The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories

Harry Gulliver

REVIEW 3 major objections 3 minor 28 references

The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories

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Pith's one-line read The injective spectrum and the torsion spectrum of a locally noetherian Grothendieck category are the same topological space.

desk verdict A genuinely useful paper whose central homeomorphism is probably right, but with a false fullness claim and two black-box imports that need opening up before I'd cite it. read the letter →

arxiv 1908.05880 v1 pith:G7T33JDH submitted 2019-08-16 math.CT math.AGmath.RA

Harry Gulliver This is my paper · ORCID

classification math.CTmath.AGmath.RA MSC 18E1516P4016S9018F20

keywords injectivespectrumtorsionGrothendieckcategoriestheoriesstructuresheafZieglertopologylocalizationnoetherianrings

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 establishes that for any locally noetherian Grothendieck category — in particular, the category of modules over a right noetherian ring — the injective spectrum and Golan's torsion spectrum are the same topological space. The identification sends each indecomposable injective object $E$ to the torsionfree class it cogenerates, $F(E)$. Because the two spectra are defined from very different data, the homeomorphism unifies two research threads and lets torsion-theoretic machinery prove topological facts about the injective spectrum, such as sobriety in the Ziegler topology. The paper also builds a sheaf of rings on the injective spectrum and shows that the two natural ways to turn a module into a sheaf agree exactly when every prime torsion theory is perfect. A reader should care because the result suggests that the injective spectrum is the noncommutative analogue of the Zariski spectrum, with prime torsion theories playing the role of prime ideals.

What carries the argument

The load-bearing object is the map $E \mapsto F(E)$, the torsionfree class cogenerated by an indecomposable injective $E$, i.e. the class of objects embedding into some product of copies of $E$. The argument that this map is surjective rests on torsion-critical objects: objects whose every proper quotient is torsion for the torsionfree class they cogenerate. Lemma 1.9, imported from Krull dimension theory, guarantees that every non-zero noetherian object in a Grothendieck category has a critical subobject $B$ such that no proper quotient of $B$ maps to $E(B)$; this produces a torsion-critical subobject inside any indecomposable injective. The sheaf machinery uses torsion-theoretic localisation: to each basic open set $[M]$ one attaches the endomorphism ring of the localisation of the ring at the torsion class generated by $M$, and then sheafifies.

What would settle it

Find a locally noetherian Grothendieck category with two distinct indecomposable injective objects $E$ and $F$ such that $F(E)=F(F)$; the homeomorphism would force $E\cong F$, so a single such example would refute Theorem 3.3. Equivalently, find an indecomposable injective $E$ whose cogenerated torsionfree class is not prime, contradicting Lemma 3.2.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.3: for a locally noetherian Grothendieck category $\mathcal{A}$, the assignment $E \mapsto F(E)$ is a homeomorphism between $\mathrm{InjSpec}(\mathcal{A})$ and $\mathrm{TorSpec}(\mathcal{A})$. Here $\mathrm{InjSpec}(\mathcal{A})$ is the space of indecomposable injective objects with the Zariski topology, and $\mathrm{TorSpec}(\mathcal{A})$ is the space of prime torsion theories with the finitary order topology. The proof shows surjectivity by proving that every prime torsionfree class is cogenerated by a single indecomposable injective, and injectivity by observing that two indecomposable injectives cogenerating the same torsionfree class specialise to each other, which forces equality because the injective spectrum is $T_0$. The homeomorphism is proved by checking that it sends basic open sets to basic open sets. This identification is then used to prove that the Ziegler topology on the injective spectrum is sober, and to characterise prime torsion theories as exactly the join-irreducible torsionfree classes, equivalently the meet-irreducible torsion classes.

Load-bearing premise

The argument depends on the classical fact that every non-zero noetherian object in a Grothendieck category contains a subobject $B$ such that no proper quotient of $B$ has a nonzero map into the injective hull of $B$; if that fact ever failed, the identification of the injective and torsion spectra would collapse.

Editorial extensions

If this is right

Topological questions about the injective spectrum can be translated into lattice-theoretic questions about torsion classes, since the points are exactly the meet-irreducible torsion classes and a basis of open sets is given by the compact elements of that lattice.
Sobriety of the injective spectrum in its Ziegler topology follows directly from the homeomorphism and the lattice characterisation of prime torsion theories.
For a right noetherian domain, the ring of global sections of the structure sheaf on the injective spectrum is precisely the original ring, recovering the commutative expectation for a large class of noncommutative rings.
For a commutative noetherian ring, the injective spectrum homeomorphism recovers the usual Zariski spectrum via the Matlis bijection, and the structure sheaf constructed here is the classical Zariski structure sheaf.
The tensor sheaf and torsion sheaf functors from modules to sheaves on the injective spectrum are naturally isomorphic precisely when every prime torsion class is perfect, which holds for commutative noetherian rings and for right noetherian right hereditary rings.

