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 $D(A)$ 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 $\mathcal_X$ 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 $(X, \mathcal_X)$, an $\mathcal_X$-module is called pseudo-coherent if for every integer $n \ge 0$, locally, there is a free presentation of finite type of length ''n''; i.e.,
:$L_n \to L_ \to \cdots \to L_0 \to F \to 0$.

A complex ''F'' of $\mathcal_X$-modules is called pseudo-coherent if, for every integer ''n'', there is locally a quasi-isomorphism $L \to F$ where ''L'' has degree bounded above and consists of finite free modules in degree $\ge n$. If the complex consists only of the zero-th degree term, then it is pseudo-coherent if and only if it is so as a module.

Roughly speaking, a pseudo-coherent complex may be thought of as a limit of perfect complexes.

See also

*Hilbert–Burch theorem
* elliptic complex (related notion; discussed at SGA 6 Exposé II, Appendix II.)

References

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Bibliography

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Abstract algebra

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