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created: 2022-04-05T23:42
updated: 2024-01-01T23:13
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date: 2021-11-05 13:33
modification date: Friday 5th November 2021 13:33:13
title: perfect complexes
aliases: [perfect complex]

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- Tags 
	- #AG/deformation-theory #higher-algebra/DAG
- Refs:
	- #todo/add-references
- Links:
	- [tor amplitude](Unsorted/tor%20amplitude.md)
	- [derived stack](Unsorted/derived%20algebraic%20geometry.md)
	- [compact generators](Unsorted/Generators%20of%20a%20category.md)
	- [constructible complex](Unsorted/constructible%20sheaf.md)

---

# perfect complexes

Idea: an analog in complexes of the notion of a finite dimensional vector space (finiteness and dualizability). One can relate $\Perf(X)$ for $X$ a [separated scheme](Unsorted/separated.md) [of finite type](of%20finite%20type) with $\perf(A)$ for $A$ a single [DGA](Unsorted/DGA.md), and if $X$ is a [smooth scheme](Unsorted/smooth%20scheme.md) then $A$ is a [smooth algebra](Unsorted/smooth%20algebra.md).

Moreover $\derivedcat{\QCoh(X)} = \Indcat\derivedcat{\Perf(X)}$, i.e. the former is compactly generated by perfect complexes.

TFAE for a *complex* $M\in \Ch(\amod)$:

-  $M$ is **perfect**.
- $M$ is in the thick subcategory $\gens{\cat A} \leq \derivedcat{\amod}$ in the (unbounded) [derived category](Unsorted/derived%20category.md) of $A$ 
	- So $M$ is built from finite colimits and direct summands of $A$
- $M$ is a [compact object](Unsorted/compact%20object%20of%20a%20category.md) of $\derivedcat{\amod}$, so $[M, \bigoplus L_i] = \bigoplus [M, L_i]$ for any collection of complexes $\ts{L_i}_{i\in I}$.
- $M$ is a [dualizable object](Unsorted/dualizable%20object%20of%20a%20category.md) of $\derivedcat{\amod}$
- $M \qiso C \in \bderivedcat{\amod}^{\fin,\proj}$, a bounded complex of finite projective $A\dash$modules.
- $M$ is a compact object in the [infty-category](Unsorted/infinity%20categories.md) of [module spectra](Unsorted/E_n%20ring%20spectrum.md) over a [ring spectrum](Unsorted/ring%20spectra.md).

The category $\Perf(A)$ is defined as the subcategory of compact objects in $\derivedcat{\amod}$. Equivalently, it is the smallest subcategory which contains $A[0]$ and is closed under taking cones, shifts, and direct summands.

> Warning: for a more general category, perfect *objects* are not always [compact](Unsorted/compact%20object%20of%20a%20category.md).



# For modules

- A module $M\in\amod$ is **perfect** in iff its image $M[0] \in \derivedcat{\amod}$ is perfect. 
	- If $A$ is Noetherian, $M$ is perfect iff $M$ is finitely generated over $A$ and of finite projective dimension.
- In this case, $M$ admits a finite projective resolution by finite projective $A\dash$modules.

# For complexes of sheaves
- A complex of sheaves is perfect iff it is locally quasi-isomorphic to a bounded complex of [locally free](locally%20free) sheaves [of finite type](of%20finite%20type).
	- The [bounded derived category](bounded%20derived%20category) $\derivedcat{\Coh(X)}^b$ has a [triangulated subcategory](triangulated%20category) consisting of perfect complexes.


# Misc

- The derived algebraic stack of perfect complexes contains the [classifying stack](classifying%20stack) $\B\GL_n \embeds \Perf$ as an open substack, so $\Perf$ is thought of as a generalization of $\B\GL_n$.



# Applications

To the moduli of perfect complexes over a [K3 surface](K3%20surfaces.md):
![](attachments/Pasted%20image%2020220516184254.png)