Reading between the lines

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

A testable extension of the paper's identification is that sobriety of the Zariski topology on the injective spectrum might be provable purely from the lattice of torsion classes: if every irreducible Zariski-closed set corresponds to a join of prime torsionfree classes with a unique maximal element, then the Zariski topology would be sober.
The paper's equivalence suggests that computability transfers between the two spectra: torsion-theoretic invariants such as Gabriel dimension or the lattice-theoretic structure of torsion classes could be read off the injective spectrum, and vice versa.
One could check in explicit noncommutative examples, such as the first Weyl algebra or other right hereditary rings, whether the perfectness condition for all prime torsion classes has a module-theoretic shadow that determines when the tensor and torsion sheaf constructions agree.

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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

Let me give you the short version: this is a real paper, with a genuinely new categorical homeomorphism and two module-sheaf functors that aren't in the earlier literature, but there are three spots a referee should push on. The load-bearing one is Theorem 3.3, InjSpec(A) ≅ TorSpec(A). The proof is short and the idea is right, but it leans on Lemma 1.9, imported from the self-cited prequel, and the compressed argument doesn't show why K(B/C) < K(B) gives Hom(B/C, E(B)) = 0. I think the gap is fillable — standard Krull-dimension exactness plus criticality should do it — but the paper currently just points to [11, 3.1.4 & 3.2.6]. Since Theorem 3.3 carries the rest of Section 3, the referee should ask for a proof or a precise statement of what is being imported.

The sheaf part is the most original. The structure sheaf on InjSpec is Gabriel's, but the tensor sheaf and torsion sheaf functors, the natural transformation between them, and the perfection criterion in Theorem 2.11 are new. The kA2 example showing that global sections can differ from R is clean and useful. Corollary 2.14 has a false line: classical localization at a multiplicative set is not full — try R = k[x], D = {x^n}; the induced map on Hom(R,R) misses x^{-1}. The conclusion is still true, since the quotient category is equivalent to Mod-D^{-1}R via the fully faithful restriction adjunction, so this is a repair, not a collapse.

Theorem 2.11's claim that the stalks are prime localisations is asserted, not proved. It looks right in locally noetherian categories because every torsion theory is finite type and the neighbourhood system is cofinal in the prime torsion class, but that identification needs to be written down. Section 4's spectrality results are honestly conditional, and the isolation of closed sets is a nice application.

Who is this for? Someone working on noncommutative spectra, torsion theories, or representation theory of noetherian rings. It deserves a serious referee: the main structural claims are likely correct and significant, but the paper needs revision on the points above before I would rely on it.

Referee Report

3 major / 3 minor

Summary. The paper studies the injective spectrum of a right noetherian ring and, more generally, of a locally noetherian Grothendieck category. It constructs a structure sheaf of finite-type localisations on the injective spectrum, defines two functors from R-modules to sheaves of modules, and investigates when these functors coincide. The central result is Theorem 3.3, which asserts a homeomorphism between the injective spectrum and Golan's torsion spectrum; from this the paper derives a lattice-theoretic characterisation of prime torsion theories, Ziegler sobriety, and consequences for spectral spaces. The paper is presented as largely independent of the author's prequel [11], with a few results imported as black boxes.

Significance. If the main claims hold, the identification of InjSpec and TorSpec in Theorem 3.3 is significant: it shows that the Zariski-type injective spectrum and Golan's torsion spectrum carry exactly the same topological information, and it gives a purely lattice-theoretic description of the points of the spectrum via Theorem 3.10. The sheaf-theoretic part provides a noncommutative analogue of the Zariski structure sheaf and a useful criterion, Theorem 2.11, for when the tensor and torsion sheaf functors agree. The paper is honest about its dependence on the prequel and explicitly records open questions in Sections 3.3 and 4.1. However, the central proof chain passes through Lemma 1.9 and an unproved stalk identification in Theorem 2.11; these points need to be supplied before the main theorem can be regarded as established.

major comments (3)

