A technique for computing oriented cohomology rings of semisimple algebraic groups

Raj Gandhi

REVIEW 2 major objections 4 minor 86 references

A technique for computing oriented cohomology rings of semisimple algebraic groups

T0 review · 2 major / 4 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read Oriented cohomology of semisimple groups admits a finite presentation built from formal Demazure operators, with explicit minimal rings for A1, A2 and B2.

desk verdict Clean algebraic pipeline that turns formal Demazure data into finite presentations of h^*(G); the six low-rank tables are new and usable. read the letter →

arxiv 2607.07900 v1 pith:I3ZQH7K7 submitted 2026-07-08 math.AG math.RA

Raj Gandhi This is my paper · ORCID

classification math.AGmath.RA MSC 14F4314L3019L4120G15

keywords orientedcohomologyformalDemazureoperatorssemisimplealgebraicgroupsgroupalgebraflagvarietiescobordismgeneratorsandrelations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that, for any oriented cohomology theory that obeys the localization axiom, the ring of a semisimple algebraic group G can be written as a quotient of a free algebra on dual formal Demazure operators by two explicitly described ideals of multiplication and Chern-class relations. The construction uses the formal group algebra of the character lattice and the coalgebra of formal Demazure operators as an algebraic model for the cohomology of the flag variety, then quotients by the image of first Chern classes of homogeneous line bundles. An algorithm turns the coproduct formulas and the twisted Leibniz rule into concrete generators and relations; applying it by hand yields minimal presentations for the six adjoint and simply-connected groups of types A1, A2 and B2. A reader who already knows Chow rings or algebraic cobordism can therefore compute the corresponding rings for these groups without new geometric arguments, and the same method is claimed to extend to other root systems.

What carries the argument

The dual of the formal affine Demazure algebra D_F^*, equipped with the product induced by the cocommutative coproduct on Demazure operators; the two ideals M (multiplication relations coming from that coproduct) and A (Chern-class relations) give the presentation of h^*(G).

What would settle it

Compute the presentation produced by Algorithm 7.11 for one of the Table-1 groups (say PGL(3)) and check whether the resulting ideal coincides with the known Chow ring or algebraic K-theory of that group; a mismatch would refute the isomorphism of Theorem 7.9.

Watch

Extended reading notes

Core claim

There is an R-algebra isomorphism h^*(G) ≈ D_F^*/(C_RJΛK_F + I_F D_F^*), equivalently a finite presentation R⟨B^*⟩/(M+A) whose relations are computed by a five-step algorithm that extracts constant terms of coproduct coefficients and Demazure actions on first Chern classes; for the adjoint and simply-connected groups of types A1, A2 and B2 the resulting presentations simplify to the six rings listed in Table 1.

Load-bearing premise

The cohomology theory must be weakly birationally invariant, satisfy localization, and have Bott–Samelson classes as a free basis of the flag variety, while the formal group algebra must be regular with respect to roots and the torsion index of the root datum must be invertible in the coefficient ring.

Editorial extensions

If this is right

Any oriented theory satisfying the listed axioms acquires an explicit finite presentation for h^*(G) once the coproduct coefficients of the Demazure operators are known.
Specialization of the free formal group law recovers the classical Chow rings and Grothendieck rings of the six groups as the rings of Table 1.
The same generators-and-relations description applies, with only larger Weyl-group data, to every other semisimple root datum whose formal group algebra is Σ-regular.
Computer algebra can output the full list of relations for higher-rank groups; hand simplification then yields minimal presentations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The algorithm is already coded for rank-2 root systems; extending the code to G2 or A3 would immediately produce candidate presentations that can be checked against known special cases.
Because the construction factors through the simply-connected cover, the method also gives presentations for intermediate groups whose character lattices sit between root and weight lattices.
The same formal-Demazure model may supply generators for the equivariant oriented cohomology of flag varieties, not only the ordinary rings of the groups themselves.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper gives a usable computational route from formal Demazure operators to a finite presentation of the oriented cohomology ring h^*(G) for split semisimple G. The central new piece is Theorem 7.9: h^*(G) ≃ D_F^* / (C + I D^*), equivalently the free-algebra presentation of Corollary 7.10 whose relations are produced by Algorithm 7.11. That quotient step, together with the six minimal presentations in Table 1, is the actual contribution; the underlying model of h^*(G/B) is already in CPZ13/CZZ16.

