Almost Absolutely Pure Modules

• Almost absolutely pure modules are defined as R–modules for which Ext^1_R(N, E) is almost zero for every finitely presented R–module, reflecting a weakened purity concept.
• They preserve key homological properties through almost isomorphisms and facilitate dualities, notably with almost flat modules, thereby extending classical FP-injective notions.
• These modules underpin the structure of almost coherent rings and play a crucial role in applications from perfectoid geometry to relative algebraic geometry.

An almost absolutely pure module is an $R$–module $E$ for which, given a commutative ring $R$ with an idempotent ideal of definition $\m\subset R$ ($\m^2 = \m$), $\Ext^1_R(N,E)$ is almost zero for every finitely presented $N$. The concept arises within almost mathematics—a framework originally motivated by perfectoid geometry—where purity and finiteness conditions are systematically weakened up to annihilation by the ideal $\m$. Almost absolutely pure modules allow homological methods akin to classical absolutely pure (FP-injective) modules to persist, but reflect the approximate nature of the ambient algebraic objects.

1. The Almost Category and Foundational Definitions

Let $R$ be a commutative ring with a fixed ideal of definition $\m$ such that $\m^2 = \m$. An $R$–module $M$ is almost zero if $\m M = 0$. The subcategory $\Sigma_R$ of almost zero modules is Serre, and the category of almost $R$–modules is the abelian quotient $\Mod^a_R = \Mod\mbox{-}R/\Sigma_R$. Morphisms in this category—almost isomorphisms—are those whose kernel and cokernel are almost zero. Concepts such as tensor products, $\Hom$, and $\Ext$ all inherit almost analogues, where, for example, $\Ext^1_R(N,M)$ is almost zero if annihilated by $\m$.

An $R$–module $E$ is almost absolutely pure if, for every finitely presented $R$–module $N$, $\Ext^1_R(N,E)$ is almost zero.

2. Characterizations of Almost Absolutely Pure Modules

Several equivalent formulations establish the nature of almost absolutely pure modules (Theorem 6.1 (Zhang, 17 Jan 2026)):

1. Ext-vanishing: $E$ is almost absolutely pure if $\Ext^1_R(N,E)$ is almost zero for all finitely presented $N$.
2. Almost Purity of Extensions: Every short exact sequence $0\to E \to B \to C \to 0$ is almost pure—remains pure modulo almost zero modules, or equivalently, is exact under tensoring with any module up to almost isomorphism.
3. Embedding Purity: If $E$ embeds into an injective module $I$, the submodule $E\subset I$ is almost pure.
4. Injective Envelope: The embedding $E\to E^+$ into the injective envelope is almost pure.
5. Lifting Property: For a finitely generated projective $P$, every finitely generated $K\subset P$ and map $f:K\to E$, and each $s\in\m$, there is a lift $g_s:P\to E$ such that $g_s\circ i = s\cdot f$.

The equivalence of these arises via diagram-chasing, tensor–Hom adjunctions, and the preservation of almost exactness under functorial image.

3. Relationship to Almost Flat and Almost Coherent Modules

There is a duality between almost absolutely pure and almost flat modules mirroring the classical Hom–tensor relationship. If $F$ is an $R$–module and $E$ an injective cogenerator, $F$ is almost flat (that is, $F^a$ is flat over $R^a$) if and only if, for every almost absolutely pure $E$, $\Hom_R(F,E)$ is almost absolutely pure. This follows from the isomorphism

$\Ext^1_R(N,\Hom_R(F,E))\cong \Hom_R(\Tor_1^R(N,F),E)$

and the cogenerating property of $E$: $\Tor_1^R(N,F)$ is killed by $\m$ for all finitely presented $N$ if and only if the above Ext group is almost zero. The vanishing of $\m\Tor_1$ characterizes almost flatness.

Almost absolutely pure modules also feature in the homological structure of almost coherent rings: $R$ is almost coherent if and only if, for all almost absolutely pure $E$ and injective $I$, $\Hom_R(E,I)$ is almost flat, or equivalently, for almost flat $F$ and injectives $I_1, I_2$, the iterated Hom $\Hom_R(\Hom_R(F,I_1),I_2)$ is almost flat.

4. Closure Properties and Examples

Several closure properties hold:

Property	Almost Absolutely Pure Modules	Reference
Direct sums/products	Closed	(Zhang, 17 Jan 2026)
Almost pure submodules	Closed	(Zhang, 17 Jan 2026)
Absolutely pure modules/injective modules	All are almost absolutely pure	(Zhang, 17 Jan 2026)
Almost pure quotients (under almost coherence)	Closed	(Zhang, 17 Jan 2026)

Closure under direct sums and products is due to the compatibility of $\Ext^1$ with colimits and products (modulo almost isomorphism for finitely presented $N$). Any almost pure submodule of an almost absolutely pure module remains almost absolutely pure. Furthermore, every absolutely pure module is almost absolutely pure, as are all injective modules.

Nontrivial examples—modules that are almost absolutely pure without being absolutely pure—arise over rings $R$ that are almost coherent but not coherent in the classical sense.

5. Homological Criteria and Categorical Significance

Almost absolutely pure modules are central in the categorical and homological analyses of almost coherent rings. Several equivalent homological criteria for almost coherence rely on them:

• $R$ is almost coherent if and only if every almost pure quotient of an almost absolutely pure module is almost absolutely pure; equivalently, every direct limit or direct product of absolutely pure modules is almost absolutely pure; or every $R$–module admits an almost absolutely pure (pre)cover.

Existence of almost absolutely pure covers implies almost coherence, as finitely generated ideals become almost-presented.

6. Role in Perfectoid Geometry and Algebraic Applications

Almost absolutely pure modules underpin key dualities and approximation phenomena in almost ring theory, particularly in settings originating from perfectoid geometry. In applications such as Scholze's proof of finiteness of $\mathbf{F}_p$-cohomology for proper rigid spaces, certain cohomology groups are shown to be almost coherent, inheriting the almost purity properties that enable the deduction of classical finiteness results. Accordingly, almost absolutely pure modules provide indispensable algebraic infrastructure for the study of homological approximation in both arithmetic geometry and relative algebraic geometry contexts (Zhang, 17 Jan 2026).