[§2.2, Theorem 2.11] The proof asserts without argument that 'the stalks are the localisations at torsionfree classes cogenerated by a single indecomposable injective; i.e., at prime torsion theories.' The stalk at a point E is a colimit of the localisations R_T(A) over all finitely presented A with (A,E)≠0, and it is not automatic that this colimit is the single localisation at T F(E); one must show that the torsion classes T(A) form a cofinal family generating T F(E) and that localisation commutes with the resulting colimit. Moreover, the identification with prime torsion theories is only proved later in Lemma 3.2, so as written the proof of Theorem 2.11 depends on a future result. Please supply the missing stalk computation or reorder the paper so that Theorem 2.11 is proved after Section 3.2.
[§1.3, Lemma 1.9] The proof of Lemma 1.9 is too compressed and leaves a load-bearing gap. From K(B/C)<K(B) the text immediately concludes (B/C,E(B))=0 by citing [11, 3.1.4 & 3.2.6]; this conclusion needs the additional fact that every nonzero submodule of a critical object has the same Krull dimension as the critical object and is itself critical, and that fact is neither stated nor proved in the present paper. In addition, Lemma 1.9 is stated for an arbitrary Grothendieck category, while the cited references [17, §6.2] concern modules over a ring; the extension to Grothendieck categories is not explained. Since Lemma 1.9 is used in Lemma 3.2 to prove the surjectivity half of the homeomorphism in Theorem 3.3, this missing argument directly affects the main theorem.
[§2.1, structure sheaf construction] The sentence 'the associated torsionfree class is F([M]), cogenerated by the indecomposable injectives in [M]' is false as written. By Lemma 1.3, F(M)=F(E(M)), and the indecomposable summands of E(M) are precisely those indecomposable injectives E with (M,E)≠0, i.e., the elements of (M), not of [M]. For M=R over a commutative domain, for instance, [R]={0}, so F([R])=0, which is not the torsionfree class cogenerated by R. The well-definedness argument for T(M) can be repaired by replacing [M] with (M), but the construction of the structure sheaf should be corrected.

minor comments (3)

[§2.2, Example 2.2] The example states that the two indecomposable injectives of kA2 with orientation 1→2 are (k→0) and (k→k), but the indecomposable injectives for this orientation are (k→k) and (0→k); the representation (k→0) is not injective. Consequently the computed stalks should be k and k rather than k and M2(k), and the ring of global sections is k⊕k, not k⊕M2(k). The conclusion of the example, that the global sections are not Morita equivalent to R, still survives with this correction, but the computation needs to be fixed.
[§3.2, Lemma 3.6] There is a typesetting artifact in the proof: 'InjSpec(A)/integerdivideC(F)' should be the set difference InjSpec(A)\C(F). This should be corrected, and the surrounding text searched for similar artifacts.
[Throughout] There are several typographical errors, including 'torison' in Lemma 3.13, 'the the family' in §4.1, and 'Respectively' capitalization in Theorem 4.2. A careful proofreading pass is recommended.

Circularity Check

2 steps flagged · score 4.0 of 10

Central homeomorphism InjSpec ≅ TorSpec is supported by black-box results from the author's own prequel [11], most critically Lemma 1.9's Hom-vanishing claim; no fitted input or definitional equivalence is present, so the circularity is moderate rather than total.

self citation load bearing [Lemma 1.9, used in Lemma 3.2 and Theorem 3.3 (Sections 1.3 and 3.2)]
"Finally, a technical Lemma which does not appear explicitly in [11], but follows from results there concerning Krull dimension and critical dimension. ... For given any proper quotient B/C, we have K(B/C) < K(B), by definition, and so (B/C, E(B)) = 0 by [11, 3.1.4 & 3.2.6]."

The proof that every indecomposable injective cogenerates a prime torsionfree class, which is the surjectivity half of the homeomorphism in Theorem 3.3, needs Lemma 1.9. That lemma is not proved here; its decisive Hom-vanishing statement '(B/C, E(B)) = 0' is deferred to the author's own prequel [11, 3.1.4 & 3.2.6], and the paper explicitly labels such prequel results as 'black box'. Thus the central identification is imported from a self-citation rather than derived in this paper. Unless [11] independently establishes those statements for arbitrary locally noetherian Grothendieck categories, the surjectivity assertion rests on an unverified dependency.
self citation load bearing [Theorem 3.3 (injectivity), using Lemma 1.5 and Proposition 1.6 (Section 1.3)]
"We recall here the relevant definitions and a small number of results from [11], which will be relevant to this paper, and may be viewed as 'black box' results for the reader ... To show injectivity, we must show that two indecomposable injectives which cogenerate the same torsionfree class are isomorphic; but this follows immediately from Lemma 1.5 and Corollary 1.6."