What works well is the care with which the pieces are assembled. The derivation chains published isomorphisms (CPZ13 Thm 13.13, GZ12 Prop 5.1, CZZ coalgebra structures) with a short, carefully checked lemma on pull-backs of homogeneous line bundles (Prop 5.9). The A1 and A2 hand simplifications are written out completely; the B2 case is longer but follows the same pattern, and the Python scripts that generate the raw relations are linked. Specializing the general presentation recovers the expected Chow and K_0 rings, which is a useful sanity check.

The soft spots are real but limited. Everything rests on Assumptions 3.8 and 6.6 (Σ-regularity, torsion index invertible, localization, Bott–Samelson basis). Those are standard for CH, K_0 and algebraic cobordism, and the paper states them up front rather than burying them. Only ranks ≤2 are fully simplified by hand; G2 is left as raw output. That is a scope choice, not a flaw in the method. The citation pattern stays inside the expected circle (CPZ, CZZ, GZ, Totaro, Levine–Morel) and is appropriate.

This is for people who already work with oriented cohomology of flag varieties or who need explicit generators-and-relations for low-rank groups. It does not open a new technology, but it is a solid, reproducible tool inside an established program. I would send it to referees without hesitation; the math is transparent and the tables are immediately usable.

Referee Report

2 major / 4 minor

Summary. The paper constructs an algebraic model for the oriented cohomology ring h^*(G) of a semisimple algebraic group G over an algebraically closed field of characteristic 0, for oriented cohomology theories h^* that satisfy the localization axiom (and related technical hypotheses). Combining the formal affine Demazure algebra D_F and its dual with the characteristic map and the Gille–Zainoulline relation h^*(G) ≅ h^*(G/B)/(im c_1|L_B), the author obtains an R-algebra isomorphism h^*(G) ≅ D_F^*/(C_{RJΛK_F} + I_F D_F^*) (Theorem 7.9) and an equivalent finite presentation R⟨B^*⟩/(M+A) (Corollary 7.10). Algorithm 7.11 extracts the relations explicitly from coproduct coefficients and constant terms of formal Demazure operators. The method is applied by hand (with machine-generated intermediate relations) to produce the minimal presentations of Table 1 for the adjoint and simply-connected groups of types A_1, A_2 and B_2.

Significance. The result supplies a uniform computational technique for oriented cohomology rings of semisimple groups that goes beyond the classical Chow-ring calculations of Grothendieck and Kac and the partial algebraic-cobordism computations of Yagita. The algebraic model is built from published isomorphisms (CPZ13, CZZ16, GZ12) under explicitly stated assumptions that hold for the principal theories of interest (CH, K_0, Ω). The low-rank presentations in Table 1 are concrete, falsifiable outputs; the linked Python scripts make the intermediate relations reproducible. If the method extends routinely to higher rank, it would become a standard tool for computing h^*(G) in the oriented-cohomology literature.

major comments (2)

The B_2 case of Example 8.5 (and the corresponding rows of Table 1) is only sketched: after listing the generators of M and A the text states that “the rest of this calculation is performed by hand and is similar to” the A_2 case. Because the claimed minimal presentation R[x]/(2x-a_{11}x^{2},2x^{2},x^{4}) is one of the paper’s main concrete outputs, the intermediate reductions that eliminate all generators except a single class of degree 1 should be written out (or placed in an appendix) so that a reader can verify the relations without re-running the code.
Assumption 6.6 (weak birational invariance + CPZ Assumption 13.2 + localization) and Assumption 3.8 (Σ-regularity and regularity of the torsion index) are load-bearing for Theorems 7.7–7.9 and Proposition 7.8. While they hold for CH, K_0 and Ω, the manuscript never states whether the resulting presentations remain valid after base change of R that may kill the torsion index or destroy Σ-regularity. A short remark clarifying the range of coefficients for which Table 1 is known to hold would strengthen the claim.

minor comments (4)

