Semiderived Categories in Homological Algebra

• Semiderived categories are triangulated categories that bridge classical derived and coderived methods, addressing challenges in curved or ind-scheme contexts.
• They employ t-adic filtrations and explicit compact generators to construct well-behaved filtered categories that restore compact generation.
• Applications include resolving issues in deformation theory and semi-infinite algebraic geometry, providing categorical resolutions and enriched monoidal structures.

A semiderived category is a triangulated category that interpolates between classical derived and (co)derived categories, enabling well-behaved homological algebra in contexts where standard approaches are obstructed, such as in the presence of curvature or in categories of quasi-coherent sheaves on non-Noetherian ind-schemes. The construction formalizes the idea, first developed by Positselski, of combining “half” derived (colimit/co) and “half” coderived (limit/contra) methods, and it plays a crucial role in the study of curved or infinitesimal deformations and in the semi-infinite algebraic geometry of infinite-dimensional spaces.

1. Definition of Semiderived Categories in Curved and Filtered Settings

Given a field $k$, a differential graded (dg) algebra $A$ over $k$, and a fixed non-negative integer $n$, one considers the cdg-algebra $A_n = A[t]/(t^{n+1})$ over $R_n = k[t]/(t^{n+1})$. The structure is determined by:

• Predifferential $d_{A_n}$, the $k$-linear extension of $d_A$ via the Leibniz rule,
• Curvature $c = d_{A_n}^2 \in t\,A_n$, which satisfies $d_{A_n}(c) = 0$ and $d^2_{A_n}(x) = [c, x]$.

The $n$-derived category $D^n(A_n)$ is constructed via the Verdier quotient:

• The homotopy category $\Hot(A_n) = H^0(A_n\text{-Mod})$ comprises (left) cdg-modules,
• Each $M \in A_n\text{-Mod}$ is equipped with a $t$-adic filtration $$0 = t^{n+1}M \subset t^nM \subset \cdots \subset tM \subset M$$,
• Associated graded pieces $\operatorname{gr}^i_t(M) = t^iM/t^{i+1}M$ inherit a square-zero differential, yielding honest complexes over $A$,
• $M$ is called $n$-acyclic if each $\operatorname{gr}^i_t(M)$ is acyclic in $D(A)$ for $i = 0, \dots, n$ (the kernel-filtration yields an equivalent criterion).

The $n$-derived category is then the quotient: $D^n(A_n) = \Hot(A_n)\Big/\{n\text{-acyclic modules}\}.$

This approach resolves the pathological “vanishing” of objects in conventional curved derived categories and restores compact generation, allowing the admissible embedding of Positselski’s semiderived category (Lehmann et al., 2024).

2. Compact Generation and Explicit Generators

Compact generation of $D^n(A_n)$ is achieved by constructing $n+1$ explicit compact “twisted” modules: $\Gamma_i, \quad i = 0, 1, \ldots, n,$ defined by:

• $A_i = A_n / t^{i+1}A_n$, $A_{-1} = 0$,
• $X_i = A_i \oplus A_{i-1}[1]$ as a graded module,
• Predifferential $d_{X_i} = d_{A_n} \oplus (-d_{A_n})$,
• Twisting by $$\gamma_i = \begin{pmatrix}0 & \pi(c/t) \ t & 0\end{pmatrix}$$, with $\pi: A_i \to A_{i-1}$ the quotient map and $c/t$ satisfying $t(c/t) = c$.

Each $\Gamma_i$ is verified to be compact in $\Hot(A_n)$ and generates $D^n(A_n)$ via cones, shifts, and coproducts. Thus, every object in $D^n(A_n)$ is built from the $\Gamma_i$.

3. Semiorthogonal Decomposition and Recollement

The category $D^n(A_n)$ possesses a semiorthogonal decomposition: $D^n(A_n) = \langle T_0, T_1, \ldots, T_n \rangle,$ where each $$T_i = \{M \in D^n(A_n) \mid \operatorname{gr}^j_t(M) \simeq 0\,\,\,\forall j \neq i\}$$ is equivalent to $D(A)$, and the projection functors $\operatorname{gr}^i_t : D^n(A_n) \to D(A)$ admit both left and right adjoints. The decomposition satisfies

• $\operatorname{Hom}_{D^n}(T_j, T_i) = 0$ for $j > i$,
• Unique Postnikov towers for each object, with cones in the $T_i$.

This structure is mirrored by a tower of recollements: $$D(A) \hookrightarrow D^1(A_1) \to D(A),\;\; D^1(A_1) \hookrightarrow D^2(A_2) \to D(A),\ldots,$$ successively adding new copies of $D(A)$ at each stage.