The injectivity half of the homeomorphism uses Lemma 1.5, which identifies the specialisation order with inclusion of torsionfree classes, and the T0 property of InjSpec(A), both recalled from [11] without a self-contained proof in this paper. Consequently, the bijectivity of the central map E ↦ F(E) depends on the same author's prequel at two separate points, not merely on the definitions. This is a load-bearing self-citation, though it is not a definitional reduction.

full rationale

The paper's main equivalence, Theorem 3.3, is not assumed as an input: TorSpec(A) is defined independently via torsion-critical objects, and the proof then attempts to show that E ↦ F(E) is a bijection. The surjectivity direction is proved in Lemma 3.2 using Lemma 1.9, whose key claim that a proper quotient B/C with smaller Krull dimension has no nonzero map to E(B) is not proved here but deferred to [11, 3.1.4 & 3.2.6]. Since [11] is the author's own prequel and the paper explicitly calls such results 'black box', the central identification rests on self-citation. The injectivity direction similarly relies on Lemma 1.5 and Proposition 1.6 from [11]. However, there is no fitted parameter being renamed as a prediction, and no step assumes the target homeomorphism itself; Lemma 1.9 is a substantive ingredient rather than a restatement of the conclusion. If the cited prequel results are independently established for arbitrary locally noetherian Grothendieck categories, the dependency is legitimate. On the face of the present text, the derivation chain contains load-bearing self-citations but stops short of definitional circularity, meriting a moderate score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper rests on standard torsion-theoretic localisation and on black-box results from the author's prequel [11]. There are no fitted parameters and no invented entities; the main unproved inputs are background theorems in torsion theory and the critical-subobject lemma.

assumptions (6)

standard math Gabriel-Popescu localisation: quotient by a Serre subcategory has a universal property, admits a right adjoint when the subcategory is a torsion class, and has the stated localisation formula.
Proposition 1.1 from [19], used throughout Sections 2 and 3 for localisation at torsion classes.
standard math Finite type torsion theories are generated by finitely presented torsion objects, and if a locally finitely presented category is locally noetherian then all torsion theories are of finite type.
Lemma 1.2 and Lemma 3.5 from [20], used to construct the sheaf-on-a-basis and to compare Zariski and Ziegler topologies.
standard math Perfection criteria for torsion classes: exactness of the localisation functor, flat epimorphism, and the Gabriel filter conditions are equivalent.
Theorem 2.10 from [25], used in the statement and proof of Theorem 2.11 and Corollary 2.16.
domain assumption Every non-zero noetherian object in a Grothendieck category has a critical subobject B such that every proper quotient B/C has no nonzero maps to E(B).
Lemma 1.9, attributed to [17] and [11]; not re-proved in detail, and used in Lemma 3.2 to show every indecomposable injective cogenerates a prime torsionfree class.
domain assumption In a locally noetherian Grothendieck category, every indecomposable injective object contains a non-zero noetherian subobject.
Invoked at the start of Lemma 3.2; it is part of what local noetherianity provides.
domain assumption The injective spectrum of a right noetherian domain is irreducible with generic point E(R_R).
Theorem 1.8 from the author's prequel [11], used in Theorem 2.3 to prove that the ring of global sections of the structure sheaf is R.

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Pith. "Pith review of The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories." pith.science (2026). https://pith.science/paper/G7T33JDH

@misc{pith2026190805880,
  author       = {Pith},
  title        = {Pith review of: The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7T33JDH}},
  note         = {Machine review of arXiv:1908.05880}
}

read the original abstract

This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure sheaf and their relation to modules over the original ring. We then explore links with the spectrum of prime torsion theories developed by Golan and use this torsion-theoretic viewpoint to prove further results about the topology.

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Gulliver

H. Gulliver. The Injective Spectrum of a Right Noetherian Ring I: Injecti ve Spectra and Krull Dimension

work page

[2] [1]

R. Block. The Irreducible Representations of the Lie Algebra sl(2) and of the Weyl Algebra. Advances in Mathematics, 39(1): pp. 69-110, 1981

work page 1981

[3] [2]

A. Duca. Injective Modules over a Principal Right and Left Ideal Doma in, with App- plications. PhD. thesis, University of Manitoba, 2007

work page 2007

[4] [3]

Eckmann and A

B. Eckmann and A. Schopf. ¨Uber Injektive Moduln . Archiv der Mathematik, 4(2): pp. 75-78, 1953

work page 1953

[5] [4]

Eklof and G

P. Eklof and G. Sabbagh. Model-Completions and Modules . Annals of Mathematical Logic, 2(3): pp. 251-195, 1971

work page 1971

[6] [5]

E. Fisher. Abelian Structures. Unpublished, 1974/5. Partly published as Abelian Struc- tures I in Abelian Group Theory , Lecture Notes in Mathematics, Vol. 616, Springer- Verlag, 1977

work page 1974

[7] [6]

P. Gabriel. Des Cat´ egories Ab´ eliennes. Bulletin de la S.M.F., tome 90: pp. 323-448, 1962

work page 1962

[8] [7]

Gabriel and R

P. Gabriel and R. Rentschler. Sur la Dimension des Anneaux et Ensembles Ordonn´ es. C.R. Acad. Sci. Paris, tome 265: pp. 712-715, 1967

work page 1967