The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull Dimension

Harry Gulliver

The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull Dimension

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Pith's one-line read The paper claims that specialisation in the injective spectrum of a right noetherian ring is governed by critical dimension: if E specialises to F, then cd(E) ≥ cd(F), with equality exactly when E and F are isomorphic.

desk verdict A solid paper on injective spectra whose main claims hold up; the only sketched compactness lemma is not actually load-bearing, and the Heisenberg and quantum-plane examples are the real payoff. read the letter →

arxiv 1908.05876 v1 pith:5SLFXCNM submitted 2019-08-16 math.RA math.AGmath.CT

Harry Gulliver This is my paper · ORCID

classification math.RAmath.AGmath.CT MSC 16D5016P6018E10

keywords injectivespectrumindecomposablemodulesZieglertopologyKrulldimensioncriticalrightnoetherianringsnoncommutativespecialisation

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 develops the injective spectrum of a right noetherian ring: the topological space of indecomposable injective right modules, which for commutative noetherian rings reproduces the Zariski spectrum. The main claim is that specialisation in this space is governed by a numeric invariant, the critical dimension of an injective module: if E specialises to F then cd(E) ≥ cd(F), and equality holds exactly when E and F are isomorphic. From this it follows that the spectrum is T0 and that every chain of specialisations has length at most the Krull dimension of a noetherian generator. The paper also proves functoriality for certain ring maps and computes the spectrum for artinian rings, 1-critical rings, and the enveloping algebra of the Heisenberg algebra, while the quantum plane shows that the space can be non-noetherian with closed points that contain no simple submodule.

What carries the argument

The load-bearing mechanism is critical dimension, cd(E), defined as the minimum Krull dimension of a non-zero subobject of E; for an indecomposable injective it is well-defined and equals the common Krull dimension of all critical subobjects. Lemma 3.4 converts a topological specialisation $E\leadsto F$ into the ordinal inequality cd(E) ≥ cd(F), with equality forcing E ≅ F, and this is what turns chains of specialisations into bounded ordinal chains. A second key ingredient is the compactness of basic open sets in the Ziegler topology (Lemma 1.2), which lets the paper pass from an infinite union of critical subquotients to a finite subcover when proving that every irreducible basic closed set is given by a single critical module.

What would settle it

Exhibit a locally noetherian Grothendieck category, for instance the module category of a right noetherian ring, with indecomposable injectives $E,F$ such that $E\leadsto F$ but $\mathrm{cd}(E)<\mathrm{cd}(F)$, or $E\leadsto F$ with $\mathrm{cd}(E)=\mathrm{cd}(F)$ and $E\not\cong F$; either would refute Lemma 3.4. Alternatively, exhibit a finitely presented object $A$ for which the basic open set $(A)$ is not compact in the Ziegler topology, contradicting Lemma 1.2 and removing the support for Lemma 3.11.

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Extended reading notes

Core claim

The central discovery is that the topology of the injective spectrum is a faithful, dimension-sensitive invariant: specialisation is monotone with respect to critical dimension, and no two distinct points can specialise to each other. Concretely, for indecomposable injectives E and F over a locally noetherian Grothendieck category, $E\leadsto F$ exactly when every non-zero map from a finitely presented object to E yields one to F, equivalently when E embeds into a product of copies of F. In that situation, cd(E) ≥ cd(F), with equality precisely when E and F are isomorphic; here cd is the critical dimension, the minimum Krull dimension of a non-zero subobject of the injective. Specialisation chains therefore become strictly descending chains of ordinals, bounding their length by the Krull dimension of a noetherian generator, and the spectrum is T0. The paper uses this to show that right noetherian domains have irreducible spectra with generic point the injective hull of the right regular module, and it computes full spectra for artinian rings, for rings whose regular module is 1-critical, and for the Heisenberg enveloping algebra, while the quantum plane provides a counterexample to the naive picture.

