perfect complex

Perfect complexes are precisely the compact objects in the unbounded derived category $D(A)$ of A-modules.See, e.g., {{harvtxt|Ben-Zvi|Francis|Nadler|2010}} They are also precisely the dualizable objects in this category.Lemma 2.6. of {{harvtxt|Kerz|Strunk|Tamme|2018}}

A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect;{{harvtxt|Lurie|2014}} see also module spectrum.

When the structure sheaf $\backslash mathcal\{O\}\_X$ 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 $(X,\; \backslash mathcal\{O\}\_X)$, an $\backslash mathcal\{O\}\_X$-module is called pseudo-coherent if for every integer $n\; \backslash ge\; 0$, locally, there is a free presentation of finite type of length n; i.e.,

:$L\_n\; \backslash to\; L\_\{n-1\}\; \backslash to\; \backslash cdots\; \backslash to\; L\_0\; \backslash to\; F\; \backslash to\; 0$.

A complex F of $\backslash mathcal\{O\}\_X$-modules is called pseudo-coherent if, for every integer n, there is locally a quasi-isomorphism $L\; \backslash to\; F$ where L has degree bounded above and consists of finite free modules in degree $\backslash ge\; n$. 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.

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

{{Reflist}}

• {{Citation|last1=Ben-Zvi|first1=David|last2=Francis|first2=John|last3=Nadler|first3=David|title=Integral transforms and Drinfeld centers in derived algebraic geometry|journal=Journal of the American Mathematical Society|

volume=23|year=2010|issue=4|pages=909–966|mr=2669705|doi=10.1090/S0894-0347-10-00669-7|arxiv=0805.0157|s2cid=2202294}}

• {{cite book

| editor-last = Berthelot

| editor-first = Pierre

| editor-link = Pierre Berthelot (mathematician)

| editor2=Alexandre Grothendieck

| editor2-link=Alexandre Grothendieck

| editor3=Luc Illusie

| editor3-link=Luc Illusie

| title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225)

| series = Lecture Notes in Mathematics

| year = 1971

| volume = 225

| publisher = Springer-Verlag

| location = Berlin; New York

| language = fr

| pages = xii+700

| no-pp = true

|doi=10.1007/BFb0066283

|isbn= 978-3-540-05647-8

| mr = 0354655

}}

• {{cite journal

|doi=10.1007/s00222-017-0752-2

|title=Algebraic K-theory and descent for blow-ups

|date=2018

|last1=Kerz |first1=Moritz

|last2=Strunk |first2=Florian

|last3=Tamme |first3=Georg

|journal=Inventiones Mathematicae

|volume=211

|issue=2

|pages=523–577

|arxiv=1611.08466

|bibcode=2018InMat.211..523K }}

• {{cite web

|last1=Lurie |first1=Jacob

|title=Algebraic K-Theory and Manifold Topology (Math 281), Lecture 19: K-Theory of Ring Spectra.

|url=https://www.math.ias.edu/~lurie/281notes/Lecture19-Rings.pdf

|date=2014}}

• {{cite web |title=Determinantal identities for perfect complexes |url=https://mathoverflow.net/questions/354214/determinantal-identities-for-perfect-complexes |website=MathOverflow}}
• {{cite web |title=An alternative definition of pseudo-coherent complex |url=https://mathoverflow.net/questions/200540/an-alternative-definition-of-pseudo-coherent-complex |website=MathOverflow}}
• {{Cite web |title=15.74 Perfect complexes |url=http://stacks.math.columbia.edu/tag/0656

|website=The Stacks project}}

• {{Cite web |title=perfect module |url=https://ncatlab.org/nlab/show/perfect+module |website=ncatlab.org}}

Category:Abstract algebra

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