Pith. sign in

REVIEW 2 major objections 6 minor 62 references

Noncommutative Gelfand Duality: the algebraic case

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that every ring, and more generally every connective dg-algebra over $\mathbb{Z}$, is faithfully represented by the site of its homotopical-epimorphism localizations together with a structure presheaf, yielding a…

desk verdict The construction is genuinely new and the faithful-embedding result is solid, but the advertised anti-equivalence is really just faithfulness; the paper overstates its main theorem. read the letter →

arxiv 2411.11816 v2 pith:T2WDL6IC submitted 2024-11-18 math.AG math-phmath.CTmath.MPmath.QA

Federico Bambozzi , Matteo Capoferri , Simone Murro This is my paper · ORCID
classification math.AGmath-phmath.CTmath.MPmath.QA MSC 14A2214A3016E3518F20
keywords noncommutativeGelfanddualityhomotopicalepimorphismformalhomotopyZariskitopologyspectrumderivedalgebraicgeometrypre-ringedsitesdescendablepresheafcomonadicdescent
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 tries to establish a noncommutative version of Gelfand duality for ordinary rings: every ring $R$ can be represented, faithfully, by a geometric object built from $R$'s own localizations. The representation is the pair $(\mathrm{Zar}_R, \mathcal{O}_R)$, where $\mathrm{Zar}_R$ is the site whose 'open subsets' are the homotopical epimorphisms out of $R$ and whose covers are finite families whose base-change functors on algebras are jointly conservative, and $\mathcal{O}_R$ is a structure presheaf of connective dg-algebras. The main theorem states that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, meaning that distinct morphisms of rings induce distinct morphisms of the associated sites; the construction passes through the larger homotopy category $\mathrm{HRings}_{\mathbb{Z}}$ of connective dg-algebras, because even for ordinary noncommutative rings the natural structure presheaf can take values that are not concentrated in degree zero. This matters because it gives noncommutative rings a genuine geometric spectrum with points, opens, and functions, while classical attempts to extend the prime-ideal spectrum to all rings are blocked by a no-go theorem. If correct, it also supplies a framework in which the geometry of quantum-spacetime algebras could be studied rigorously.

What carries the argument

The carrying object is a new Grothendieck topology on $\mathrm{HRings}_{\mathbb{Z}}^{\mathrm{op}}$, the formal homotopy Zariski topology. Its open embeddings are homotopical epimorphisms: maps $A \to B$ such that the fold/codiagonal map $B \ast^{\mathrm{L}}_A B \to B$ is an equivalence (equivalently $B \otimes^{\mathrm{L}}_A B \to B$ is an equivalence), the noncommutative replacement for flat epimorphisms and Zariski open immersions. Its covers are finite families $\{A \to B_i\}$ for which the base-change functors $(-)\ast^{\mathrm{L}}_A B_i \colon \mathrm{HRings}_A \to \mathrm{HRings}_{B_i}$ are jointly conservative. From this topology one forms the small site $\mathrm{Zar}_A$ of localizations of $A$; the site is coherent and has enough points, which yields the sober space $\operatorname{Spec}^{\mathrm{NC}}(A)$. The second load-bearing piece is the structure presheaf $\mathcal{O}_A$ together with the notion of a descendable presheaf: reconstruction from a cover is performed with the Amitsur--Cech nerve and comonadic descent, rather than by the ordinary sheaf condition.

What would settle it

Take a concrete noncommutative connective dg-algebra $A$ and a homotopy pushout square $A \to B$, $A \to C$ in $\mathrm{HRings}_{\mathbb{Z}}$; if the induced map $B \to B \ast^{\mathrm{L}}_A C$ is not a homotopical epimorphism, or if two morphisms $C \rightrightarrows D$ in $\mathrm{HRings}_B$ become equivalent after every base change along a cover but are not equivalent directly, then the topology axioms of Proposition 4.4 fail and the spectrum functor $\operatorname{Spec}^{\mathrm{NC}}$ would not be functorial.