Load-bearing premise

The load-bearing premise is Lemma 1.2, whose proof is only sketched: the basic open sets $(A)$ for finitely presented $A$ must be compact open in the Ziegler topology, and Lemma 3.11 needs this compactness to reduce an infinite union of critical subquotients to a finite subcover.

Editorial extensions

If this is right

For a right artinian ring, the injective spectrum is finite and discrete, matching Krull dimension 0.
For a right noetherian ring whose regular module is 1-critical, the spectrum consists of one generic point together with the closed points of simple modules, and every non-empty open set contains the generic point; this covers 1-dimensional noetherian domains.
Every specialisation chain in the spectrum has length at most d, where d is the Krull dimension of a noetherian generator; when the spectrum is sober this bounds its topological dimension by d.
A right noetherian domain has an irreducible spectrum whose generic point is the injective hull of $R_R$, and $R_R$ is critical.
For the Heisenberg enveloping algebra, the spectrum decomposes into closed fibres parameterised by the central character, with lines over each non-zero value, an affine plane at zero, and a line of generic points whose closures specialise across the fibres.

Reading between the lines

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

If specialisation is as rigid as Lemma 3.4 says, the injective spectrum can serve as a dimension-theoretic invariant for noncommutative rings, with Krull dimension recoverable from the longest specialisation chain; a natural next test is whether the monotonicity survives in locally coherent categories, where the basic open sets may fail to be compact.
The Heisenberg computation suggests a geometric picture of a base line of central characters with a line of generic points over it; one testable extension is whether analogous rational-parameter families of simple modules in other enveloping algebras produce the same pattern of closures crossing fibres.
The quantum-plane example shows that the converse to the closed-point lemma fails exactly when the injective spectrum is not noetherian; a possible repair would be to replace ordinary irreducibility by a dimension-relative notion and ask whether a generic-point theorem returns.

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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

Read Gulliver's injective spectrum paper. Short version: the central dimension-specialisation story is real, and the main results survive scrutiny. The one lemma everyone will worry about, Lemma 1.2, is only sketched, but it does not carry the weight of the paper. I checked the dependence: Lemma 3.4 uses only Lemma 2.1, Lemma 3.2, and Proposition 3.1. The compactness of basic opens enters only in Lemma 3.11, which is not used later. So even if that Ziegler-spectrum compactness proof is left as a sketch, the core of Sections 3 and 4 stands.

What's genuinely new: the specialisation criterion in Lemma 2.1, the monotonicity of critical dimension along specialisation with equality iff isomorphism (Lemma 3.4), and the finiteness of indecomposable injectives of maximal critical dimension (Theorem 3.14). The examples are the best part. The Heisenberg enveloping algebra gets a detailed picture: fibres over k, the plane at z=0, and a line of generics specialising across fibres. The quantum plane counterexamples are not decorative; they kill plausible conjectures, showing a closed point of critical dimension 1 with no simple submodule and a critical module whose hull is not generic in its own closed set. That is useful, concrete information.

Soft spots, in proportion. Lemma 1.2 is under-proved, but as noted it is not needed for the central claims. Still, because the paper is foundational in spirit, the author should either give a full proof or explicitly flag Lemma 3.11 as dispensable. There is a small wrinkle in Lemma 2.1(3): the proof invokes a single noetherian generator, which need not exist in an arbitrary locally noetherian category. The argument repairs itself by taking a noetherian object from a generating set, and for the main setting of right noetherian rings, R_R itself is a generator, so no real harm. The quantum plane section is intricate and not machine-checked; I did not find an error, but a referee should go through those annihilator computations line by line.

Citation pattern is clean. Pappacena's weak Zariski topology is explicitly distinguished, and the self-citation to the sequel is harmless. This is a solid PhD-derived paper. It is for noncommutative geometers and anyone who works with injective spectra or Krull dimension in module categories. It deserves a serious referee and, with attention to Lemma 1.2 and the quantum plane details, publication.