4. Semiderived Categories for Torsion Sheaves on Ind-Schemes

For a flat affine morphism $\pi: X \to S$ of ind-Noetherian ind-schemes (with dualizing complex $\omega_S^\bullet$), the semiderived category of quasi-coherent torsion sheaves is constructed as follows:

• The homotopy category $K(X\text{-tors})$ of complexes,
• A complex is $\pi$-coacyclic if its direct image $\pi_*\mathcal{N}^\bullet$ is contractible (coacyclic) on $S$,
• The semiderived category is the Verdier quotient: $$D^{\mathrm{si}}_\pi(X\text{-tors}) = \frac{K(X\text{-tors})}{\{\pi\text{-coacyclic complexes}\}}.$$

When $\pi$ is affine, $\pi$-coacyclicity coincides with Becker- or Positselski-coacyclicity, so $D^{\mathrm{si}}_\pi(X\text{-tors})$ interpolates between the classical derived and coderived categories. Under mild Noetherian hypotheses, $$D^{\mathrm{co}}(X\text{-tors}) \simeq K(X\text{-tors}_{\mathrm{inj}})$$ is compactly generated, typically corresponding to the “cohomological half” of the semi-infinite decomposition (Positselski, 2021).

5. Monoidal Structures: The Semitensor Product

A salient feature is the semitensor product: $$-\;\overset{\rm si}{\otimes}_{S}\;- : D^{\rm si}_\pi(X\text{-tors}) \times D^{\rm si}_\pi(X\text{-tors}) \to D^{\rm si}_\pi(X\text{-tors}),$$ which is “a mixture of” the cotensor product along the base $S$ and the derived tensor product along the fibers of $\pi$. Explicitly:

• On torsion sheaves $\mathcal{M}, \mathcal{G}$, $\mathcal{M} \boxtimes_S \mathcal{G}$ is defined on pro-systems, then passed to torsion sheaves,
• For pro-sheaf $\mathcal{G}$ and torsion sheaf $\mathcal{M}$, $$\mathcal{G} \boxtimes_S \mathcal{M} \simeq \pi_*(\pi^*\mathcal{G} \otimes_{\mathcal{O}_X} \pi^*\mathcal{M})$$,
• For “relatively homotopy flat” resolutions, one sets $$\mathcal{M} \overset{\rm si}{\otimes}_S \mathcal{N} = \mathcal{G}_{\mathrm{fl}} \overset{\flat}{\otimes} \mathcal{M}_\mathrm{si}$$,
• The unit object is $\pi^*\omega_S^\bullet$.

Associativity, base change, and duality compatibilities are rigorously verified. For instance, under a smooth base change, the semiderived categories are identified and the unit $\pi^*\omega_S^\bullet$ is preserved (Positselski, 2021).

6. Embedding of Positselski's Semiderived Category

Positselski’s semiderived category, constructed by taking the homotopy category of $R_n$-free $A_n$-modules and quotienting out "semiacyclic" modules (those whose reduction mod $t$ is acyclic), embeds as an admissible subcategory in the filtered $n$-derived category $D^n(A_n)$.

• The inclusion functor is fully faithful and admits both left (free/cohomological resolution) and right (cofree resolution) adjoints,
• Every semiacyclic $R_n$-free module is $n$-acyclic, establishing the well-definedness of the embedding,
• For any $A_n$-module $M$, there is a canonical exact triangle in $\Hot(A_n)$ with $M^{\mathrm{fr}}$ free and $C$ contraacyclic, producing the adjunctions upon passage to $D^n(A_n)$.

This embedding provides a precise categorical mechanism situating Positselski’s “semi-infinite” techniques within a more robust filtered/curved formalism (Lehmann et al., 2024).

7. Examples and Applications

• For $n=1$, $D^1(A_1)$ is a categorified square-zero extension, $D^1(A_1) = \langle D(A), D(A) \rangle$, with recollement directly recovering the sequence $0 \to A \to A[t]/(t^2) \to A \to 0$.
• For $A = k$ and a central deformation by $f$, $A_1 = k[u,u^{-1}]$ with $c = fu$ realizes the theory of 2-periodic modules (matrix factorizations) in a filtered perspective.
• If $A$ is smooth, $D^n(A_n)$ remains homologically smooth, and contains $D(A_n)$ fully faithfully, providing a non-commutative “resolution” of possibly singular dg-algebras $A_n$.

In infinite-dimensional geometry, the semiderived category and associated semitensor product support Tate affine spaces, cotangent bundles of infinite projective spaces, and loop groups, yielding well-behaved tensor triangulated categories $(D^{\mathrm{si}}_\pi(X\text{-tors}),\,\si\otimes_S,\,\pi^* \omega_S^\bullet)$ even when units are acyclic but nonetheless generate the structure.

This filtered formalism avoids the deficiencies of existing curved derived frameworks and enables new categorical resolutions, positioning the semiderived category as central in modern approaches to both deformation theory and semi-infinite algebraic geometry (Lehmann et al., 2024, Positselski, 2021).