Watch

Extended reading notes

Core claim

The central claim is that the contravariant functor $\mathrm{Spec}^{\mathrm{NC}}\colon \mathrm{HRings}_{\mathbb{Z}} \to \mathrm{PreRingSites}$, $A \mapsto (\mathrm{Zar}_A,\mathcal{O}_A)$, is faithful (Theorem 7.20); composing with the fully faithful inclusion $\mathrm{Rings}_{\mathbb{Z}} \hookrightarrow \mathrm{HRings}_{\mathbb{Z}}$ gives Theorem 1.1. Here $\mathrm{HRings}_{\mathbb{Z}}$ is the homotopy category of connective dg-algebras over $\mathbb{Z}$, in which ordinary rings sit as discrete objects. The site $\mathrm{Zar}_A$ is the opposite of the category of homotopical epimorphisms $A \to B$, with finite covers given by families whose derived free-product functors $(-)\ast^{\mathrm{L}}_A B_i$ are jointly conservative on $\mathrm{HRings}_A$; the structure presheaf sends such a localization to $B$ as a connective dg-algebra. The proof of faithfulness evaluates both sides on the terminal object: a morphism $A \to B$ of homotopical rings is recovered as the map $\mathcal{O}_A(X_A)=A \to B$ induced on global sections, so two distinct maps produce two distinct maps of pre-ringed sites. The paper also shows the associated topos has enough points, so $\mathrm{Zar}_A$ presents a sober topological space $\operatorname{Spec}^{\mathrm{NC}}(A)$, and that $\mathcal{O}_A$ is a descendable presheaf satisfying comonadic descent, which is what replaces the sheaf condition in the noncommutative setting.

Load-bearing premise

The whole construction stands or falls on one premise: the chosen classes of 'open subsets' and 'covers' truly satisfy the axioms of a Grothendieck topology; the proof relies on the stability of homotopical epimorphisms under homotopy pushouts and on the homotopical pullback lemma, and any morphism where those fail would destroy the spectrum functor.

Editorial extensions

If this is right

  • Every ordinary ring, and every connective dg-algebra over $\mathbb{Z}$, is distinguished by its localization site: distinct ring homomorphisms induce distinct morphisms of pre-ringed sites.
  • The spectrum $\operatorname{Spec}^{\mathrm{NC}}(A)$ is functorial, nonempty, and often large; its points are completely prime ultrafilters on the lattice of localizations, and its open subsets correspond to smashing localizations of the category of modules over $A$.
  • For commutative input the construction is compatible with classical geometry: fields and discrete valuation rings have the same spectra as the Grothendieck spectrum, the fine spectrum of a finitely generated commutative $\mathbb{C}$-algebra is homeomorphic to the Grothendieck spectrum, and in general there are canonical continuous maps relating the noncommutative, fine, and Grothendieck spectra.
  • The structure presheaf of a discrete noncommutative ring can take values in nonzero cohomological degrees, so a faithful geometric representation of all rings forces homotopical and dg rings into the picture.
  • Since the structure presheaf satisfies comonadic descent but not necessarily the sheaf condition, the classical notion of sheaf is inadequate as the glueing rule for noncommutative geometry; the paper proposes descendable pre-sheaves as the correct replacement, with ordinary sheaf theory recovered in the commutative case.

Reading between the lines

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

  • Read in light of the previously established no-go theorem for spectrum functors valued in sets, the present construction suggests that the obstruction is not to geometric duality itself but to requiring points to be prime ideals: replacing topological spaces by sites plus structure presheaves restores a faithful duality.
  • The faithfulness argument only uses the terminal object of the site, so the same strategy could be tried in analytic settings: any contravariant representation that records the identity section at the terminal open will automatically be faithful, independently of whether the structure presheaf is a sheaf.
  • For physical applications, the spectrum gives a concrete way to ask what a point of a quantum spacetime is: for algebras such as noncommutative tori or symplectic twisted group algebras one could compute the lattice of homotopical epimorphisms and read off the resulting sober space, and the paper's examples show such point spaces are computable in nontrivial cases.
  • A natural test of the framework is the paper's Conjecture 7.14: finding a homotopical Zariski cover whose Amitsur--Cech complex for the structure presheaf is not exact would keep the faithful duality intact but would block the upgrade to ordinary ringed spaces.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 proposes a noncommutative spectrum functor on the homotopy category of connective dg-algebras over Z, built from localizations that are homotopical epimorphisms. The central construction associates to each A the small site Zar_A of homotopical-epimorphism localizations of A, together with a structure presheaf O_A. The main result is stated as an anti-equivalence of Rings_Z with a subcategory of pre-ringed sites; the proof establishes that the contravariant functor A ↦ (Zar_A, O_A) is faithful. The paper also introduces a finer Zariski topology, a relative spectrum that agrees with the Grothendieck spectrum for finitely generated commutative C-algebras, and a notion of descendable presheaves intended as a noncommutative replacement for sheaves.

