The cotangent complex exists to “linearize deformation theory”.
Construction idea: For a ring map \(A\to B\), you choose a simplicial resolution \(P_\bullet \to B\) and then you set
\begin{align*} L_{B/A} = \Omega_{P_\bullet/A}\otimes_{P_\bullet} B, \end{align*}which is a complex in nonpositive cohomological degrees and is well defined up to quasi isomorphism. The construction is compatible with localization and etale maps, hence it sheafifies to \(L_{X/Y}\in D^{\leq 0}_{qcoh}(X)\) for schemes; for algebraic stacks it’s harder, apparently you lose the resolution picture above but keep the universal property.
See
- Alpher Appendix C.5
- The Stacks project Tag 08T2
- When \(f:X\to Y\) is affine, you get an explicit construction due to Lichtenbaum-Schlessinger.
Properties you actually use:
- Degree zero: \(h^0(L_{X/Y}) = \Omega_{X/Y}\)
- Transitivity: for \(X\xrightarrow{f} Y\to Z\) you get a distinguished triangle \(Lf^*L_{Y/Z}\to L_{X/Z} \to L_{X/Y}\to Lf^*L_{Y/Z}[1]\).
- Base change holds for flat and derived base change
- Deformation control: for \(g:T\to X\) and a square-zero extension \(T\hookrightarrow \overline T\) with ideal \(\mathcal J\), BF construct a class \(\omega(g) \in \Ext^1(g^*L^\bullet_X, \mathcal J)\) using that \(\tau_{\geq -1}L^\bullet_{T/\overline T} = \mathcal J[1]\); the extension of \(g\) exists if and only if \(\omega(g) = 0\) and extensions then form a torsor under \(\Ext^0(g^*L^\bullet_X, \mathcal J)\). Obstructions to extending then live in \(\Ext^2\).
- When \(f\) is smooth, \(L_{X/Y} = \Omega_{X/Y}[0]\). That is, smoothness kills everything below degree 0.
The following theorem of Illusie from 1971 globalizes an earlier result of Michel Andre and established the existence and properties of the cotangent complex.
Theorem
For every morphism \(f:X\to Y\) of schemes (resp. finite type morphisms of noetherian schemes), there exists a complex
\begin{align*} L_{X/Y}: \dots \to L^{-1}_{X/Y} \to L^0_{X/Y}\to 0 \end{align*}of flat \(\mathcal O_X\)-modules with quasi-coherent (resp. coherent) cohomology, whose image in \(D^-_{\mathsf{QCoh}}(\mathcal O_X)\) (resp. \(D^-_{\mathsf{Coh}}(\mathcal O_X)\)) is also denoted by \(L_{X/Y}\). It satisfies the following properties:
- \(H^0(X,L_{X/Y}) \cong \Omega_{X/Y}\)
- \(f\) is smooth if and only if \(f\) is locally of finite presentation nd \(L_{X/Y}\) is a perfect complex supported in degree \(0\). In this case \(L_{X/Y}\) is quasi-isomorphic to the complex where the vector bundle \(\Omega_{X/Y}\) sits in degree \(0\).
- If \(f\) is flat and finitely presented, then \(f\) is syntomic if and only if \(L_{X/Y}\) is a perfect complex supported in degrees \([-1,0]\). Explicitly, if \(f\) factors as a local complete intersection \(X\hookrightarrow \widetilde Y\) defined by a sheaf of ideals \(I\) and a smooth morphism \(\widetilde Y\to Y\), then \(L_{X/Y}\) is quasi-isomorphic to \(0\to I/I^2 \xrightarrow{d} \Omega_{X/Y} \to 0\) (with \(\Omega_{X/Y}\) in degree 0);
If
is a cartesian diagram with either \(f\) or \(g\) flat (or more generally \(f\) and \(g\) are tor-independent), then there is a quasi-isomorphism \(g'^*L_{X/Y}\to L_{X'/Y'}\). (Note that without any flatness condition \(g'^*\Omega_{X/Y} \cong \Omega_{X'/Y'}\).)
If \(X\xrightarrow{f} Y\to Z\) i a composition of morphisms of schemes, then these is an exact triangle in \(D^-_{\mathsf{QCoh}}(\mathcal O_X)\)
\begin{align*} f^{*}L_{Y/Z}\to L_{X/Z}\to L_{X/Y}\to f^{*}L_{Y/Z}[1] \end{align*}which induces a long exact sequence on cohomology (see Alper Appendix C.5.1 for this one because the tikzcd is gonna be gross).
We then call \(L_{f}\) or \(L_{X/Y}\) the cotangent complex of \(f:X\to Y\).