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Coherent sheaves on projective spaces and a lifting of the integral form of the Cartan subalgebra for quantum sl(2)

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Pith's one-line read The graded algebra that lifts the integral Cartan subalgebra of quantum sl(2) is the Grothendieck ring of G_m-equivariant coherent sheaves on projective spaces, with multiplication given by equivariant pushforward.

desk verdict Genuinely new geometric model of the integral lifting of U^0_A, with a fixable freeness proof gap that the author should patch before publication. read the letter →

arxiv 2608.09711 v1 pith:A2CCGXNP submitted 2026-08-10 math.QA math.RT

Mikhail Khovanov This is my paper · ORCID
classification math.QAmath.RT MSC 17B3719L4714F0505A30
keywords quantumsl(2)CartansubalgebracategorificationequivariantcoherentsheavesGrothendieckringprojectivespacesbinomialcoefficientsK-theory
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

This paper proves that a graded algebra R that surjects onto the integral form of the quantum-sl(2) Cartan subalgebra is actually the Grothendieck ring of a symmetric monoidal category of G_m-equivariant coherent sheaves on projective spaces. The identification takes the generator K_{c,n} to the class of the equivariant line bundle O(-c) on P^n, and under it the purely algebraic multiplication of R becomes the equivariant pushforward of external tensor products along the maps P^n × P^m → $P^{{n+m}}$. Since the pushforward of an external tensor product of line bundles is shown to split explicitly into a direct sum of line bundles with quantum-binomial multiplicities, the q-binomial multiplication formula is lifted to actual direct sum decompositions of equivariant vector bundles. A non-equivariant specialization recovers a description of the quotient ring eR in terms of ordinary coherent sheaves on projective spaces.

What carries the argument

The machinery rests on the symmetrizing addition maps f_{n,m}: P^n × P^m → $P^{{n+m}}$, which are finite, flat, and G_m-equivariant, and on the equivariant pushforward functor (f_{n,m})_*. The main technical theorem, Theorem 4.4, computes this pushforward on external tensor products of line bundles: for -m ≤ j-i ≤ n it gives f_*(O_{P^n}(i) ⊠ O_{P^m}(j)) ≅ ⊕_{k=0}^m {j-i+m \choose k}{i-j+n \choose m-k} O_{$P^{{n+m}}$}(i+k-m), with balanced quantum binomials as multiplicities. The splitting is proved using Horrocks' Criterion, a cohomological condition that forces a vector bundle on projective space to split into line bundles, together with the equivariant Künneth formula and the balanced quantum Chu–Vandermonde identity; the multiplicities match the coefficients in relation (10), so the algebraic multiplication rule is the shadow of an actual decomposition of bundles.

What would settle it

Compute the module presented by the bK operators of Theorem 2.2 for small n and m (for example n=m=2) and verify explicitly that the operators commute and satisfy relation (10); if a nonzero Z[q,$q^{{-1}}$]-linear dependence among K_{0,2}, K_{1,2}, and K_{2,2} emerges, the claimed basis of R fails and the isomorphism R ≅ ⊕ $K^{{G_m}}$_0(P^n) collapses, since the target's line-bundle classes are independent.

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Extended reading notes

Core claim

The central discovery is an isomorphism of graded Z[q,$q^{{-1}}$]-algebras R ≅ ⊕_{n≥0} $K^{{G_m}}$_0(P^n), sending K_{c,n} to the symbol of the equivariant line bundle O(-c) on P^n. The multiplication in R, defined by the quantum-binomial relation (10), is realized geometrically: for 0≤c≤n and 0≤b≤m, the class K_{c,n}K_{b,m} is the equivariant pushforward along f_{n,m} of the external tensor product of the corresponding line bundles, and Theorem 4.4 gives an explicit decomposition of that pushforward into a direct sum of equivariant line bundles with multiplicities given by balanced quantum binomials. The same geometric picture categorifies relations (9) via exact sequences of equivariant line bundles on P^n. Setting q=1 yields the non-equivariant isomorphism eR ≅ ⊕_n K_0(P^n), where the multiplication is governed by classical binomial coefficients.

