries the argument

The local representing objects A_M for simple modules M, which replace localization at maximal ideals to define associative varieties and support the subsequent differential geometry when the base ring is C^∞(R^n).

If this is right

Classical varieties defined over algebraically closed fields extend directly to arbitrary base fields.
Associative algebras admit a well-defined notion of varieties via local representations of simple modules.
Riemannian metrics and connections become definable on these associative varieties.
Algebraic geodesic curves arise naturally as solutions to the geodesic equation in this setting.
Real differential geometry, including its analytic aspects, enters the theory of associative algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may supply algebraic models for smooth non-commutative spaces that recover classical manifolds when the algebra is commutative.
Explicit calculations on low-dimensional associative algebras such as matrix rings could verify whether the defined geodesics match expected lengths or curvatures.
The same local-representation replacement might extend to other structures such as symplectic forms or complex structures on associative varieties.
This points toward a possible dictionary between module-theoretic data over associative algebras and classical differential-geometric invariants.

Load-bearing premise

The local representing objects A_M for simple modules provide a faithful generalization of localization at maximal ideals, and substituting the smooth function ring for the polynomial ring preserves enough algebraic structure to define a consistent Riemannian metric, connections, and geodesics.

What would settle it

A concrete associative algebra A and simple module M where the local object A_M yields no well-defined derivation or metric compatible with the module action, or where candidate algebraic geodesics fail to satisfy the first-order differential equation expected from the connection.

read the original abstract

We prove that the classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields $k.$ Then we prove that for associative algebras $A$, there exist local representing objects $A_M$ for simple modules $M.$ Replacing the localization in maximal ideals in the commutative situation with the local representations in simple modules in the associative, we define an associative generalization of varieties. Now we realize that replacing $\mathbb R[x_1,\dots,x_n]=\mathbb R[n]$ with $C^\infty(\mathbb R^n),$ we can do differential geometry for associative $\mathbb R[n]$-algebras. This says that we can define a Riemannian geometry on associative varieties. This gives us the definition of connections and algebraic geodesic curves, introducing real geometry into associative 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 sketches associative varieties via simple-module local objects and claims Riemannian geometry on them, but the central constructions are asserted without explicit definitions or commutative reduction.

read the letter

The main point to take away is that this paper replaces localization at maximal ideals with local representing objects for simple modules to define associative varieties, then swaps the polynomial ring for smooth functions to introduce Riemannian geometry, connections, and algebraic geodesics on those objects. It also claims to show that classical varieties work over arbitrary fields as a starting step. That framing is new in its specific combination, and it does a reasonable job of pointing toward a possible bridge between commutative algebraic geometry and non-commutative algebra that might interest people in deformation theory or quantum algebra. The author is clearly trying to bring real differential geometry into the associative setting, which is an honest attempt at something useful if the details hold up. The soft spot is that the local objects A_M are introduced by assertion rather than by giving a concrete construction, a universal property, or a direct check that they recover ordinary localization when the algebra is commutative. Without that reduction, it is difficult to confirm that the later substitution of C^∞ functions preserves the algebraic data needed for a well-defined metric, connection, or geodesic equation. The existence claims for A_M and the Riemannian structures are stated at a high level with no derivations or examples visible, so the argument stays at the level of an unverified analogy. This paper is for readers already working in non-commutative algebraic geometry who are willing to fill in missing steps themselves. A casual reader or someone needing reproducible theorems will get little out of it. I would send it to peer review so that referees can ask for the explicit definition of A_M and the commutative check; the idea has enough potential that it is worth seeing whether the gaps can be closed.

Referee Report

3 major / 1 minor

Summary. The manuscript asserts that classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields k. For associative algebras A it claims the existence of local representing objects A_M for simple modules M; these objects are said to replace localization at maximal ideals, thereby defining an associative generalization of varieties. Substituting the polynomial ring R[n] with C^∞(R^n) is then asserted to permit the definition of Riemannian geometry on these associative varieties, including connections and algebraic geodesic curves.

Significance. If the local representing objects A_M were explicitly constructed, shown to satisfy a universal property, and verified to recover ordinary localization in the commutative case, the work would supply a potentially useful bridge between algebraic geometry and non-commutative differential geometry. At present the absence of any such construction or reduction check leaves the central claims unverified and therefore of limited significance.

major comments (3)

Abstract: the existence of local representing objects A_M for simple modules is asserted without any definition, explicit construction, or universal property. This omission is load-bearing, since the subsequent definition of associative varieties and the Riemannian extension both rest on the claim that A_M faithfully replaces localization at maximal ideals.
Abstract: no argument is supplied showing that, when A is commutative, A_M recovers the usual localization A_m at the annihilator of M. Without this commutative reduction the claimed generalization cannot be checked against the classical theory.
Abstract: the substitution of R[n] by C^∞(R^n) is said to allow the definition of connections and algebraic geodesic curves, yet no derivation is given for the resulting bilinear forms, covariant derivatives, or geodesic equations, nor is any discussion provided of the analytic hypotheses needed to make these objects well-defined.

minor comments (1)

The abstract contains awkward phrasing (e.g., 'we realize that replacing') that reduces readability; a minor rewrite for clarity would help.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, agreeing that additional detail is required for clarity and verifiability. We will revise the manuscript accordingly.

read point-by-point responses

Referee: Abstract: the existence of local representing objects A_M for simple modules is asserted without any definition, explicit construction, or universal property. This omission is load-bearing, since the subsequent definition of associative varieties and the Riemannian extension both rest on the claim that A_M faithfully replaces localization at maximal ideals.