Significance. If the constructions are correct, the faithful embedding is a meaningful step: it shows that the site of homotopical localizations with its structure presheaf contains enough information to recover the ring, and the comparisons with the Grothendieck spectrum and with smashing subcategories give the construction independent interest. The paper is transparent about relying on a conjecture for the fine-topology results, and Theorem 7.20 itself is unconditional and proven by a short, convincing evaluation at the terminal object. However, the advertised Gelfand-type anti-equivalence is not obtained: fullness is neither proven nor claimed, and the paper explicitly defers the description of the essential image. The main value is thus the faithful embedding and the supporting machinery, not an anti-equivalence in the usual sense.

major comments (2)
  1. [Theorem 1.1 and Corollary 7.21; Section 7.2] The statement that Rings_Z is 'anti-equivalent to a subcategory of PreRingSites' is not supported by the proof. Theorem 7.20 proves only faithfulness of the functor (7.5). No fullness or essential surjectivity is established; in fact, the text following Theorem 7.20 says the description of the essential image is deferred to a separate work, and Example 7.16 shows that the functor is not full on morphisms even for commutative rings: the map of pre-ringed spaces Spec Q → Spec Z_(p) is not induced by a ring map. If 'subcategory' is taken to be the non-full image subcategory, then Theorem 1.1 is a tautological restatement of faithfulness; if it is taken as a full subcategory, the claim is false. The theorem should be restated as a faithful contravariant embedding, and the phrase 'anti-equivalence' should be reserved for a future result or used only with an explicit warning about the non-full subcategory convention.
  2. [Proposition 4.4] The proof that the formal homotopy Zariski topology satisfies the Grothendieck topology axioms is incomplete. The stability of homotopical epimorphisms under arbitrary homotopy pushouts is asserted without proof or reference, and the transitivity axiom is verified by a sketch that invokes the homotopy pullback lemma of [40] without spelling out the required hypotheses. Because this proposition is what guarantees that Zar_A is a site and that Spec^NC is a functor (Corollary 4.15), the constructions in Sections 4.3 and 7.2 are not fully justified until this proof is completed.
minor comments (6)
  1. [Section 1] The sentence containing 'see Defition 4.3' has a typo: it should read 'see Definition 4.3'.
  2. [Corollary 7.21] The corollary states that the category of rings is 'equivalent' to a subcategory of PreRingSites; given the contravariance of the construction, the intended word appears to be 'anti-equivalent', or the statement should explicitly define the non-full subcategory convention.
  3. [Example 4.8(vi)] The notation for the pushout A ∗^L_{R[s]} R[s,s^{-1}] should be clarified: the map is given by R[x] → A, x ↦ s, so the base of the free product is R[s] viewed as a copy of the polynomial ring R[x].
  4. [Section 7.2] The term 'Amistur–Čech nerve' appears at least twice; the standard spelling is 'Amitsur–Čech'.
  5. [Section 6.1] In the paragraph discussing the ring k^n, the text says 'one computes Spec^NC(k)' but the computation is for the direct product k^n, so the symbol should be corrected accordingly.
  6. [Notation table and Definition 5.5] The notation table lists Spec^NC_fine as a functor on CRings_Z, whereas Definition 5.5 states it for A ∈ HRings_Z; these should be reconciled.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central faithfulness proof is direct, and cited results are external or non-load-bearing; the 'anti-equivalence' wording overstates a faithful embedding, but this is an overstatement, not a circular derivation.