Load-bearing premise

The whole identification rests on the claim that the elements K_{c,n} with 0≤c≤n form an independent basis of R over the Laurent polynomial ring Z[q,$q^{{-1}}$]; the paper leaves the direct verification of this independence to a computation it omits, and its alternate proof goes through the very isomorphism being established.

Editorial extensions

If this is right

  • The q-binomial multiplication formula (10) is upgraded: each product K_{c,n}K_{b,m} equals a sum of classes K_{k,n+m} with coefficients that are the ranks of explicit direct-summand line bundles, not just formal coefficients.
  • The category Coh^{G_m}(P) = ⊕_n Coh^{G_m}(P^n) becomes a symmetric monoidal abelian category with Grothendieck ring R, providing a categorification of the ring that lifts the integral Cartan subalgebra of quantum sl(2).
  • The non-equivariant specialization gives an isomorphism eR ≅ ⊕_n K_0(P^n), categorifying the classical binomial multiplication rule.
  • The natural surjection R → U^0_A has a nontrivial kernel containing elements such as K_{c+2,n} - (q^n+q^{-n})K_{c+1,n}+K_{c,n}-K_{c,n-2}, which vanish in U^0_A but are nonzero in R, distinguishing the categorified ring from the Lusztig integral Cartan algebra itself.
  • The equivariant Euler form on R is computed as (K_{c,n},K_{d,m}) = δ_{n,m}{n+c-d \choose n}, with multiplication adjoint to a comultiplication ∆(K_{c,n}) = Σ_k K_{c,k}⊗K_{c,n-k}, giving a self-dual structure on the categorified algebra.

Reading between the lines

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

  • If the isomorphism is accepted, the Harder–Narasimhan argument used in the proof suggests the direct-sum decompositions are canonical rather than merely cohomological: the direct sums in (35) reflect an intrinsic filtration of the pushforward bundle, so the categorification is strong, not just an equality of Grothendieck classes.
  • Remark 3.11 connects the pushforward bundles to exterior powers of secant (Schwarzenberger) bundles on symmetric powers of P^1; this opens a bridge between the quantum Cartan algebra and classical vector bundles, potentially yielding new splitting or stability statements for those bundles.
  • The SL_2-equivariant version in Remark 4.7 indicates that the τ-invariant subring R^τ is the Grothendieck ring of SL_2-equivariant coherent sheaves on the disjoint union of P^n, suggesting a categorification over the representation ring of SL_2 rather than over Laurent polynomials in q.
  • The general scheme—define a ring by quantum-binomial relations, find a quotient map whose pushforward splits via vanishing of intermediate cohomology, and match multiplicities via a hypergeometric identity—could serve as a template for categorifying the Cartan parts of other quantum groups, using higher-dimensional toric or flag varieties in place of projective spaces.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper defines a graded Z[q,q^{-1}]-algebra R with generators K_{c,n} and relations (9) and (10), and identifies it with the direct sum of G_m-equivariant Grothendieck groups of projective spaces, sending K_{c,n} to [O_{P^n}(-c)]. Multiplication in R is thereby realized as the equivariant pushforward along the addition maps P^n × P^m → P^{n+m}. The main technical content is Theorem 4.4, which gives an explicit direct-sum decomposition of these pushforwards using Horrocks' criterion and the balanced q-Chu-Vandermonde identity. A non-equivariant analogue (Theorem 1.2) is proved in Section 3. The paper also shows that R surjects onto Lusztig's integral form U^0_A of the Cartan subalgebra of quantum sl(2). The overall strategy is compelling, but the proof of the main isomorphism, as written, omits the injectivity step and contains a circular dependence on Theorem 2.2.