My recommendation: send it to peer review, and ask for a complete proof or explicit de-emphasis of Lemma 1.2, plus a careful check of Section 4.4.

Referee Report

0 major / 5 minor

Summary. This paper develops basic topological properties of the injective spectrum of a right noetherian ring and, more generally, of a locally noetherian Grothendieck category. The spectrum is the set of indecomposable injectives with the dual-Ziegler topology. The main results are: a specialisation criterion (Lemma 2.1), functoriality for flat epimorphisms and central quotients (Corollary 2.7 and Theorem 2.9), a link between specialisation and Gabriel-Rentschler critical dimension (Lemma 3.4), the T0 property and a bound on specialisation chains by the Krull dimension of a noetherian generator (Corollaries 3.5 and 3.6), and finiteness of points of maximal dimension (Theorem 3.14). The final sections give detailed computations for artinian rings, 1-critical rings, the Heisenberg enveloping algebra, and the quantum plane, the last providing counterexamples to natural conjectures.

Significance. The central contribution is the clean statement that specialisation in the injective spectrum is controlled by critical dimension: a non-trivial specialisation strictly lowers critical dimension, which yields T0-ness and a sharp upper bound on chains. If the claims hold, the injective spectrum becomes a useful dimension-sensitive invariant for noncommutative noetherian rings. The paper is explicit about the limits of the theory; the quantum plane example is a valuable falsification of the expectation that closed points are injective hulls of simples and that critical modules give irreducible basic closed sets. The proofs of the main dimension results are largely self-contained, and the examples are detailed, with the Heisenberg case giving a nearly complete picture of the topology. I found no load-bearing mathematical error, but several local proof details require attention.

minor comments (5)

[§2.1, Lemma 2.1] In the proof of (2⇒3), the phrase 'there is a noetherian generating object G' should be read carefully: for a general locally noetherian category there need not be a single noetherian generator. The argument is valid if G is taken from a generating set of noetherian objects, so please rephrase to remove the ambiguity. In (3⇒4), 'fix a generator G' is acceptable because a Grothendieck category has a generator, but the text should not suggest that this generator is noetherian.
[§1.2, Lemma 1.2] The proof of Lemma 1.2 is only a sketch. Since Lemma 3.11 depends on the compactness of the basic open sets, please provide a complete proof or a precise chain of references rather than a sketch; as written this is an incompletely proved stated lemma, although it is not used in the main dimension results.
[§3.2, Theorem 3.15] In the reduction to cyclic modules, the sentence 'if (mR,E)=0 for all E... then (M,E)=0' is not justified and is false in general. The needed claim [mR]≠∅ follows directly from the chosen φ with φ(m)≠0 and the fact that E(RR) is an indecomposable injective, so this passage should be corrected.
[§4.4, Quantum plane] The strictness of the descending chain of closed sets (Mλ^(n)) is asserted but not demonstrated. It follows, for instance, because E(Mλ^(n)) lies in (Mλ^(n)) but not in (Mλ^(n+1)); please add a sentence making this explicit.
[General editorial] There are several typos and uncited references: the abstract has 'G rothendieck' and 'functoriali ty'; §4.3 refers to 'Lemma 2.9' where 'Theorem 2.9' is meant; and the bibliography contains entries that never appear in the text, including [2], [5], [8], [10], [13], [14], [19], [22], [23], [25], and [26]. Please cite or remove them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central specialisation-versus-critical-dimension theorem is derived from the definitions and standard lemmas, not assumed.