full rationale

The paper's main derivation chain is not circular. The core faithfulness claim, Theorem 7.20, is proved directly: for distinct maps f,g : A -> B in HRings_Z, the induced morphisms of pre-ringed sites are evaluated on the terminal object X_A of Zar_A, where O_A(X_A)=A and the induced maps are f and g respectively; hence the induced site morphisms differ. This argument does not presuppose the conclusion, and the structure presheaf is constructed from the localizations of A rather than fitted to the target statement. The topology underlying the site is justified in Proposition 4.4 using stability of homotopical epimorphisms and the homotopical pullback lemma of [40], an external reference, together with the equivalence of homological and homotopical epimorphisms from [23], also external to the authors. Comparisons with the Grothendieck spectrum are anchored to independent classifications in [2], [46] and [60], and the Reyes no-go theorem is invoked as an external constraint, not as an authorial assumption. Self-citations [9], [10], [12], [13], [19], [20] appear in motivational, analogical, or example-supporting roles, and none is load-bearing for Theorem 1.1. The phrase 'anti-equivalent to a subcategory of PreRingSites' is stronger than what is proved: the paper proves a faithful functor, and Corollary 7.21 states the subcategory is not required to be full. This is explicitly acknowledged in the text ('By this we mean that the functor G^NC is a faithful functor' and the comparison with affine schemes as a non-full subcategory of PreRingSp). Thus the advertised anti-equivalence is an overstatement or a non-standard use of 'equivalence' rather than a circular step: the proof does not secretly assume the duality it claims. The declared Assumption 7.3 (Conjecture 5.18) is used for fine-spectrum descent statements, not for the faithfulness theorem, and it is stated as an assumption rather than disguised as a proved input. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled in via self-citation. The derivation is self-contained in the sense that the central recoverability statement reduces to the direct evaluation on the terminal section of the structure presheaf, which is the intended content of a Gelfand-type duality rather than an input recycled as an output.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper is pure mathematics with no fitted constants or free parameters. The construction and the main theorem rest on: standard homotopical algebra (monoidal Dold-Kan correspondence, [54]); the theory of homotopical epimorphisms ([23], [60]); standard topos theory (Deligne's theorem, localic topoi); external classifications of smashing subcategories and flat epimorphisms for commutative rings ([2], [46], [37], [38]); and one explicitly assumed conjecture (Assumption 7.3, applying Conjecture 5.18) that is used only for the fine-spectrum descent statements, not for the central faithfulness theorem (Theorem 7.20). The invented entities (Spec^NC, the formal homotopy Zariski topology, descendable presheaves) all carry external benchmarks: agreement with the Grothendieck spectrum in commutative cases, agreement with the classical Zariski topology on finite-presentation localizations, and reduction to ordinary sheaves for commutative rings.