Significance. If the main theorem is established, the paper gives a genuine geometric categorification of a lifting of Lusztig's integral Cartan subalgebra: the quantum-binomial multiplication in R is lifted to direct-sum decompositions of equivariant vector bundles. The explicit pushforward computation in Theorem 4.4 is a concrete, checkable result with independent value, and the q=1 version in Section 3 is proved in detail. The geometric computations are presented with enough care that the central identification does not appear to be circular. However, the missing injectivity proof is a load-bearing gap that must be repaired before the main claim is fully supported.

major comments (2)
  1. [§4, Corollary 4.5 and Theorem 1.1] The proof establishes that the defining relations (9) and (10) are respected by the map K_{c,n} ↦ [O_{P^n}(-c)] (relation (9) from the Koszul resolution on P^n, relation (10) from Theorem 4.4) and that the resulting map is surjective, because K^{G_m}_0(P^n) is free on [O(0)],...,[O(-n)]. This gives a well-defined surjection but not injectivity. Injectivity can be proved without Theorem 2.2: relation (9) shows R_n is spanned by K_{0,n},...,K_{n,n}; if an element ∑ a_s K_{s,n} maps to zero, then ∑ a_s [O(-s)]=0, and linear independence of the basis forces all a_s=0. This argument should be stated explicitly in the proof of Corollary 4.5. As written, the text only checks the relations and then asserts the isomorphism, leaving injectivity unsupported except for the deferred Theorem 2.2.
  2. [§2, Theorem 2.2] The proof of freeness of R is not complete. The module-theoretic verification is deferred with the comment that it is 'a tedious computation' and the details are omitted, and the alternative proof offered invokes Theorem 1.1, which is circular because the proof of Theorem 1.1 currently depends on freeness for injectivity. Since the injectivity argument described above does not require Theorem 2.2, the cleanest remedy is to insert that argument into Section 4 and then derive Theorem 2.2 as a corollary. Alternatively, the omitted q-Chu-Vandermonde/Sears computation should be supplied in an appendix.
minor comments (6)
  1. [§2, first paragraph] There is a typo: 'definion' should be 'definition', and the arrow in (1) is typeset as 'R− → U0 A' with unusual spacing.
  2. [§3.2, proof of Theorem 3.3] After writing E ≅ ⊕_k O(a_k)^{⊕ C_k}, the sequence notation '. . . a_k < a_{k+1} < . . .' should specify the finite index set and the bounds on k.
  3. [§3.2, Remark 3.8] The displayed chain of adjoint functors appears to have a typo: it reads '(f_{n,m})_*, (f_{n,m})_*, (f_{n,m})^!' but should presumably read '(f_{n,m})^*, (f_{n,m})_*, (f_{n,m})^!'.
  4. [§4, proof of Theorem 4.4] In the character computation, the statement that 'the right-hand side is the character of the global sections of the equivariantly split bundle E′' is only justified for t ≥ -min(i,j); this domain is already introduced, but the sentence should explicitly repeat the constraint to avoid ambiguity.
  5. [§3.2, Example 3.10 and §4, Example 4.6] Both examples state nontrivial isomorphisms with proofs omitted. Since they are not needed for the main theorem this is acceptable, but adding a reference or a one-sentence indication of the proof of (30) would improve readability.
  6. [§5, proof of Proposition 2.4] The notation is slightly confusing: F_n is first defined as an operator, and then F_{c,n}=F_n(1) is the evaluation on the constant function; please distinguish the operator from its evaluation more explicitly.

Circularity Check

1 steps flagged · score 4.0 of 10

A circular alternative proof of freeness appears in Theorem 2.2, but the central geometric identification has independent content and the loop can be broken.

  1. other [Section 2, Theorem 2.2 proof; Section 4, Corollary 4.5]
    "Alternatively, the theorem follows from the identification of R via Theorem 1.1 and an explicit computation of the pushforward of the external tensor product of suitable equivariant line bundles in Theorem 4.4. ... Note also that relation (9) matches the corresponding exact sequences involving equivariant bundles OPn (−c−k). The following corollary is stated as Theorem 1.1 in the introduction."

