Geometrically Pure-Injective Objects

• Geometrically pure-injective objects are defined via local purity tests in Grothendieck, tensor-triangulated, and derived categories, extending classical pure-injectivity by requiring purity after localization.
• Their characterization leverages idempotent and tensor methods to establish splitting properties and existence of pure-injective envelopes, applying tools like the coherator functor and pushforwards.
• Applications include the study of smashing ideals, spatial Ziegler spectra, and connections to absolute purity and Noetherian geometry, enhancing insights in modern homological algebra.

Geometrically pure-injective objects are those objects in Grothendieck categories, tensor-triangulated categories (tt-categories), and derived categories, which generalize the conventional notion of pure-injectivity by testing purity or injectivity locally (at stalks, tensor-primes, or affine patches). This concept refines ordinary purity by incorporating a geometric viewpoint: exact triangles or sequences are required to be pure after all suitable localizations. Recent developments establish comprehensive characterizations, existence theorems for envelopes, closedness properties, and connections to model-theoretic spectra and smashing ideal frames (Gómez et al., 28 Jan 2026, Enochs et al., 2013).

1. Definition and Local Criteria

In rigidly-compactly generated tt-categories $\mathcal T$, with subcategory of compact (dualizable) objects $\mathcal T^c$, geometric purity for objects and exact triangles is formulated via finite localizations at primes $\mathcal{P} \in \mathrm{Spc}(\mathcal{T}^c)$. The tt-stalk at $\mathcal{P}$ is the localized category $$\mathcal{T}_{\mathcal{P}} = \mathcal{T}/\mathrm{loc}(\mathcal{P})$$, and the geometric localization functor $\iota_{\mathcal{P}}^*$ is exact and preserves coproducts. An exact triangle

$x \to y \to z \to \Sigma x$

in $\mathcal{T}$ is geometrically pure (g-pure) if for all $\mathcal{P}$, the triangle

$$\iota_{\mathcal{P}}^*(x) \to \iota_{\mathcal{P}}^*(y) \to \iota_{\mathcal{P}}^*(z) \to \Sigma \iota_{\mathcal{P}}^*(x)$$

is pure in $\mathcal{T}_{\mathcal{P}}$. For Grothendieck categories (e.g., $O_X$-modules or $\mathrm{Qcoh}(X)$), the stalkwise criterion [(Enochs et al., 2013), Prop. 1.2] requires, for a short exact sequence $$0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$$, that the sequence induced at each stalk is pure in the appropriate local module category.

A quasi-coherent sheaf $\mathcal{E} \in \mathrm{Qcoh}(X)$ is geometrically pure-injective if, for every geometrically pure sequence, the induced sequence of $\mathrm{Hom}$ functors is exact or, equivalently, every pure monomorphism $\mathcal{E} \to \mathcal{M}$ splits [(Enochs et al., 2013), Def. 2.1].

2. Idempotent and Tensor Characterizations

The theory leverages idempotent objects $\epsilon_{(Y)}$ and $\varphi_{(Y)}$ associated to Thomason subsets $Y \subset \mathrm{Spc}(\mathcal{T}^c)$, satisfying relations such as $\varphi_{(Y)} \otimes \epsilon_{(Y)} \simeq 0$ and $$\epsilon_{(Y)} \otimes \epsilon_{(Y)} \simeq \epsilon_{(Y)}$$ (Gómez et al., 28 Jan 2026). For a prime $\mathcal{P}$, geometric purity can be equivalently tested by tensoring with $\varphi_{\mathcal{P}}$: a map $f : x \to y$ is g-pure if $f \otimes \varphi_{\mathcal{P}}$ is pure for all $\mathcal{P}$. In this framework, geometric-pure-injectivity is characterized by the splitting property for every g-pure monomorphism. This facilitates analysis via pushforwards and localizations.

3. Envelopes, Closure, and Structural Properties

A major result is the existence of pure-injective envelopes: for every $\mathcal{F} \in \mathrm{Qcoh}(X)$, there exists a pure monomorphism $$\eta_{\mathcal{F}} : \mathcal{F} \to \mathrm{PE}(\mathcal{F})$$ where $\mathrm{PE}(\mathcal{F})$ is geometrically pure-injective [(Enochs et al., 2013), Thm. 4.10]. The construction uses character modules and the coherator functor to transfer injectivity and purity from module categories to sheaves. The class of pure-injectives is closed under pure subobjects, direct products, and direct summands, but not under arbitrary sums or limits.