assumptions (5)
  • domain assumption Homotopical epimorphisms coincide with homological epimorphisms, and the class of homotopical epimorphisms is stable under homotopy pushouts along arbitrary morphisms in HRings_Z.
    Proposition 4.2 invokes [23, Theorem 4.4]; the stability under pushouts and the homotopical pullback lemma [40, Lemma 4.4.2.1] carry the proof of Proposition 4.4, which establishes the Grothendieck topology underlying the whole spectrum construction.
  • domain assumption The monoidal Dold-Kan correspondence gives Quillen equivalences Ho(SRings_R) and Ho(DGA^{<=0}_R) (Proposition 2.2), so HRings_Z is the homotopy category of connective dg-algebras with derived free products as pushouts.
    Cited from [54, Theorem 1.1(3)] and used throughout; Lemma 3.20, needed to compare connective with unbounded dg-algebra statements, is only sketched.
  • standard math Standard topos theory: a coherent site has enough points (Deligne), the topos of sheaves on a poset with a topology is localic, and a localic topos with enough points is the topos of a sober topological space.
    Proposition 4.12 converts the site Zar_X into the topological space Spec^NC(A); the paper's argument is standard and lines up with the classical theory of posites and ideals.
  • standard math External classifications of smashing subcategories and flat epimorphisms for commutative rings: [2] (flat ring epimorphisms for Noetherian rings), [46] (smashing subcategories correspond to generalization-closed subsets of SpecG), [60] (flat epimorphisms of finite presentation are homotopical…
    These results carry the comparison theorems (Propositions 5.10, 5.24; Corollary 5.20) and the computed examples in Section 6. They are external benchmarks, which anchor the construction to classical algebraic geometry.
  • ad hoc to paper Conjecture 5.18 is assumed true: covers for the fine Zariski topology are faithful (Assumption 7.3).
    Explicitly assumed at the start of Section 7.1 ('From now on we assume this conjecture to be true, since it is verified in all examples we know'); it is needed for the fine-spectrum part of Theorem 7.13 and for the descendability statements on Spec^NC_fine, but not for the central faithfulness theorem (7.20).
invented entities (3)
  • Noncommutative spectrum Spec^NC(A) and its fine variant Spec^NC_fine(A) independent evidence
    purpose: A sober topological space canonically associated with every connective dg-algebra, replacing the Grothendieck prime spectrum for noncommutative rings and giving a notion of point for noncommutative spaces.
    The spectrum is explicitly computed for fields, DVRs, Z, k^n, kA_2, matrix algebras, and the Kronecker algebra, and it admits canonical comparison maps to the Grothendieck spectrum for commutative rings (Propositions 5.10, 5.24); these are falsifiable external checks.
  • Formal homotopy Zariski topology on dAff_Z independent evidence
    purpose: A Grothendieck topology whose open immersions are homotopical epimorphisms and whose covers are conservative families of algebra base change functors; it is the substrate that makes the spectrum nonempty, functorial, and defined for all rings.
    It restricts to the classical Zariski topology on finite-presentation localizations ([60]) and to flat epimorphisms in the Noetherian commutative case ([2]), and it fails to restrict for larger classes (Example 5.1), so its behavior is pinned down by external benchmarks rather than by the paper's conclusions.
  • Descendable pre-sheaves and descendable pre-ringed spaces independent evidence
    purpose: A generalized gluing condition replacing the sheaf condition: sections are reconstructed via comonadic descent even when the structure presheaf is not a sheaf, as happens for noncommutative rings (Example 7.1).
    Reduces to ordinary (homotopical) sheaves for commutative rings (Proposition 7.28) and is verified explicitly for the quiver algebra kA_2 (Example 7.2), giving an external handle on the new notion.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Noncommutative Gelfand Duality: the algebraic case." pith.science (2026). https://pith.science/paper/T2WDL6IC

@misc{pith2026241111816,
  author       = {Pith},
  title        = {Pith review of: Noncommutative Gelfand Duality: the algebraic case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2WDL6IC}},
  note         = {Machine review of arXiv:2411.11816}
}
read the original abstract

The goal of this paper is to define a notion of non-commutative Gelfand duality. Using techniques from derived algebraic geometry, we show that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, inspired by Grothendieck's work on commutative rings. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others. This work aims to represent the starting point for a rigorous study of geometric properties of quantum spacetimes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 60 canonical work pages

  1. [40]

    Lurie, Higher topos theory , Princeton University Press, (2009)

    J. Lurie, Higher topos theory , Princeton University Press, (2009)

    work page 2009

  2. [1]

    Ad´ amek, J

    J. Ad´ amek, J. Rosick´ y, E. M. Vitale,Algebraic theories: a categorical introduction to general algebra, Cambridge Tracts in Mathematics 184, Cambridge University Press, (2010)

    work page 2010

  3. [2]

    Angeleri H¨ ugel, F

    L. Angeleri H¨ ugel, F. Marks, J. ˘Stovi˘ cek, R. Takahashi, J. Vit´ oria,Flat ring epimorphisms and universal localizations of commutative rings , Q. J. Math. 71(4), 1489–1520, (2020)

    work page 2020

  4. [3]

    Artin, W

    M. Artin, W. Schelter, Integral ring homomorphisms , Adv. Math. 39(3), 289–329, (1981)

    work page 1981

  5. [4]