    Theorem 2.2 asserts that R is free with basis {K_s}, i.e. that the K_{c,n} with 0≤c≤n are independent. Its stated alternative proof derives this freeness from Theorem 1.1 and Theorem 4.4. But the route to Theorem 1.1 in Section 4 checks that the defining relations (9)-(10) hold in ⊕_n K^{G_m}(P^n) via exact sequences and Theorem 4.4, and then concludes the isomorphism. The injectivity of K_{c,n} ↦ [O(-c)] is exactly the independence assertion of Theorem 2.2. Thus the alternative proof of Theorem 2.2 invokes a theorem whose isomorphism step requires Theorem 2.2 unless a separate injectivity argument is supplied. The paper does not supply that argument at this point, so the written dependency is circular; the loop is breakable, which prevents this from being fatal.

full rationale

The main theorem is not a renamed input or a fitted prediction: Theorem 4.4 computes equivariant pushforwards from Horrocks' criterion and character-valued Hilbert functions, and the multiplicities match the q-binomial coefficients in relation (10); relation (9) is matched by standard Euler/Koszul exact sequences. These checks are independent of the ring structure being proved. However, the paper leaves Theorem 2.2 with an omitted 'tedious computation' and gives an alternative proof referring to Theorem 1.1. As written, that alternative proof is circular unless one adds the available but unstated argument that a well-defined surjective map from R onto the free module ⊕ K^{G_m}(P^n) forces injectivity. Because the circularity is confined to a lemma and the central geometric content is independently derived, the score is moderate rather than high. No load-bearing self-citation appears; Proposition 2.5 is cited to [8]/[16], works by other authors.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted parameters appear; q is a formal variable, not a fitted constant. The proof of the main theorem relies on the freeness assertion for R (unproved in the text), plus standard tools in equivariant K-theory and algebraic geometry. No new physical or algebraic entities are postulated.

assumptions (7)
  • ad hoc to paper R is a free Z[q,q^{-1}]-module with basis {K_{c,n}: 0<=c<=n} (Theorem 2.2).
    The proof is omitted ('We omit the details'); the alternative proof cites Theorem 1.1, which in turn relies on this freeness, creating a circular dependency. Injectivity of the main isomorphism depends on this assertion.
  • standard math Horrocks' criterion for splitting of vector bundles on projective spaces.
    Used in Theorems 3.3 and 4.4 to conclude E is a direct sum of line bundles after intermediate cohomology vanishes. Cited to [19, Theorem 2.3.1].
  • standard math Equivariant projective bundle theorem gives a basis of K^{G_m}(P^n) as [O(-i)], 0<=i<=n.
    Used in Section 4 to identify the target of the isomorphism and its freeness. Cited to [17, Theorem 10] and [26].
  • standard math Harder-Narasimhan filtration exists and is unique; it is G-invariant under the G_m action.
    Used in Theorem 4.4 to make the splitting equivariant. Cited to [9].
  • standard math G_m is linearly reductive, so taking invariants is exact.
    Used in Theorem 4.4 to split equivariant extensions. Standard for tori.
  • standard math f_{n,m}: P^n times P^m to P^{n+m} is finite, flat and surjective (Miracle flatness).
    Ensures pushforward of coherent sheaves is exact and preserves vector bundles. Proposition 3.1, cited to [5] and Stacks.
  • domain assumption There exist equivariant exact sequences realizing relation (9) as alternating sums of line bundle classes.
    The paper asserts these sequences exist ('Note also that relation (9) matches the corresponding exact sequences...') without giving details; they are likely standard Koszul resolutions.

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Pith. "Pith review of Coherent sheaves on projective spaces and a lifting of the integral form of the Cartan subalgebra for quantum sl(2)." pith.science (2026). https://pith.science/paper/A2CCGXNP

@misc{pith2026260809711,
  author       = {Pith},
  title        = {Pith review of: Coherent sheaves on projective spaces and a lifting of the integral form of the Cartan subalgebra for quantum sl(2)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2CCGXNP}},
  note         = {Machine review of arXiv:2608.09711}
}
read the original abstract

We categorify a lifting of the integral form of the Cartan subalgebra for quantum sl(2), via categories of equivariant coherent sheaves on projective spaces.

