Perfect complex

In algebra, a perfect complex of modules over a commutative ring A is an object in the derived category of A-modules that is quasi-isomorphic to a bounded complex of finite projective A-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if A is Noetherian, a module over A is perfect if and only if it is finitely generated and of finite projective dimension.

Other characterizations

Perfect complexes are precisely the compact objects in the unbounded derived category

D

(

A

)

{\displaystyle D(A)}

of A-modules. They are also precisely the dualizable objects in this category.

A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect; see also module spectrum.

Pseudo-coherent sheaf

When the structure sheaf

O

X

{\displaystyle {\mathcal {O}}_{X}}

is not coherent, working with coherent sheaves has awkwardness (namely the kernel of a finite presentation can fail to be coherent). Because of this, SGA 6 Expo I introduces the notion of a pseudo-coherent sheaf.

By definition, given a ringed space

(

X

,

O

X

)

{\displaystyle (X,{\mathcal {O}}_{X})}

, an

O

X

{\displaystyle {\mathcal {O}}_{X}}

-module is called pseudo-coherent if for every integer

n

≥

0

{\displaystyle n\geq 0}

, locally, there is a free presentation of finite type of length n; i.e.,

L

n

→

L

n

−

1

→

⋯

→

L

0

→

F

→

0

{\displaystyle L_{n}\to L_{n-1}\to \cdots \to L_{0}\to F\to 0}

.

A complex F of

O

X

{\displaystyle {\mathcal {O}}_{X}}

-modules is called pseudo-coherent if, for every integer n, there is locally a quasi-isomorphism

L

→

F

{\displaystyle L\to F}

where L has degree bounded above and consists of finite free modules in degree

≥

n

{\displaystyle \geq n}

.