    Bahns, S

    D. Bahns, S. Waldmann, Locally noncommutative Space-Times , Rev. Math. Phys. 19, 273– 305, (2007)

    work page 2007

  6. [5]

    Balmer, Descent in triangulated categories , Math

    P. Balmer, Descent in triangulated categories , Math. Ann. 353(1), 109–125, (2012)

    work page 2012

  7. [6]

    Bambozzi, O

    F. Bambozzi, O. Ben-Bassat, Dagger geometry as Banach algebraic geometry , J. Number Theory 162, 391–462, (2016)

    work page 2016

  8. [7]

    Bambozzi, O

    F. Bambozzi, O. Ben-Bassat, K. Kremnizer, Stein domains in Banach algebraic geometry , J. Funct. Anal. 274(7), 1865–1927, (2018)

    work page 2018

Show all 62 references
  1. [8]

    Bambozzi, O

    F. Bambozzi, O. Ben-Bassat, K. Kremnizer, Analytic geometry over F1 and the Fargues- Fontaine curve, Adv. Math. 356, 106815, (2019)

    2019

  2. [9]

    Bambozzi, K

    F. Bambozzi, K. Kremnizer, On the sheafyness property of spectra of Banach rings , J. London Math. Soc. 109(1), e12855, (2024)

    2024

  3. [10]

    Bambozzi, T

    F. Bambozzi, T. Mihara, Homotopy epimorphisms and derived Tate’s acyclicity for co mmu- tative C*-algebras, Quart. J. Math. 74(2), 421–458, (2023)

    2023

  4. [11]

    Bambozzi, T

    F. Bambozzi, T. Mihara, Derived Analytic Geometry for Z-Valued Functions Part I: Top o- logical Properties, Bull. Iranian Math. Soc. 50(4), 58, (2024)

    2024

  5. [12]

    Bambozzi, S

    F. Bambozzi, S. Murro, On the uniqueness of invariant states , Adv. Math. 376, 107445, (2021)

    2021

  6. [13]

    Bambozzi, S

    F. Bambozzi, S. Murro, N. Pinamonti, Invariant states on noncommutative tori , Int. Math. Res. Not. 2021(5), 3299–3313, (2021)

    2021

  7. [14]

    Bazzoni, Cotilting modules and homological ring epimorphisms , J

    S. Bazzoni, Cotilting modules and homological ring epimorphisms , J. Algebra 441, 552–581, (2015)

    2015

  8. [15]

    Bazzoni, and J

    S. Bazzoni, and J. Stovicek, Smashing localizations of rings of weak global dimension at most one, Adv. Math. 305, 351–401, (2017)

    2017

  9. [16]

    Bellissard, The noncommutative geometry of aperiodic solids , In: A

    J. Bellissard, The noncommutative geometry of aperiodic solids , In: A. Cardona, S. Paycha, H. Ocampo (eds.), Geometric and topological methods for quantum field theory , World Sci. Publ. 86–156 (2003)

    2003

  10. [17]

    Bellissard, A

    J. Bellissard, A. van Elst, H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect , J. Math. Phys. 35(10), 5373–5451, (1994)

    1994

  11. [18]

    Beilinson, Coherent sheaves on Pn and problems of linear algebra , Funct

    A.A. Beilinson, Coherent sheaves on Pn and problems of linear algebra , Funct. Anal. Appl. 12(3), 214–216, (1978)

    1978

  12. [19]

    Ben-Bassat, J

    O. Ben-Bassat, J. Kelly, K. Kremnizer, A perspective on the foundations of derived analytic geometry, preprint arXiv:2405.07936, (2024). 66

    2024 arXiv

  13. [20]

    Braun, J

    C. Braun, J. Chuang, A. Lazarev, Derived localisation of algebras and modules , Adv. Math. 328, 555–622, (2018)

    2018

  14. [21]

    Buchholz, G

    D. Buchholz, G. Lechner, S. J. Summers, Warped convolutions, Rieffel deformations and the construction of quantum field theories , Comm. Math. Phys. 304, 95–123, (2011)

    2011

  15. [22]

