A cotangent complex is a certain spectrum object which exerts full control of the linear-order deformation & obstruction theory in a moduli problem. Consequently, a construction of a cotangent complex constitutes a complete understanding of the deformation theory of the situation.
Thereβs both a local and global version of this idea. Let $X$ be a derived prestack. One can seek the following objects:
$\mathbb{L}_{X, x} \in QCoh(U)$, for a βpointβ $U \xrightarrow{x} X \in Aff_{/X}$. The local/point-wise cotangent complex.
$\mathbb{L}_X \in QCoh(X)$. The global cotangent complex.
In the case that $X$ is a nice geometric object (e.g. an n-geometric stack), $\mathbb {L}_{X, x}$ can be viewed as the (derived) cotangent space at a point, and $\mathbb {L}_X$ the global cotangent bundle.
That being said, the global cotangent complex exist for a much broader class of prestacks $X$ than just the geometric ones, allowing one to talk about the βexistence of a good deformation theory everywhereβ even when the moduli problem is not representable.
Before diving into the various variants of cotangent complexes, we summarize their relationships here.
Weβve already explained the global/local distinction above, we briefly discuss the absolute/relative distinction now.
In algebraic geometry, relative notions are notions about families. e.g. flatness/smoothness/etaleness over a base scheme/stack. Relative cotangent complexes encode deformation theory of families. Of course these are analogous to relative cotangent bundles in differential geometry.
We also note that even if one is only interested in finding an absolute cotangent complex, some of the most useful computational lemmas require you to compute relative cotangent complexes. Hence the notion is truly ubiquitous in deformation theory.
We begin by discussing what it means for the cotangent complex to control deformation/obstruction theory.
Let
The $\infty$-groupoid encoding all extensions of $x$ along $\iota_M$ is called the space of derivations. This can be written as a (homotopy) fiber diagram in the $\infty$-category of spaces:
Formation of these homotopy fibers is functorial in $M \in QCoh(U)$, hence determines a functor
A cotangent complex for $X$ and $x$ is a object in $QCoh(U)$ corepresenting the functor $Der_{X, x}$. Such an object is denoted $\mathbb {L}_{X, x}$
Hence, a cotangent complex controls, up to homotopy, the deformation theory of $x$ in $X$ by the following mechanism.
Hence this construction translates the geometric situation on the left to the linear situation on the right.
An object $\mathbb{L}_X \in QCoh(X)$ is said to be a global cotangent complex if for all $x: U \rightarrow X \in Aff_{/X}$, $x^*\mathbb {L}_X \in QCoh(U)$ is a cotangent local complex at $x$.
Hence, one can informally write the condition as existence of a compatible system of equivalences $x^*\mathbb {L}_X \simeq \mathbb {L}_{X, x}$.
Both the local and global notions above can be relativizedβ¦
Let $X \xrightarrow{a} Y \xrightarrow{b} Z \in Fun(\Delta^2, pSt)$ such that each arrow has a global relative cotangent complex, the following diagram
is a fiber (and hence cofiber) diagram in the stable $\infty$ category $QCoh(X)$.
For example, taking $Z \simeq pt$ exhibits the $\mathbb{L}_{X/Y}$ as the cofiber of $f^*\mathbb{L}_{Y} \to \mathbb{L}_{X}$
There are several ways to demonstrate the existence of cotangent complexes via explicit constructions:
Some elaboration of these ideas are found in the article cotangent complex.
Last revised on February 26, 2020 at 12:51:06. See the history of this page for a list of all contributions to it.