Perfect Unipotent Spectra

• Perfect unipotent spectra are defined as the stabilization of Toën’s category of perfect affine stacks over a field, encoding Frobenius-invariant unipotent homotopy groups.
• They provide a framework for reformulating Artin–Mazur formal groups and syntomic cohomology, lifting classical arithmetic duality into a derived setting.
• A key recognition theorem identifies bounded-below perfect unipotent spectra with modules over the skew-polynomial Laurent algebra kₛ[F,F⁻¹], enabling concrete algebraic computations.

Searching arXiv for the specified paper and closely related work to ground the article in current literature. Perfect unipotent spectra are the perfect-field variant of unipotent spectra, a construction defined as the stabilization of Toën’s category of affine stacks. They provide unipotent stable homotopy groups and unipotent homology, viewed as invariants for schemes valued in unipotent group schemes, and they serve as the ambient category in which Artin–Mazur formal groups, syntomic cohomology, and Milne-style arithmetic duality can be reformulated without the vanishing assumptions classically imposed in parts of the subject. In the setting of a perfect field of characteristic $p>0$, the resulting category $Sp^{U,\mathrm{perf}}_k$ has a bounded-below $t$-structure with heart the abelian category of perfect commutative unipotent group schemes, admits a recognition theorem via modules over $k_{\sigma}[F,F^{-1}]$, and supports involutive dualities for bounded quasi-finite type $F_p$- and $\mathbf Z$-module objects (Mondal et al., 7 Oct 2025).

1. Unipotent spectra as stabilized affine stacks

Let $A$ be a commutative ring, and write $AffSt_A$ for Toën’s $\infty$-category of pointed affine stacks over $A$. The category $Sp^{U,\mathrm{perf}}_k$0 carries the loop endofunctor $Sp^{U,\mathrm{perf}}_k$1 sending $Sp^{U,\mathrm{perf}}_k$2. The stable $Sp^{U,\mathrm{perf}}_k$3-category of unipotent spectra over $Sp^{U,\mathrm{perf}}_k$4 is defined by

$Sp^{U,\mathrm{perf}}_k$5

and equivalently

$Sp^{U,\mathrm{perf}}_k$6

the $Sp^{U,\mathrm{perf}}_k$7-category of spectrum-objects in pointed affine stacks (Mondal et al., 7 Oct 2025).

Concretely, an object $Sp^{U,\mathrm{perf}}_k$8 is a sequence $Sp^{U,\mathrm{perf}}_k$9 with $t$0 and each $t$1 an affine stack. The construction comes equipped with a left adjoint

$t$2

and a right adjoint

$t$3

When $t$4 is a field, the bounded-below part $t$5 admits a natural $t$6-structure whose heart is the abelian category of commutative unipotent affine $t$7-group schemes. For $t$8, the objects $t$9 are the unipotent stable homotopy group schemes, and they lie in that heart for $k_{\sigma}[F,F^{-1}]$0. Moreover, $k_{\sigma}[F,F^{-1}]$1 if and only if all $k_{\sigma}[F,F^{-1}]$2. This positions unipotent spectra as a stable enlargement of affine-stack geometry in which ordinary unipotent group schemes become homotopical coefficients rather than merely discrete coefficients.

Two basic examples illustrate the scope of the construction. First, for any spectrum $k_{\sigma}[F,F^{-1}]$3, one defines its unipotent completion $k_{\sigma}[F,F^{-1}]$4, adjoint to the inclusion into $k_{\sigma}[F,F^{-1}]$5. Second, if $k_{\sigma}[F,F^{-1}]$6 is a commutative unipotent group scheme, then its Eilenberg–Mac Lane affine stacks $k_{\sigma}[F,F^{-1}]$7 form a unipotent spectrum, still denoted $k_{\sigma}[F,F^{-1}]$8. These examples show that the theory interpolates between ordinary stable homotopy theory and affine-group geometry.

