• Tags
  • #AG/deformation-theory #higher-algebra/DAG
• Refs:
  • #todo/add-references
• Links:
  • tor amplitude
  • derived stack
  • compact generators
  • constructible complex

perfect complexes

Idea: an analog in complexes of the notion of a finite dimensional vector space (finiteness and dualizability). One can relate \(\mathsf{Perf}(X)\) for \(X\) a separated scheme of finite type with \({\mathrm{perf}}(A)\) for \(A\) a single DGA, and if \(X\) is a smooth scheme then \(A\) is a smooth algebra.

Moreover $\mathbf{D} {{\mathsf{QCoh}}(X)} = \mathsf{Ind} \mathbf{D} {\mathsf{Perf}(X)} $, i.e. the former is compactly generated by perfect complexes.

TFAE for a complex \(M\in \mathsf{Ch}({\mathsf{A}{\hbox{-}}\mathsf{Mod}})\):

• \(M\) is perfect.
• \(M\) is in the thick subcategory $\left\langle{\mathsf{A}}\right\rangle \leq \mathbf{D} {{\mathsf{A}{\hbox{-}}\mathsf{Mod}}} $ in the (unbounded) derived category of \(A\)
  • So \(M\) is built from finite colimits and direct summands of \(A\)
• \(M\) is a compact object of $\mathbf{D} {{\mathsf{A}{\hbox{-}}\mathsf{Mod}}} $, so \([M, \bigoplus L_i] = \bigoplus [M, L_i]\) for any collection of complexes \(\left\{{L_i}\right\}_{i\in I}\).
• \(M\) is a dualizable object of $\mathbf{D} {{\mathsf{A}{\hbox{-}}\mathsf{Mod}}} $
• \(M \simeq C \in \mathbf{D}^b {{\mathsf{A}{\hbox{-}}\mathsf{Mod}}} ^{{\mathrm{fin}},\mathop{\mathrm{proj}}}\), a bounded complex of finite projective \(A{\hbox{-}}\)modules.
• \(M\) is a compact object in the infty-category of module spectra over a ring spectrum.

The category \(\mathsf{Perf}(A)\) is defined as the subcategory of compact objects in $\mathbf{D} {{\mathsf{A}{\hbox{-}}\mathsf{Mod}}} $. 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.

For modules

• A module \(M\in{\mathsf{A}{\hbox{-}}\mathsf{Mod}}\) is perfect in iff its image $M[0] \in \mathbf{D} {{\mathsf{A}{\hbox{-}}\mathsf{Mod}}} $ 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 \(A{\hbox{-}}\)modules.

For complexes of sheaves

• A complex of sheaves is perfect iff it is locally quasi-isomorphic to a bounded complex of locally free sheaves of finite type.
  • The bounded derived category \(\mathbf{D} {{\mathsf{Coh}}(X)} ^b\) has a triangulated subcategory consisting of perfect complexes.

Misc

• The derived algebraic stack of perfect complexes contains the classifying stack \({\mathbf{B}}\operatorname{GL}_n \hookrightarrow\mathsf{Perf}\) as an open substack, so \(\mathsf{Perf}\) is thought of as a generalization of \({\mathbf{B}}\operatorname{GL}_n\).

Applications

To the moduli of perfect complexes over a K3 surface: