Small dg categories, perfect modules, and Morita deformation

Let $k$ be a field. A small $k$-linear cohomological dg category $\mathcal{A}$ has a set of objects, a cochain complex $\mathcal{A}(a,b)$ of $k$-vector spaces for each ordered pair of objects, closed degree-zero identity elements, and associative unital degree-zero composition chain maps. The cochain differential has degree $+1$ and composition obeys the signed Leibniz rule of a cohomological dg algebra. A right dg $\mathcal{A}$-module is a $k$-linear enriched functor ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}\to \mathrm{C}\mathrm{h}(k)$, where $\mathrm{C}\mathrm{h}(k)$ is the category of cochain complexes with the total tensor differential and Koszul signs. For a commutative $k$-algebra $R$, define $R$-linear dg categories and their right modules by the same formulas with $R$-modules and $R$-linear maps in place of $k$-vector spaces and $k$-linear maps.

Write $D(\mathcal{A})$ for the homotopy-coherent localization of the category of right dg $\mathcal{A}$-modules at the objectwise quasi-isomorphisms: it is characterized by the property that functors out of it are functors out of the module category that send those quasi-isomorphisms to equivalences. Its morphisms are derived mapping spaces. For each object $a$, the representable module is $h_a=\mathcal{A}(-,a)$. A full subcategory is stable here if it is closed under shifts and finite limits and colimits; it is idempotent-complete if every idempotent endomorphism splits. Define $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(\mathcal{A})$ to be the smallest stable idempotent-complete full subcategory of $D(\mathcal{A})$ containing every representable module. These formulas are definitions and do not assert a separate compact-generation theorem.

For a dg functor $F:\mathcal{A}\to \mathcal{B}$, let $F_{!}⊣F^{\ast }$ denote derived extension and restriction of scalars on module categories. The functor is a Morita equivalence when this adjunction induces an equivalence $D(\mathcal{A})\simeq D(\mathcal{B})$. If compact generation by representables has separately been proved in a given setting, this can equivalently be checked on $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(\mathcal{A})\to \mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}(\mathcal{B})$; that equivalence is not assumed as part of this definition.

For a $k$-scheme $X$, let $D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(X)$ denote the full subcategory of the derived category of ${\mathcal{O}}_X$-modules consisting of complexes which, on an open cover of $X$, are quasi-isomorphic to bounded complexes of finite-rank locally free ${\mathcal{O}}_X$-modules. A specified dg enhancement of perfect complexes on $X$, denoted ${\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}}_{dg}(X)$, is a small $k$-linear dg category ${\mathcal{A}}_X$ together with a $k$-linear triangulated equivalence $h\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}({\mathcal{A}}_X)\simeq D_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}(X)$. Here $h\mathcal{C}$ denotes the homotopy category of an $\mathrm{\infty }$-category $\mathcal{C}$: its objects are those of $\mathcal{C}$ and its morphism sets are ${\pi }_0$ of the mapping spaces. The right-hand side is the ordinary derived category, so this comparison is explicitly made on homotopy categories. The notation records this chosen enhancement; it does not assert existence or uniqueness of one, and any comparison between choices is made only after a Morita equivalence is supplied.

Let $R$ be a finite-dimensional commutative local Artin $k$-algebra with residue map $ϵ:R\to k$. An admissible marked Morita deformation of a small $k$-linear dg category ${\mathcal{A}}_0$ over $R$ is a pair $({\mathcal{A}}_R,\eta )$ where ${\mathcal{A}}_R$ is a small $R$-linear dg category, ${\mathcal{A}}_R{\otimes }_R^{\mathbf{L}}k$ is its derived reduction, and the marking $\eta$ is a specified equivalence between ${\mathcal{A}}_R{\otimes }_R^{\mathbf{L}}k$ and ${\mathcal{A}}_0$ in the localization of small $k$-linear dg categories at Morita equivalences. Thus the marking includes an equivalence $D({\mathcal{A}}_R{\otimes }_R^{\mathbf{L}}k)\simeq D({\mathcal{A}}_0)$ induced by Morita equivalences and its specified restriction $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}({\mathcal{A}}_R{\otimes }_R^{\mathbf{L}}k)\simeq \mathrm{P}\mathrm{e}\mathrm{r}\mathrm{f}({\mathcal{A}}_0)$. Both are required data here; a bare equivalence of the perfect subcategories is not being used as an unproved criterion for Morita equivalence. Derived tensor products are computed using cofibrant or K-flat replacements of mapping complexes; this is a model choice and imposes no additional condition on the Morita class. A morphism is an $R$-linear Morita equivalence compatible with the markings. The Morita deformation groupoid is the maximal subgroupoid of the homotopy-coherent localization at Morita equivalences of these marked objects and compatible morphisms; its set of connected components is the associated set-valued deformation functor.

The underlying dg category is taken up to Morita equivalence, not quasi-equivalence of chosen dg presentations and not gauge equivalence of a fixed coderivation carrier. The definition leaves existence of model structures, compact generation, and comparison with any geometric deformation functor to separate results.

Cohomological differential graded algebra

Cohomological left and right differential graded modules

Model categories and Quillen adjunctions

Derived category of an abelian category

The source uses derived module categories and Morita equivalences over a field. A prior finite-field dg-category definition is not a characteristic-zero supplier for the Morimura application. This definition makes the marked reduction explicit and does not identify Morita equivalence with quasi-equivalence.