2. Passage to the perfect setting

The qualifier “perfect” refers to the Frobenius-stable geometry available over a perfect field $k_{\sigma}[F,F^{-1}]$9 of characteristic $F_p$0. A derived $F_p$1-algebra $F_p$2 is called perfect if its Frobenius $F_p$3 is an equivalence. Writing $F_p$4 for the full subcategory of perfect coconnective derived $F_p$5-algebras and $F_p$6 for their opposite via $F_p$7, one defines the stabilization

$F_p$8

Its objects are spectrum-objects all of whose levels are perfect affine stacks, that is, Frobenius-isomorphic affine stacks (Mondal et al., 7 Oct 2025).

This definition isolates the Frobenius-invariant portion of unipotent stable geometry. The bounded-below part of $F_p$9 again carries a $\mathbf Z$0-structure, but now its heart is the abelian category of perfect commutative unipotent group schemes over $\mathbf Z$1. In this sense, perfect unipotent spectra are not merely unipotent spectra over a perfect field; they are spectra internal to the Frobenius-completed geometry of affine stacks.

A central conceptual consequence is that perfect commutative unipotent group schemes appear as the discrete objects of the theory, while higher perfect unipotent spectra encode derived extensions, delooped group objects, and duality operations not visible at the level of ordinary affine groups alone. This suggests a derived framework in which Frobenius perfection and unipotent stabilization are treated simultaneously rather than sequentially.

3. Recognition, coefficients, and homotopical structure

The perfect theory admits a recognition theorem that identifies bounded-below perfect unipotent spectra with a full subcategory of modules over a specific endomorphism algebra. Writing

$\mathbf Z$2

the functor

$\mathbf Z$3

exhibits $\mathbf Z$4 as a full subcategory of $\mathbf Z$5-$\mathbf Z$6. An analogous statement holds for perfect unipotent $\mathbf Z$7- or $\mathbf Z$8-modules (Mondal et al., 7 Oct 2025).

This recognition result supplies a concrete algebraic model for a category defined abstractly as a stabilization. In particular, it shows that the perfect unipotent theory is accessible through derived module categories over the skew-polynomial Laurent algebra $\mathbf Z$9. A plausible implication is that computations in perfect unipotent spectra can often be reduced to algebraic calculations after passage through the $A$0-functor.

The homotopical structure of the category remains controlled by its $A$1-structure. Since the heart consists of perfect commutative unipotent group schemes, the homotopy objects $A$2 function as the primary algebraic invariants of a bounded-below perfect unipotent spectrum. Their role parallels that of homotopy groups in ordinary stable homotopy theory, but with values in perfect unipotent group schemes rather than abelian groups. This is the precise sense in which unipotent stable homotopy groups are new invariants for schemes.

4. Quasi-finite type and involutive dualities

Within perfect affine group schemes, a perfect affine group scheme $A$3 over $A$4 is quasi-finite type if

$A$5

for some finite-type affine group $A$6. Equivalently, $A$7 is a cocompact object in perfect affine groups. One then writes

$A$8

for the full subcategory of quasi-finite type perfect unipotent spectra, meaning those $A$9 for which all $AffSt_A$0 are perfect quasi-finite type unipotent groups. This is a stable subcategory and is closed under extensions (Mondal et al., 7 Oct 2025).

The quasi-finite type condition is the natural finiteness hypothesis for duality. In $AffSt_A$1-$AffSt_A$2, perfect quasi-finite type bounded unipotent $AffSt_A$3-modules admit an involutive linear duality

$AffSt_A$4

This extends Milne’s duality on perfect unipotent groups. On composition factors one checks that

$AffSt_A$5

and from this one deduces that every $AffSt_A$6 built from these is dualizable, with double dual equal to $AffSt_A$7.

Likewise, quasi-finite type perfect unipotent $AffSt_A$8-modules which are bounded admit an involutive duality

$AffSt_A$9

These dualities are significant because they generalize arithmetic duality from the level of individual perfect unipotent groups to bounded derived objects in a stable $\infty$0-category. The extension is not merely formal: it is precisely the mechanism used to lift classical pairings in syntomic cohomology to equivalences of perfect unipotent spectra.