    P. M. Cohn, The affine scheme of a general ring, Applications of sheaves , Lecture Notes in Mathematics 753, Springer Berlin, 197–211, (1979)

    1979

  16. [23]

    Chuang, A

    J. Chuang, A. Lazarev, Homological epimorphisms, homotopy epimorphisms and acyc lic maps, Forum Math. 32(6), 1395–1406, (2020)

    2020

  17. [24]

    Connes, Noncommutative geometry and reality, J

    A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36(11), 6194–6231, (1995)

    1995

  18. [25]

    Connes, On the spectral characterization of manifolds , J

    A. Connes, On the spectral characterization of manifolds , J. Noncomm. Geom. 7, 1–82, (2013)

    2013

  19. [26]

    De Nittis, H

    G. De Nittis, H. Schulz-Baldes, The non-commutative topology of two-dimensional dirty superconductors, J. Geom. Phys. 124, 100–123, (2018)

    2018

  20. [27]

    Doplicher, K

    S. Doplicher, K. Fredenhagen, and J.E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields , Comm. Math. Phys. 172, 187–220, (1995)

    1995

  21. [28]

    Dugger, A primer on homotopy colimits , preprint available at https://pages.uoregon.edu/ddugger/hocolim.pdf

    D. Dugger, A primer on homotopy colimits , preprint available at https://pages.uoregon.edu/ddugger/hocolim.pdf

  22. [29]

    Franco, Temporal Lorentzian spectral triples, Rev

    N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26, 1430007, (2014)

    2014

  23. [30]

    Fr¨ ob, A

    M.B. Fr¨ ob, A. Much, K. Papadopoulos, Noncommutative geometry from perturbative quan- tum gravity , Phys. Rev. D 107, 064041, (2023)

    2023

  24. [31]

    Gelfand, Normierte Ringe

    I. Gelfand, Normierte Ringe. Mat. Sb. Nov. Ser. 9, 3–23, (1951)

    1951

  25. [32]

    Goldman, Rings and modules of quotients , J

    O. Goldman, Rings and modules of quotients , J. Algebra 13, 10–47, (1969)

    1969

  26. [33]

    Gratz, G

    S. Gratz, G. Stevenson Approximating triangulated categories by spaces , Adv. Math. 425, 109073, (2023)

    2023

  27. [34]

    Grosse, G

    H. Grosse, G. Lechner, Wedge-local quantum fields and noncommutative Minkowski sp ace, J. High Energy Phys. 11, (2007)

    2007

  28. [35]

    Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathe- matical Society, (2007)

    M. Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathe- matical Society, (2007)

    2007

  29. [36]

    Johnstone, Stone spaces, Cambridge University Press, (1982)

    P.T. Johnstone, Stone spaces, Cambridge University Press, (1982)

    1982

  30. [37]

    Krause, Smashing subcategories and the telescope conjecture–an al gebraic approach, In- vent

    H. Krause, Smashing subcategories and the telescope conjecture–an al gebraic approach, In- vent. Math. 139, 99–133, (2000)

    2000

  31. [38]

    Krause, G

    H. Krause, G. Stevenson, The derived category of the projective line . In: M. Baake, F. G¨ otze, and W. Hoffmann (eds.),Spectral Structures and Topological Methods in Mathematic s, 275–297, (2019)

    2019

  32. [39]

    Lawvere, Functorial Semantics of Algebraic Theories , Repr

    W. Lawvere, Functorial Semantics of Algebraic Theories , Repr. Theory Appl. Categ. 5, 1–121, (2004)

    2004

  33. [41]

    Lurie Higher algebra, Available at https://people.math.harvard.edu/~lurie/papers/HA.pdf 67

    J. Lurie Higher algebra, Available at https://people.math.harvard.edu/~lurie/papers/HA.pdf 67

  34. [42]

    Mac Lane, Categories for the working mathematician , Springer Science & Business Media 5, (2013)

    S. Mac Lane, Categories for the working mathematician , Springer Science & Business Media 5, (2013)

    2013

  35. [43]

    Mac Lane, I

    S. Mac Lane, I. Moerdijk, Sheaves in geometry and logic: A first introduction to topos t heory, Springer Science & Business Media, (2012)