Authors: We agree the abstract is too terse on this point. The manuscript defines A_M via the representation of simple modules over the associative algebra A, but to make the construction explicit and state the universal property, we will add a dedicated subsection in the revision with the full definition and proof of existence. revision: yes
Referee: Abstract: no argument is supplied showing that, when A is commutative, A_M recovers the usual localization A_m at the annihilator of M. Without this commutative reduction the claimed generalization cannot be checked against the classical theory.

Authors: This is a fair criticism. We will insert a proposition in the revised version proving that, for commutative A, A_M is isomorphic to the localization of A at the maximal ideal annihilating M, thereby recovering the classical case. revision: yes
Referee: Abstract: the substitution of R[n] by C^∞(R^n) is said to allow the definition of connections and algebraic geodesic curves, yet no derivation is given for the resulting bilinear forms, covariant derivatives, or geodesic equations, nor is any discussion provided of the analytic hypotheses needed to make these objects well-defined.

Authors: We accept that the derivations and hypotheses are missing from the current text. In the revision we will supply explicit formulas for the bilinear forms, covariant derivatives, and geodesic equations obtained after the substitution, together with the required smoothness and analytic conditions on the functions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation proceeds by stated existence proofs and definitional replacement.

full rationale

The abstract asserts proofs of existence for local representing objects A_M and a subsequent definitional replacement of localization by these objects to obtain associative varieties, followed by a ring substitution to enable Riemannian geometry. No equations, self-citations, or explicit constructions are supplied in the provided text that reduce any claimed result (such as the recovery of classical localization when A is commutative, or the preservation of derivations and metrics under the C^∞ substitution) to a tautology or to a fitted input by construction. The chain therefore remains a sequence of independent claims rather than a self-referential loop, consistent with a normal non-circular definitional extension.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on two unproven assertions in the abstract: that varieties extend to arbitrary fields and that simple-module local objects exist and suffice for the generalization. No free parameters appear; the invented entities are the associative varieties themselves and the algebraic geodesics.

axioms (2)

domain assumption Classical algebraic varieties over algebraically closed fields can be defined over arbitrary fields k.
Stated as the first result to be proved.
domain assumption For associative algebras A there exist local representing objects A_M for simple modules M.
Stated as the second result to be proved; used to replace maximal-ideal localization.

invented entities (2)

associative variety no independent evidence
purpose: Generalization of algebraic variety to associative algebras via simple-module local objects
Defined by the replacement construction; no independent existence proof supplied beyond the claim.
algebraic geodesic curve no independent evidence
purpose: Geodesic notion inside the associative Riemannian geometry
Introduced as part of the new geometry; no external verification given.

pith-pipeline@v0.9.0 · 5416 in / 1531 out tokens · 56851 ms · 2026-05-10T16:22:17.834801+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Hartshorne, Algebraic Geometry, GTM, Vol 52, Springer, 1977

R. Hartshorne, Algebraic Geometry, GTM, Vol 52, Springer, 1977

work page 1977
[2]

O. A. Laudal, Mathematical models in science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2021

work page 2021
[3]

Lee, Introduction to Riemannian Manifolds, Springer, Graduate Texts in Mathematics, 2018

John M. Lee, Introduction to Riemannian Manifolds, Springer, Graduate Texts in Mathematics, 2018

work page 2018
[4]

Arvid Siqveland, Associative Algebraic Geometry, World Scientific Pub- lishing Co. Pte. Ltd., Hackensack, NJ, 2023 ISBN: 977-1-80061-354-6 13

work page 2023

[1] [1]

Hartshorne, Algebraic Geometry, GTM, Vol 52, Springer, 1977

R. Hartshorne, Algebraic Geometry, GTM, Vol 52, Springer, 1977

work page 1977

[2] [2]

O. A. Laudal, Mathematical models in science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2021

work page 2021

[3] [3]

Lee, Introduction to Riemannian Manifolds, Springer, Graduate Texts in Mathematics, 2018

John M. Lee, Introduction to Riemannian Manifolds, Springer, Graduate Texts in Mathematics, 2018

work page 2018

[4] [4]

Arvid Siqveland, Associative Algebraic Geometry, World Scientific Pub- lishing Co. Pte. Ltd., Hackensack, NJ, 2023 ISBN: 977-1-80061-354-6 13

work page 2023