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Works this paper leans on

27 extracted references · 13 canonical work pages

  1. [1]

    Linearization of algebraic group actions

    Michel Brion. “Linearization of algebraic group actions”. In:Handbook of group actions. Vol. IV. Vol. 41. Adv. Lect. Math. (ALM). Int. Press, Somerville, MA, 2018, pp. 291–340.isbn: 978-1-57146-365-4

    work page 2018

  2. [2]

    A combinatorial property ofq-Eulerian numbers

    L. Carlitz. “A combinatorial property ofq-Eulerian numbers”. In:Amer. Math. Monthly82 (1975), pp. 51–54.issn: 0002-9890,1930-0972.doi:10.2307/2319133

    work page doi:10.2307/2319133 1975

  3. [3]

    q-Bernoulli and Eulerian numbers

    — “q-Bernoulli and Eulerian numbers”. In:Trans. Amer. Math. Soc.76 (1954), pp. 332–350. issn: 0002-9947,1088-6850.doi:10.2307/1990772

    work page doi:10.2307/1990772 1954

  4. [4]

    Modern Birkh¨ auser Classics

    Neil Chriss and Victor Ginzburg.Representation theory and complex geometry. Modern Birkh¨ auser Classics. Reprint of the 1997 edition. Birkh¨ auser Boston, Ltd., Boston, MA, 2010, pp. x+495. isbn: 978-0-8176-4937-1.doi:10.1007/978-0-8176-4938-8

    doi:10.1007/978-0-8176-4938-8 1997

  5. [5]

    David Eisenbud.Commutative algebra. Vol. 150. Graduate Texts in Mathematics. With a view toward algebraic geometry. Springer-Verlag, New York, 1995, pp. xvi+785.isbn: 0-387-94268-8; 0-387-94269-6.doi:10.1007/978-1-4612-5350-1

    doi:10.1007/978-1-4612-5350-1 1995

  6. [6]

    A field-independent filtration of plethystic modules for $\mathrm{SL}_2(\mathbb{F})$ that categorifies a product rule for the Cartan subalgebra of $\mathcal{U}_q(\mathfrak{sl}_2)$

    ´Alvaro Guti´ errez,´Alvaro L. Mart ´ ınez, Micha l Szwej, and Mark Wildon.A field-independent filtration of plethystic modules forSL 2(F)that categorifies a product rule for the Cartan subalgebra ofU q(sl2). 2026. arXiv:2607.06749 [math.RT]

    work page Pith review arXiv 2026

  7. [7]

    A new bijective proof of the q-Pfaff–Saalsch¨ utz identity with applications to quantum groups

    — “A new bijective proof of the q-Pfaff–Saalsch¨ utz identity with applications to quantum groups”. In:European Journal of Combinatorics133 (Mar. 2026), p. 104321.issn: 0195-6698. doi:10.1016/j.ejc.2025.104321

    arXiv 2026

  8. [8]

    —Modular isomorphisms ofSL 2(F)-plethysms for Weyl modules labelled by hook partitions

    work page

Show all 27 references
  1. [9]

    Daniel Huybrechts and Manfred Lehn.The geometry of moduli spaces of sheaves. Second. Cam- bridge Mathematical Library. Cambridge University Press, Cambridge, 2010, pp. xviii+325.isbn: 978-0-521-13420-0.doi:10.1017/CBO9780511711985

    2010 doi

  2. [10]

    A categorification of quantum sl(n)

    Mikhail Khovanov and Aaron D. Lauda. “A categorification of quantum sl(n)”. In:Quantum Topol.1.1 (2010), pp. 1–92.issn: 1663-487X,1664-073X.doi:10.4171/QT/1

    2010 doi

  3. [11]

    A categorification of quantum sl(2)

    Aaron D. Lauda. “A categorification of quantum sl(2)”. In:Adv. Math.225.6 (2010), pp. 3327– 3424.issn: 0001-8708,1090-2082.doi:10.1016/j.aim.2010.06.003

    2010 doi

  4. [12]

    Quantum deformations of certain simple modules over enveloping algebras

    G. Lusztig. “Quantum deformations of certain simple modules over enveloping algebras”. In:Adv. in Math.70.2 (1988), pp. 237–249.issn: 0001-8708.doi:10.1016/0001-8708(88)90056-4

    1988 doi

  5. [13]

    Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra

    George Lusztig. “Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra”. In:J. Amer. Math. Soc.3.1 (1990), pp. 257–296.issn: 0894-0347,1088-6834.doi:10. 2307/1990988

    1990

  6. [15]