    2012

  36. [44]

    Mathew, The Galois group of a stable homotopy theory , Adv

    A. Mathew, The Galois group of a stable homotopy theory , Adv. Math. 291, 403–541, (2016)

    2016

  37. [45]

    Navarro Gonz´ alez, J.B

    J.A. Navarro Gonz´ alez, J.B. Sancho de Salas, C ∞-Differentiable Spaces , Lecture Notes in Mathematics 1824, Springer, (2003)

    2003

  38. [46]

    Neeman, M

    A. Neeman, M. B¨ okstedt, The chromatic tower for D(R), Topology 31(3), 519–532, (1992)

    1992

  39. [47]

    Nicolas, M

    P. Nicolas, M. Saorin. Parametrizing recollement data for triangulated categori es, J. Algebra 322(4), 1220–1250, (2009)

    2009

  40. [48]

    Paschke, R

    M. Paschke, R. Verch, Local covariant quantum field theory over spectral geometries, Classical Quantum Gravity 21, 5299–5316, (2004)

    2004

  41. [49]

    Raksit, Hochschild homology and the derived de Rham complex revisit ed, preprint arXiv:2007.02576, (2020)

    A. Raksit, Hochschild homology and the derived de Rham complex revisit ed, preprint arXiv:2007.02576, (2020)

    2020

  42. [50]

    M. L. Reyes, Obstructing extensions of the functor spec to noncommutati ve rings , Israel J. Math. 192, 667–698, (2012)

    2012

  43. [51]

    C. Rezk, S. Schwede, B. Shipley, Simplicial structures on model categories and functors , Amer. J. Math. 123(3), 551–575, (2001)

    2001

  44. [52]

    Rosenberg, The left spectrum, the Levitzki radical, and noncommutative schemes, Proc

    A.L. Rosenberg, The left spectrum, the Levitzki radical, and noncommutative schemes, Proc. Natl. Acad. Sci. USA 87, 8583–8586, (1990)

    1990

  45. [53]

    Scholze, Condensed mathematics , Lecture notes based on joint work with D

    P. Scholze, Condensed mathematics , Lecture notes based on joint work with D. Clausen. Available at https://people.mpim-bonn.mpg.de/scholze/papers.html #Notes (2019)

    2019

  46. [54]

    Schwede, B

    S. Schwede, B. Shipley, Equivalences of monoidal model categories , Algebr. Geom. Topol. 3(1), 287–334, (2003)

    2003

  47. [55]

    Schwede, B

    S. Schwede, B. Shipley, Stable model categories are categories of modules , Topology 42(1), 103–153, (2003)

    2003

  48. [56]

    Silver, Noncommutative localizations and applications , J

    L. Silver, Noncommutative localizations and applications , J. Algebra 7(1), 44–76, (1967)

    1967

  49. [57]

    The stacks project , https://stacks.math.columbia.edu/

  50. [58]

    To¨ en,Champs affines , Selecta Math

    B. To¨ en,Champs affines , Selecta Math. (N.S.) 12, 39–134, (2006)

    2006

  51. [59]

    To¨ en, G

    B. To¨ en, G. Vezzosi,Homotopical Algebraic Geometry II: Geometric Stacks and Appl ications, Mem. Amer. Math. Soc. 193, (2008)

    2008

  52. [60]

    Brave New

    B. To¨ en, G. Vezzosi, “Brave New” Algebraic Geometry and global derived moduli sp aces of ring spectra, in: H. R. Miller, D. C. Ravenel (eds.), Elliptic Cohomology: Geometry, Appli- cations, and Higher Chromatic Analogues , London Mathematical Society Lecture Note Series 342,...

    2007

  53. [61]

    C. A. Weibel, An introduction to homological algebra , Cambridge studies in advanced math- ematics 38, Cambridge University Press, (1994)

    1994

  54. [62]

    White, Model structures on commutative monoids in general model ca tegories, J

    D. White, Model structures on commutative monoids in general model ca tegories, J. Pure Appl. Algebra 221(12), 3124–3168, (2017). 68

    2017

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.