nLab pure morphism

In category theory

Definition

Given a regular cardinal $\kappa$, a morphism $f \colon A \longrightarrow B$ in a category $C$ is $\kappa$-pure (or $\kappa$-universally injective) if for every commutative square

$$\array{ P & \overset{g} {\longrightarrow} & Q \\ \mathllap{^u} \big\downarrow && \big\downarrow \mathrlap{^v} \\ A & \underset{f} {\longrightarrow} & B }$$

in which $P$ and $Q$ are $\kappa$-presentable objects, the morphism $u \colon P\to A$ factors through $g$, i.e. there is some $h: Q\to A$ with $u = h\circ g$.

Remark

Def. does not require that also the morphism $v$ is factored, hence it does not express a lifting property.

Proposition

• In a $\kappa$-accessible category $C$ every $\kappa$-pure morphism is a monomorphism, hence exhibits a pure subobject.

• In a locally $\kappa$-presentable category $\kappa$-pure morphisms are, moreover, regular monomorphisms,

• and in fact coincide with the $\kappa$-directed colimits of split monomorphisms in the arrow category $C^2 = Arr(C)$;

• more generally this characterization holds in all $\kappa$-accessible categories admitting pushouts.

(Adámek, Hub & Tholen 1996).

In ring theory and algebraic geometry

Definition

Given a unital, possibly commutative, ring ring $R$, a homomorphism $f \colon M\to M'$ of left $R$-modules is pure in the sense of Def. if tensoring the exact sequence of left $R$-modules

$$0 \to Ker f \longrightarrow M \overset{f} {\longrightarrow} M' \longrightarrow Coker f \to 0$$

with any right $R$-module $N$ (from the left) yields a exact sequence of abelian groups.

Remark

Beware that in algebraic geometry the category-theoretic terminology of Def. clashes with other terminology in a couple of ways:

1. What are called pure morphisms of schemes (Raynaud & Gruson 1971) in algebraic geometry are not pure morphisms, according to Def. , in the category of schemes.

   Instead, a “pure morphism of schemes” $f \colon X \longrightarrow Y$ is taken to be one whose corresponding morphism of structure sheaves $\mathcal{O}_Y \longrightarrow f_\ast \mathcal{O}_X$ is a pure morphism, in the sense of Def. , in the category of quasicoherent $\mathcal{O}_Y$-modules.

2. What are called universally injective morphisms of schemes (or radicial morphisms (Grothendieck 1960, Stacks Project 01S2) are also not pure morphisms in the category of schemes according to Def. .

   Instead, a radicial morphism is one whose underlying continuous map remains injective after any base change.

Remark

Grothendieck has proved that faithfully flat morphisms of commutative schemes are of effective descent for the categories of quasicoherent $\mathcal{O}$-modules. But this was not entirely optimal, as there is in fact a more general class than faithfully flat morphisms which satisfy the effective descent. For a local case of commutative rings, Joyal & Tierney (1984) have then proved (unpublished) that the effective descent morphisms for modules are precisely the pure morphisms of rings (or dually of affine schemes) according to Def .

This result can be extracted also from their Memoirs volume on Galois theory. Janelidze & Tholen (2004) have reproved this theorem as a corollary of a result for noncommutative rings obtained using Beck’s comonadicity theorem.

Related concepts

• pure subobject

References

Descent along pure morphisms

The following paper, published as a CRAS note, was the first with the result on that pure morphisms are effective descent:

• Jean-Pierre Olivier, Descente par morphismes purs, C. R. Acad. Sci. Paris Sér. A-B 271 (1970) A821–A823 scan on Gallica

The result is proved (within a larger context)

• André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309 (1984) vol. 51, pages vii+71

Clean proofs are by Mesablishvili and by Janelidze,

• Bachuki Mesablishvili, Pure morphisms of commutative rings are effective descent morphisms for modules – a new proof, Theory and Appl. of Categories 7, 2000, No. 3, 38-42, tac

• Bachuki Mesablishvili, Pure morphisms are effective for modules, Applied Categorical Structures 21 (2013), 801–809. arXiv, doi.

• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge Univ. Press 2003; errata pdf

• George Janelidze, Walter Tholen, Facets of descent III: monadic descent for rings and algebras, Appl. Categ. Structures 12 5-6 (2004) 461-477 [MR2005i:13019, doi]

More recent attention to descent along pure morphisms:

• Yves André, Luisa Fiorot, On the canonical, fpqc, and finite topologies on affine schemes. The state of the art, in Ann. Sc. Norm. Super. Pisa Cl. Sci. arXiv:1912.04957

• Yves André, ‘’On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)’’, IAS Number Theory Seminar, April 29, 2021. Seminar slides (PDF).

Ziegler spectrum and connections to model theory

The Ziegler spectrum of indecomposable pure injectives has been introduced in

• M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984) 149 – 213.

A textbook account:

• Mike Prest, Purity, spectra and localisation, Enc. Math. Appl. 121, Camb. Univ. Press 2011, 798 pages; publishers book page

The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is studied in

• G. Garkusha, M. Prest, Triangulated categories and the Ziegler spectrum, Algebras & Repres. Theory 8, 499–523 (2005) doi

Other articles:

• M. Prest, P. Rothmaler, M. Ziegler, Absolutely pure and flat modules and “indiscrete” rings, J. Alg. 174:2 (1995) 349-372 doi

• Christian U. Jensen, Helmut Lenzing, Model theoretic algebra: with particular emphasis on fields, rings, modules, Algebra, Logic and Applications 2, Gordon and Breach 1989.

• I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74:3 (1997) 503-558 doi

Other category theoretic articles

• Ivo Herzog, Pure-injective envelopes, Journal of Algebra and Its Applications 2(4) (2003), 397-402 pdf

• Jiří Adámek, H. Hub, Walter Tholen, On pure morphisms in accessible categories, J. Pure Appl. Alg. 107, 1 (1996), pp 1-8, doi

• Michel Hébert, Purity and injectivity in accessible categories, doi

• W.W. Crawley-Boevey, Locally finitely presented additive categories, Communications in Algebra 22(5)(1994), 1641-1674.

• Rosanna Laking, Purity in compactly generated derivators and t-structures with Grothendieck hearts, Math. Zeitschrift, doi (2019)

• Leonid Positselski: On pure monomorphisms and pure epimorphisms in accessible categories, Theory and applications of categories 45 7 (2026) 246–288 [tac:45-07]

Morphisms of schemes

The notion of “universally injective” or “radicial” morphisms of schemes:

• Alexander Grothendieck: Éléments de géométrie algébrique: I. Le langage des schémas, Publ. Math. IHÉS 4 (1960) 5–228 [doi:10.1007/BF02684778]

• The Stacks project authors: Tag 01S2: Radicial and universally injective morphisms [tag/01S2]

The notion of “pure morphisms of schemes” as used in algebraic geometry:

• Michel Raynaud, Laurent Gruson: Critères de platitude et de projectivité, Invent Math 13 (1971) 1–89 [