In tt-categories, every pure-injective is a retract of a g-pure-injective object, and g-pure-injectivity strictly strengthens ordinary pure-injectivity except in the local case (i.e. when $\mathrm{Spc}(\mathcal{T}^c)$ is a single point).

4. Structure of Indecomposable Geometrically Pure-Injectives

For $D(\mathrm{Qcoh}(\mathbb{P}^1))$, the indecomposable g-pure-injective objects are precisely the images of the pushforward functor from pure-injectives in tt-stalks. Specifically, given $x$ indecomposable g-pinj in $\mathcal{T}$, there exists unique $\mathcal{P}$ and pure-injective $y \in \mathcal{T}_{\mathcal{P}}$ such that $x \simeq \iota_{\mathcal{P}_+}(y)$ [(Gómez et al., 28 Jan 2026), Thm. 4.1]. In the context of quasi-coherent sheaves on $\mathbb{P}^1$, the indecomposable g-pure-injectives are the torsion sheaves, Prüfer sheaves, adic completions, and the constant sheaf at the generic point. Line bundles $\mathcal{O}(n)$ are not g-pure-injective, as they sit in non-split g-pure exact triangles [(Gómez et al., 28 Jan 2026), Prop. 5.7].

5. Spectra and Model-Theoretic Frameworks

The Ziegler spectrum $\mathrm{Zg}(\mathcal{T})$ consists of isomorphism classes of indecomposable pure-injectives with closed sets determined by intersections with definable subcategories. The geometric Ziegler spectrum $\mathrm{GZg}(\mathcal{T})$ is the subspace corresponding to indecomposable g-pure-injective objects [(Gómez et al., 28 Jan 2026), §5]. Closed subsets take the form $\mathrm{Def} \cap \mathrm{gpinj}(\mathcal{T})$, for definable tt-ideals. For a finite quasi-compact open cover $\{U_i\}$ of $\mathrm{Spc}(\mathcal{T}^c)$, the map

$$\bigsqcup_i \mathrm{GZg}(\mathcal{T}(U_i)) \to \mathrm{GZg}(\mathcal{T})$$

is a topological quotient. When $\mathrm{GZg}(\mathcal{T}(U_i))$ is closed in $\mathrm{Zg}(\mathcal{T}(U_i))$ for all $i$, $\mathrm{GZg}(\mathcal{T})$ is closed in $\mathrm{Zg}(\mathcal{T})$.

6. Applications to Smashing Ideals and Spatiality

Smashing $\otimes$-ideals form the frame $\mathrm{Sm}_\otimes(\mathcal{T})$, which is order-reversing bijective to the poset of definable tt-ideals (Gómez et al., 28 Jan 2026). A frame is spatial if it is the lattice of open sets of some topological space. Prior attempts to realize spatiality via the full Ziegler spectrum encountered counterexamples due to excess points [Balchin–Stevenson]. Restricting to the geometric subspace $\mathrm{GZg}(\mathcal{T})$ resolves these issues. If the local geometric Ziegler spectrum ttZg$(\mathcal{T}_\mathcal{P})$ realizes the definable tt-ideals, the global tt-closed sets define a genuine topology on gpinj$(\mathcal{T})$, yielding a spatial locale dual to $\mathrm{Sm}_\otimes(\mathcal{T})$ [(Gómez et al., 28 Jan 2026), Thm. 6.4].

7. Connections to Absolute Purity and Noetherian Geometry

Absolute purity is formulated as the property that a sheaf $\mathcal{F}$ is a pure subobject of every sheaf containing it, or equivalently, that $\mathrm{Ext}^1_A(F, M) = 0$ for every finitely presented $F$ (Enochs et al., 2013). Locally absolutely pure sheaves, tested on affine opens, agree with global absolute purity on locally coherent schemes. On Noetherian schemes, injectivity and absolute purity coincide.

The equivalence

$$X \text{ locally Noetherian} \iff \text{every locally absolutely pure sheaf in } \mathrm{Qcoh}(X) \text{ is locally injective}$$

provides a purely homological criterion for Noetherianness of closed subschemes of projective space [(Enochs et al., 2013), Prop. 5.11]. This suggests deep interplay between geometric purity, injective envelopes, and foundational aspects of algebraic geometry.

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Principal References:

• "Geometric purity and the frame of smashing ideals" (Gómez et al., 28 Jan 2026)
• "Pure injective and absolutely pure sheaves" (Enochs et al., 2013)