Formal Superscheme in Supergeometry

• Formal superscheme is a locally ringed superspace defined by a topological space and a sheaf of complete, Z₂-graded supercommutative k-algebras, generalizing classical formal schemes.
• The theory organizes morphisms and local structures through functor-of-points, profinite superalgebras, and faithfully flat descent, ensuring compatibility of the superstructure.
• Applications include deformation theory and moduli problems, with key insights into superdimension theory incorporating both even and odd components.

A formal superscheme over a field $k$ of characteristic not equal to 2 is a geometric object generalizing classical formal schemes by incorporating supersymmetry via $\mathbb{Z}_2$-graded algebraic structures and linearly topologized supercommutative $k$-algebras. The theory organizes these spaces and their morphisms through equivalent frameworks involving locally ringed superspaces, functorial constructions, profinite topological superalgebras, and super-coalgebraic techniques. Central results include descent theory in terms of faithfully flat morphisms and a superdimension theory with even and odd components, capturing the distinct algebraic and topological subtleties arising in the supersymmetric context (Saenz et al., 10 Nov 2025).

1. Definition and Local Structure

A formal superscheme over $k$ is specified by a pair $(|X|,\O_X)$, where $|X|$ is a topological space and $\O_X$ is a sheaf of complete linearly topologized supercommutative $k$-algebras. Locally, $|X|$ admits open charts $U\cong\Spf A$ with $A=A_{\bar0}\oplus A_{\bar1}$ a topological $k$-superalgebra, topologized via an ideal of definition $I\subset A$ such that $A\cong\varprojlim_nA/I^n$. The underlying topological space is defined as the set of prime superideals containing $I$, equipped with the Zariski topology from the system $\{A/I^n\}$. The structure sheaf on basic opens $D(f)\subset\Spf A$, for $f\in A^h$, consists of the inverse limit $\O_X(D(f)) = \varprojlim_n (A/I^n)[f^{-1}]$. Each local chart can be recovered as the inductive limit $U = \varinjlim_n\Spec(A/I^{n+1})$ functorially on finite-dimensional test superalgebras.

Admissible thickenings globally structure the formal superscheme as an inductive system $X_0\hook X_1\hook\cdots$ of closed immersions, with each $X_n$ an ordinary affine superscheme of finite type, and $X=\varinjlim X_n$. The transition maps correspond to thickenings defined by nilpotent ideals, reflecting the formal geometric nature of the theory (Saenz et al., 10 Nov 2025).

2. Underlying Topology and Sheaf Properties

The geometric realization of a formal super-functor $X$ is constructed by expressing $X$ as an inductive limit $\varinjlim_\lambda\SSpf A_\lambda$ of finite affine superschemes, yielding the underlying topological space as $|X| = \varprojlim_\lambda |\SSpf A_\lambda| = \varprojlim_\lambda \Spec(\bar A_\lambda)$, with $\bar A_\lambda = A_\lambda/(A_\lambda^2)$. The structure sheaf is assembled as the inverse limit $\O_X = \varprojlim_n\O_{X_n}$, ensuring completeness and compatibility with the $I$-adic topology at each open $D(f)$. This construction realizes the formal superscheme as a locally ringed superspace equipped with a sheaf of complete (linearly topologized) superalgebras (Saenz et al., 10 Nov 2025).

3. Functor‐of‐Points and Profinite Superalgebras

The functor-of-points perspective treats local models $\Spf(A)$ as functors $\PSSpf A: SAlgf\to\text{Sets}$, mapping a finite-dimensional $k$-superalgebra $R$ to the set of continuous homomorphisms $\Hom_{PSAlg}(A,R)$, with $R$ discrete. Profinite topological $k$-superalgebras $A = \varprojlim_n A_n$ correspond contravariantly to formal superschemes via the anti-equivalence $\PSSpf(-)$ (Theorem 3.16 in (Saenz et al., 10 Nov 2025)). For $I$-adic profinite $A=\varprojlim A/I^n$, the associated formal super-functor is the colimit $\PSSpf A = \varinjlim \SSpf(A/I^n)$. This viewpoint allows a direct translation between topological superalgebraic data and the functorial geometry of formal superschemes.