5. Syntomic cohomology and refinement of Poincaré duality

Let $\infty$1 be proper lci of dimension $\infty$2. For each integer $\infty$3 and each $\infty$4, the functor on perfect $\infty$5-schemes

$\infty$6

is representable by a perfect unipotent $\infty$7-module spectrum

$\infty$8

and it is quasi-finite type whenever $\infty$9 is smooth. Similarly, there is a $A$0-complete object $A$1 (Mondal et al., 7 Oct 2025).

This representability statement upgrades syntomic cohomology from a complex of $A$2- or $A$3-modules to a geometric object in the perfect unipotent stable category. The refinement is structural rather than merely notational: the coefficient theory now lives in a category carrying its own $A$4-structure, duality, and unipotent homotopy groups.

When $A$5 is smooth proper of dimension $A$6, the Milne/Poincaré pairing on mod $A$7 syntomic cohomology lifts to an equivalence of perfect unipotent $A$8-module spectra

$A$9

There are parallel equivalences for $Sp^{U,\mathrm{perf}}_k$00- and $Sp^{U,\mathrm{perf}}_k$01-complete coefficients: $Sp^{U,\mathrm{perf}}_k$02

$Sp^{U,\mathrm{perf}}_k$03

In particular, syntomic cohomology is promoted from a complex of $Sp^{U,\mathrm{perf}}_k$04- or $Sp^{U,\mathrm{perf}}_k$05-modules to a perfect unipotent spectrum with built-in arithmetic duality. This is the point at which the theory connects directly to the paper’s other stated applications: recovery of Artin–Mazur formal groups without vanishing assumptions, and extension of Milne’s arithmetic duality theorems to the category of perfect unipotent spectra.

6. Relation to neighboring uses of “unipotent” and terminological scope

The phrase “perfect unipotent spectra” has a specific meaning in the affine-stack stabilization of $Sp^{U,\mathrm{perf}}_k$06, and it should be distinguished from other uses of “unipotent” in recent literature. In the representation-theoretic setting of $Sp^{U,\mathrm{perf}}_k$07-adic $Sp^{U,\mathrm{perf}}_k$08, the “unipotent spectrum” is the full subcategory of the stable derived category of smooth $Sp^{U,\mathrm{perf}}_k$09-representations supported on inertial parameters of unipotent type; its compact, or “perfect,” objects coincide with $Sp^{U,\mathrm{perf}}_k$10, and this category is identified with perfect dg modules over a dg Schur algebra (Berry, 2024). Here, “perfect” refers to perfect complexes over a dg algebra, not to Frobenius-perfect affine stacks.

A different use appears in tensor triangular geometry for finite-dimensional unipotent Hopf algebras. There, the thick subcategory

$Sp^{U,\mathrm{perf}}_k$11

is described as “unipotent” because it is generated as a thick category by its unit, and its Balmer spectrum is compared with $Sp^{U,\mathrm{perf}}_k$12 and conjecturally with $Sp^{U,\mathrm{perf}}_k$13 (Solberg et al., 13 Nov 2025). In that setting, “unipotent” is a condition on the tensor-triangular generation behavior of a monoidal triangulated category, not a synonym for stabilization of affine stacks.

These comparisons clarify a common source of ambiguity. The perfect unipotent spectra of $Sp^{U,\mathrm{perf}}_k$14 belong to derived algebraic geometry and arithmetic geometry; they are constructed from perfect affine stacks and analyzed through Frobenius perfection, $Sp^{U,\mathrm{perf}}_k$15-structures, and Milne-style duality. The representation-theoretic and tensor-triangular usages are mathematically substantive, but they concern different categories, different notions of support, and different finiteness conditions. This suggests that the adjective “unipotent” is stable across several active research areas, whereas the noun “spectrum” is highly context-dependent.