full rationale

I walked the claimed derivation chain for the paper's central claim, Lemma 3.4, which asserts that if E specialises to F then cd(E) is at least cd(F), with equality iff E is isomorphic to F. The proof is self-contained from the definitions: it takes a cd(E)-critical subobject C of E, uses Lemma 2.1 to obtain a non-zero map C to F, and then argues through criticality and Lemma 3.2. If the map is not an embedding, its image is a proper quotient of the critical object C, so its Krull dimension is strictly less than cd(E), forcing cd(F) < cd(E); if it is an embedding, then F = E(C) = E. This is a genuine derivation, not a restatement of an input. Corollary 3.5 (T0-ness) and Corollary 3.6 (bounding specialisation chains by K(G)+1) follow directly from Lemma 3.4 and Proposition 3.1(3), with no fitted parameters or hidden equivalences. The only self-citation, the sequel paper [11], is about sheaves and torsion theories and is explicitly not used in this paper's results. The compactness lemma (Lemma 1.2) is sketched with references to external results of Ziegler, Prest, and Prest--Rajani; even if one were concerned about the sketch, Lemma 1.2 is used only in Lemma 3.11, which is not used in the subsequent central claims of the paper, so it is not load-bearing for the main specialisation/dimension results. I found no step where a prediction is equivalent by construction to an input, no parameter fitted to a subset and then renamed as a prediction, and no load-bearing argument that reduces to a self-citation. The paper is an honest derivation from standard notions of injective spectrum, specialisation, and Krull dimension.

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

No fitted parameters and no new postulated entities. The paper relies on standard results in module theory, Krull dimension, and Ziegler spectra; the only non-self-contained input is the sketched Lemma 1.2.

assumptions (7)

standard math Matlis bijection between prime ideals and indecomposable injective modules for commutative noetherian rings.
Cited as Theorem 1.1 and used to justify that the injective spectrum extends the Zariski spectrum.
domain assumption Ziegler spectrum compactness: the basic open sets (A) for A finitely presented are compact open, and the injective spectrum is a closed subset of the full Ziegler spectrum.
Stated as Lemma 1.2 with only a sketch proof; references [28], [21], [20]. Used in Lemma 3.11.
standard math Eklof-Sabbagh criterion: an R-module is injective iff every consistent system of linear equations has a solution.
Theorem 1.3, imported from model theory; used in the Heisenberg algebra example to produce elements with prescribed annihilators.
domain assumption Goldie's theorem: a right noetherian domain is right Ore and has uniform dimension one.
Used in Theorem 3.15 to show E(R_R) is an indecomposable injective and is generic.
standard math Artin-Wedderburn theorem: a semisimple artinian ring is a finite product of matrix rings over division rings.
Used in Proposition 4.1 to show a right artinian ring has finitely many simple modules.
standard math Krull dimension facts for Grothendieck categories (existence for noetherian objects, critical subobjects, uniformity of critical objects) from [17].
Collected in Proposition 3.1 and used throughout Section 3.
standard math Block's classification of simple modules over the first Weyl algebra.
Cited in the remark after Theorem 4.3 to compare cardinalities of spectra.

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Pith. "Pith review of The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull Dimension." pith.science (2026). https://pith.science/paper/5SLFXCNM

@misc{pith2026190805876,
  author       = {Pith},
  title        = {Pith review of: The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull Dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SLFXCNM}},
  note         = {Machine review of arXiv:1908.05876}
}

abstract

The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the sense of Gabriel and Rentschler). Finally, we use these results to compute a number of examples.

Figures

Figures reproduced from arXiv: 1908.05876 by the authors.

**Figure 1.** There are further points in the line of generics, with different closures cutting across the fibres. For instance, given any rational function f(z)/g(z) ∈ k(z) (in lowest terms), the right ideal (p − f(z)/g(z))HS = (g(z)p − f(z))HS is maximal in HS, hence there is a point Ef /g := E(H/(g(z)p − f(z))H) in the line of generics. For any α such that g(α) 6= 0, we can take the ∩-irreducible right ideal (g(α)p − f(α))Hα o… view at source ↗

**Figure 1.** The injective spectrum of the Heisenberg algebra, shown [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 5 canonical work pages

[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

1981
[2]

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

2007
[3]

Eckmann and A

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

1953
[4]

Eklof and G

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

1971
[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. 27

work page 1974
[6]

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

1962
[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

1967
[8]