In the abstract and Introduction the localization axiom is mentioned, but the full list of hypotheses (Assumptions 3.8 and 6.6) appears only later; a one-sentence pointer in the abstract would help the reader.
Notation for the dual basis elements Δ_I_w^* is introduced in §4 and then reused heavily in §7–8; a brief reminder at the beginning of Algorithm 7.11 would improve readability.
The GitHub link [Gana] is given only as a bare URL in the references; adding a short description of what the scripts compute would make the computational claims easier to check.
Typographical inconsistencies appear in the Cartan-matrix displays of Examples 2.2–2.5 (spacing and alignment) and in a few places where “Λ” is rendered as “Λ” versus plain “Lambda”.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algebraic model and low-rank presentations are derived from published independent constructions under stated assumptions, not forced by definition or self-fit.

full rationale

The load-bearing chain is: (i) formal group algebra and formal Demazure algebra D_F with its coalgebra structure (CPZ13, CZZ16, under Assumption 3.8); (ii) the geometric identification h^*(G) ≃ h^*(G/B)/(im c_1|L_B) from GZ12 (requiring localization); (iii) the algebraic model h^*(G/B) ≃ (ϵD_F)^* from CPZ13 (under Assumption 6.6 / CPZ 13.2); (iv) the quotient presentation of Theorem 7.9 / Corollary 7.10 obtained by identifying the image of the algebraic Chern class map with the ideal C + I D^*; (v) Algorithm 7.11 extracting constant terms of coproduct and Demazure evaluations, followed by hand simplification to Table 1. None of these steps is definitionally equivalent to its input: D_F and its dual are constructed independently of h^*(G); the GZ12 and CPZ13 isomorphisms are published theorems with their own proofs and do not presuppose the generators-and-relations form of h^*(G); the author (Gandhi) does not overlap with the cited authors, so there is no self-citation chain; no parameters are fitted to data and then re-predicted; and the Table 1 specializations are transparent specializations of the general presentation, not renamings of an empirical pattern. Assumptions 3.8 and 6.6 are stated explicitly and hold for the standard theories (CH, K_0, Ω). The derivation is therefore self-contained against its external benchmarks.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central claim rests on standard formal-group and root-datum machinery plus two domain assumptions (localization and Σ-regularity/torsion-index invertibility) that restrict the class of theories and groups. No free parameters are fitted; the only invented entities are the algebraic models already present in the cited literature, re-used here for the group itself.

assumptions (3)

domain assumption h^* is a weakly birationally invariant oriented cohomology theory satisfying the localization axiom (Def. 6.3) and CPZ Assumption 13.2 (Bott–Samelson classes form an R-basis of h^*(G/B)).
Invoked as Assumption 6.6; required for the isomorphism of Theorem 7.7 and the exact sequence of Proposition 7.8.
domain assumption The formal group algebra RJΛK_F is Σ-regular and the torsion index t is regular in R.
Assumption 3.8; needed for freeness of the Demazure algebra (Thm 3.17, 4.2) and for the coalgebra structure used in §7.
standard math Standard axioms of one-dimensional commutative formal group laws and of semisimple root data.
Recalled in §§2–3; used throughout to define formal Demazure operators.

invented entities (1)

Algebraic model D_F^* / (C + I D^*) for h^*(G) independent evidence
purpose: Provides a purely algebraic presentation of the oriented cohomology ring of the group itself.
Constructed in Theorem 7.9 by quotienting the dual of the formal affine Demazure algebra by the ideal generated by Chern classes; independent evidence is the recovery of known Chow and K-theory rings as special cases.

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Pith. "Pith review of A technique for computing oriented cohomology rings of semisimple algebraic groups." pith.science (2026). https://pith.science/paper/I3ZQH7K7

@misc{pith2026260707900,
  author       = {Pith},
  title        = {Pith review of: A technique for computing oriented cohomology rings of semisimple algebraic groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3ZQH7K7}},
  note         = {Machine review of arXiv:2607.07900}
}

abstract

We present a technique for computing a finite set of generators and relations for the ring $\mathrm{h}^*(G)$ in terms of formal Demazure operators, where $\mathrm{h}^*$ is an oriented cohomology theory satisfying the localization axiom and $G$ is a semisimple algebraic group. Using this technique, we give minimal presentations for the oriented cohomology rings of the adjoint and simply-connected groups of types $A_1$, $A_2$, and $B_2$.

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Pith tools

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[1] [1]

, TITLE =

Aluffi, P., Marcolli, M. , TITLE =. Int. J. Geom. Methods Mod. Phys. , FJOURNAL =. 2011 , NUMBER =. doi:10.1142/S0219887811005099 , URL =

work page doi:10.1142/s0219887811005099 2011

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