    Lectures on theK-functor in algebraic geometry

    Ju. ˜I. Manin. “Lectures on theK-functor in algebraic geometry”. In:Uspehi Mat. Nauk24.5(149) (1969), pp. 3–86.issn: 0042-1316

    1969

  7. [16]

    SL 2-plethysms and Cartan categorification

    Alvaro Luis Martinez Ruiz. “SL 2-plethysms and Cartan categorification”. PhD thesis. Columbia University, 2025.doi:10.7916/wfhx-dr37

    2025 doi

  8. [17]

    EquivariantK-theory

    Alexander S. Merkurjev. “EquivariantK-theory”. In:Handbook ofK-theory. Vol. 1, 2. Springer, Berlin, 2005, pp. 925–954.isbn: 978-3-540-23019-9; 3-540-23019-X.doi:10.1007/978-3-540- 27855-9\_18

    2005 doi

  9. [18]

    Mumford, J

    D. Mumford, J. Fogarty, and F. Kirwan.Geometric invariant theory. Third. Vol. 34. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, 1994, pp. xiv+292.isbn: 3-540-56963-4

    1994

  10. [19]

    Modern Birkh¨ auser Classics

    Christian Okonek, Michael Schneider, and Heinz Spindler.Vector bundles on complex projective spaces. Modern Birkh¨ auser Classics. Corrected reprint of the 1988 edition, With an appendix by S. I. Gelfand. Birkh¨ auser/Springer Basel AG, Basel, 2011, pp. viii+239.isbn: 978-3-03...

    1988

  11. [20]

    Lectures on K-theoretic computations in enumerative geometry

    Andrei Okounkov. “Lectures on K-theoretic computations in enumerative geometry”. In:Geom- etry of moduli spaces and representation theory. Vol. 24. IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2017, pp. 251–380.isbn: 978-1-4704-3574-5.doi:10.1090/pcms/024

    2017 doi

  12. [21]

    Kyle Petersen.Eulerian numbers

    T. Kyle Petersen.Eulerian numbers. Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. [Birkh¨ auser Advanced Texts: Basel Textbooks]. With a foreword by Richard Stanley. Birkh¨ auser/Springer, New York, 2015, pp. xviii+456.isbn: 978-1-4939-3090-6; 978-1-4939-3091-3.doi:10.1007/...

    2015 doi

  13. [22]

    Hermite reciprocity and Schwarzenberger bundles

    Claudiu Raicu and Steven V. Sam. “Hermite reciprocity and Schwarzenberger bundles”. In: Commutative algebra. Springer, Cham, [2021]©2021, pp. 689–721.isbn: 978-3-030-89693-5; 978-3-030-89694-2.doi:10.1007/978-3-030-89694-2\_23

    2021 doi

  14. [23]

    Raphael Rouquier.2-Kac-Moody algebras. 2008. arXiv:0812.5023 [math.RT]

    2008 arXiv

  15. [24]

    Quiver Hecke algebras and 2-Lie algebras

    Rapha¨ el Rouquier. “Quiver Hecke algebras and 2-Lie algebras”. In:Algebra Colloq.19.2 (2012), pp. 359–410.issn: 1005-3867,0219-1733.doi:10.1142/S1005386712000247

    2012 doi

  16. [25]

    The Stacks project authors.The Stacks project.https://stacks.math.columbia.edu. 2026

    2026

  17. [26]

    AlgebraicK-theory of group scheme actions

    R. W. Thomason. “AlgebraicK-theory of group scheme actions”. In:Algebraic topology and algebraicK-theory (Princeton, N.J., 1983). Vol. 113. Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563.isbn: 0-691-08415-7; 0-691-08426-2

    1983

  18. [27]

    Weibel.TheK-book

    Charles A. Weibel.TheK-book. Vol. 145. Graduate Studies in Mathematics. An introduction to algebraicK-theory. American Mathematical Society, Providence, RI, 2013, pp. xii+618.isbn: 978-0-8218-9132-2.doi:10.1090/gsm/145. Department of Mathematics, Johns Hopkins University, Balt...

    2013 doi

  19. [2026]

    arXiv:2509.01490 [math.RT]

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