J. Golan. Topologies on the Torsion-Theoretic Spectrum of a Noncommu tative Ring . Paciﬁc Journal of Mathematics, 51(2): pp. 439-450, 1974

1974

Show all 28 references

[9]

Gordon and J.C

R. Gordon and J.C. Robson. Krull Dimension . American Mathematical Society Mem- oir 133, 1973

1973
[10]

Grothendieck and J

A. Grothendieck and J. Dieudonn´ e. ´El´ ements de G´ eometrie Alg´ ebrique: I. Le Langage des Sch´ emas. Publications Math´ ematiques de l’I.H.E.S., tome 4, 1960

1960
[11]

Gulliver

H. Gulliver. The Injective Spectrum of a Right Noetherian Ring II: Sheave s and Tor- sion Theories
[12]

Hochster

M. Hochster. Prime Ideal Structure in Commutative Rings . Transactions of the Amer- ican Mathematical Society, 142: pp. 43-60, 1969

1969
[13]

R. Kanda. Classifying Serre Subcategories via Atom Spectrum . Advances in Mathe- matics, 231(3-4): pp. 1572-1588, 2012

2012
[14]

R. Kanda. Atom Spectra of Grothendieck Categories . Master’s thesis, Nagoya Univer- sity, 2013

2013
[15]

T.Y. Lam. Lectures on Modules and Rings . Springer, 1999

1999
[16]

E. Matlis. Injective Modules over Noetherian Rings . Paciﬁc Journal of Mathematics, 8(3): pp. 511-528, 1958

1958
[17]

McConnell and J.C

J.C. McConnell and J.C. Robson. Noncommutative Noetherian Rings . Chichester Wi- ley, 1987

1987
[18]

Pappacena

C. Pappacena. The Injective Spectrum of a Noncommutative Space. Journal of Algebra, 250(2): pp. 559-602, 2002

2002
[19]

N. Popescu. Abelian Categories with Applications to Rings and Modules . Academic Press, 1973

1973
[20]

M. Prest. Purity, Spectra, and Localisation . Cambridge University Press, 2008

2008
[21]

Prest and R

M. Prest and R. Rajani. Structure Sheaves of Deﬁnable Additive Categories . Journal of Pure and Applied Algebra, 214: pp. 1370-1383, 2010

2010
[22]

Prest and R

M. Prest and R. Wisbauer. Local Finite Presentation of Categories σ[M]. Colloq. Math., 99(2): pp. 189-202, 2004

2004
[23]

R. Rajani. Model-Theoretic Imaginaries and Ziegler Spectra in Genera l Categories . Doctoral Thesis, University of Manchester, 2007

2007
[24]

L. Silver. Noncommutative Localizations and Applications . Journal of Algebra, 7(1): pp. 44-76, 1976. 28

1976
[25]

Stenstr¨ om.Rings of Quotients

B. Stenstr¨ om.Rings of Quotients . Springer-Verlag, 1975

1975
[26]

H. Storrer. On Goldman ’s Primary Decomposition. Lectures on Rings and Modules (Tulane Univ. Ring and Operator Theory Year, 1970-1921, Vol. I) p p. 617-661. Lecture Notes in Mathematics, 246. Springer, 1972

1970
[27]

Wisbauer

R. Wisbauer. Foundations of Module and Ring Theory . Gordon and Breach, 1991

1991
[28]

M. Ziegler. Model Theory of Modules . Ann. Pure Appl. Logic, 26(2): pp. 149-213, 1984. 29

1984

Pith tools

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

[1] [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

1981

[2] [2]

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

2007

[3] [3]

Eckmann and A

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

1953

[4] [4]

Eklof and G

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

1971

[5] [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. 27

work page 1974

[6] [6]

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

1962

[7] [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

1967

[8] [8]

J. Golan. Topologies on the Torsion-Theoretic Spectrum of a Noncommu tative Ring . Paciﬁc Journal of Mathematics, 51(2): pp. 439-450, 1974

1974