4. Morphisms, Flat Descent, and Fiber Dimensions

Morphisms between formal superschemes admit several equivalent descriptions: as maps of locally ringed superspaces, as natural transformations of the corresponding functors, as continuous maps of profinite topological superalgebras (or dually, super-coalgebra maps), or as compatible families of maps between thickenings. Key classes of morphisms—closed/open immersions, (faithfully) flat morphisms—are defined by algebraic or coalgebraic properties: flatness is characterized by exactness of pullback on supercomodules, while faithful flatness adds surjectivity on topological spaces.

The faithfully flat descent theorem extends Takeuchi's descent for formal schemes to the super-setting: for a faithfully flat morphism $f:X\to Y$, the Čech–Hopf complex

$$X\times_Y X\;\rightrightarrows\;X\;\xrightarrow{f}\;Y$$

forms an equalizer in the category of formal superschemes. In the coalgebraic language, this corresponds to an exact sequence involving the bar-cotensor complex of supercomodules, central to the descent formalism (Saenz et al., 10 Nov 2025).

The fiber-dimension theorem for locally algebraic formal superschemes relates the (even, odd) superdimension at a point $x\in|X|$ to that at $y=f(x)\in|Y|$ and the fiber over $y$. For Noetherian local superrings $R, S$ associated to $y, x$, and $D=O_{f^{-1}(y),x}$,

• Even part: $$sdim_{x,\bar0}X\leq sdim_{y,\bar0}Y + sdim_{x,\bar0}f^{-1}(y)$$, with equality when $f$ is flat at $x$.
• Odd part (under mild regularity): $$sdim_{x,\bar1}X\geq sdim_{y,\bar1}Y + sdim_{x,\bar1}f^{-1}(y)$$.

These statements generalize classical fiber dimension lemmas to the super-setting, incorporating the Masuoka–Zubkov theory of odd Krull dimension (Saenz et al., 10 Nov 2025).

5. Illustrative Example: Formal Neighborhood in Affine Superspace

Consider $A = k[x_1,\dots,x_m\mid\theta_1,\dots,\theta_n]$, the coordinate superalgebra of affine superspace $\mathbb{A}^{m|n}$. With maximal ideal $$\mathfrak{m} = (x_1,\dots,x_m,\theta_1,\dots,\theta_n)$$ and the $\mathfrak{m}$-adic topology, the formal neighborhood of the origin is described by

$\Spf\hat{A},\quad \hat{A} = \varprojlim_{r\geq 1}A/\mathfrak{m}^r \cong k\llbracket x_1,\dots,x_m\mid\theta_1,\dots,\theta_n\rrbracket,$

where $$\hat A_{\bar 0}=k \llbracket x_1,\dots,x_m \rrbracket$$ and $$\hat A_{\bar 1}=k\llbracket x_1,\dots,x_m\rrbracket\cdot\langle\theta_1,\dots,\theta_n\rangle$$. The underlying topological space is a single point; the local ring captures the structure of a local formal superscheme of dimension $(m|n)$. This provides the prototypical example illustrating the general formalism (Saenz et al., 10 Nov 2025).

6. Generalizations, Applications, and Further Directions

Formal superschemes generalize classical formal schemes along three axes:

1. All rings/promorphisms are replaced by super-rings and super-maps with Koszul sign rules.
2. Categorical constructions (tensor products, Hom, cotensor) observe super-symmetry conventions.
3. Dimension theory gains an odd component via maximal odd parameter systems.

Research directions include the deformation theory of superschemes, moduli problems in the supersymmetric regime, formal supergroups and their quotients (as in Takahashi–Masuoka), the paper of étale, smooth, and unramified morphisms via supermodules of differentials, and the analysis of “constant” or “étale” formal superschemes (generalizing Tate spaces). Notably, the Krull superdimension can be developed coalgebraically, paralleling Sweedler’s coradical theory in the super context. The foundational treatment by Felipe Saenz and Joel Torres del Valle gives a unified algebraic-supergeometric framework for further exploration in supersymmetric geometry and arithmetic (Saenz et al., 10 Nov 2025).