Subsection 2.3. (0MZT) · ScalingStacks

2.3. Locally unital algebras

It is sometimes convenient to pass from additive categories, such as the heart of a weight structure $\mathcal{C}^{w=0},$ to categories of modules over some algebra with idempotents. We use this perspective to prove a small variation on Corollary 2.12.

0MXM

Definition 2.13. A (non-unital) algebra $A$ with a distinguished set of idempotents $\{e_{i}|i\in I\}$ is called locally unital if the canonical map

\bigoplus_{i,j\in I}e_{i}Ae_{j}\to A

is an isomorphism. A right module over a locally unital algebra $A$ is called finitely generated (projective) if it is isomorphic to a quotient (direct summand) of a finite direct sum of the modules $e_{i}A$ for $i\in I.$ The perfect derived category of $A$ is the bounded homotopy category of the finitely generated projective right modules

\operatorname{D_{perf}}(A)=\operatorname{K}^{b}(\operatorname{mod_{fgp}-}A).

If $A$ is moreover $\mathbb{Z}$ -graded we consider the category $\operatorname{mod}^{\mathbb{Z}}\operatorname{-}A$ of graded right modules over $A$ with morphisms of degree $0$ and the graded perfect derived category

\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(A)=\operatorname{K}^{b}(\operatorname{mod}^{\mathbb{Z}}_{\operatorname{fgp}}\operatorname{-}A).

For a family of objects $\mathcal{T}$ in an idempotent-closed additive category $\mathcal{C}$ we consider the locally unital algebra with the distinguished idempotents $e_{M}=\operatorname{id}_{M}$ for $M\in\mathcal{T}$

(2.2)

\displaystyle\operatorname{End}_{\mathcal{C}}(\mathcal{T})=\bigoplus_{M,N\in\mathcal{T}}\operatorname{Hom}_{\mathcal{C}}(M,N).

By general nonsense we obtain:

0MXN

Proposition 2.14. The functor

\bigoplus_{M\in\mathcal{T}}\operatorname{Hom}_{\mathcal{C}}(M,-):\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus}\to\operatorname{mod_{fgp}-}\operatorname{End}_{\mathcal{C}}(\mathcal{T})

is an equivalence of categories.

For a graded version, assume that $\mathcal{C}$ is equipped with an autoequivalence $\langle 1\rangle.$ Then one can define the $\mathbb{Z}$ -graded locally unital algebra

(2.3)

\displaystyle\operatorname{End}_{\mathcal{C}}^{\bullet}(\mathcal{T})=\bigoplus_{M,N\in\mathcal{T}}\operatorname{Hom}^{\bullet}_{\mathcal{C}}(M,N)\text{ where }\operatorname{Hom}^{n}_{\mathcal{C}}(M,N)=\operatorname{Hom}_{\mathcal{C}}(M,N\langle n\rangle).

Again, general nonsense yields

0MXP

Proposition 2.15. The functor

\bigoplus_{M\in\mathcal{T}}\operatorname{Hom}^{\bullet}_{\mathcal{C}}(M,-):\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus,\langle\pm 1\rangle}\to\operatorname{mod}^{\mathbb{Z}}_{\operatorname{fgp}}\operatorname{-}\operatorname{End}_{\mathcal{C}}^{\bullet}(\mathcal{T})

is an equivalence of categories.

0MXQ

Corollary 2.16. In the situation of Corollary 2.12 the weight complex functor induces an equivalence

t:\langle\mathcal{T}\rangle_{\cong,\inplus,\Delta}\to\operatorname{D_{perf}}(\operatorname{End}_{\mathcal{C}}(\mathcal{T})).

If $\mathcal{C}$ admits and autoequivalence $\langle 1\rangle$ such that $\widehat{\mathcal{T}}=\bigcup_{n}\mathcal{T}\langle n\rangle$ is also tilting then the weight complex functor induces an equivalence

t:\langle\widehat{\mathcal{T}}\rangle_{\cong,\inplus,\Delta}\to\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(\operatorname{End}^{\bullet}_{\mathcal{C}}(\mathcal{T})).