Cotangent complex

In mathematics the cotangent complex $\mathbf {L} _{X/Y}^{\bullet }$ is roughly a universal linearization of a morphism of geometric or algebraic objects $X\to Y$ . They are defined in certain derived categories of sheaves for a space $X$ , or a morphism of spaces $X\to Y$ and control their deformation theory.[1][2] Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M. André independently came up with a definition that works in all cases.

Motivation

Suppose that $X$ and $Y$ are algebraic varieties and that $f:X\to Y$ is a morphism between them. The cotangent complex of $f$ is a more universal version of the relative Kähler differentials $\Omega _{X/Y}$ . The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If $Z$ is another variety, and if $g:Y\to Z$ is another morphism, then there is an exact sequence

f^{*}\Omega _{Y/Z}\to \Omega _{X/Z}\to \Omega _{X/Y}\to 0.

In some sense, therefore, relative Kähler differentials are a right exact functor. (Literally this is not true, however, because the category of algebraic varieties is not an abelian category, and therefore right-exactness is not defined.) In fact, prior to the definition of the cotangent complex, there were several definitions of functors that might extend the sequence further to the left, such as the Lichtenbaum–Schlessinger functors $T^{i}$ and imperfection modules. Most of these were motivated by deformation theory.

This sequence is exact on the left if the morphism $f$ is smooth. If Ω admitted a first derived functor, then exactness on the left would imply that the connecting homomorphism vanished, and this would certainly be true if the first derived functor of f, whatever it was, vanished. Therefore, a reasonable speculation is that the first derived functor of a smooth morphism vanishes. Furthermore, when any of the functors which extended the sequence of Kähler differentials were applied to a smooth morphism, they too vanished, which suggested that the cotangent complex of a smooth morphism might be equivalent to the Kähler differentials.

Another natural exact sequence related to Kähler differentials is the conormal exact sequence. If f is a closed immersion with ideal sheaf I, then there is an exact sequence

I/I^{2}\to f^{*}\Omega _{Y/Z}\to \Omega _{X/Z}\to 0.

This is an extension of the exact sequence above: There is a new term on the left, the conormal sheaf of f, and the relative differentials Ω_X/Y have vanished because a closed immersion is formally unramified. If f is the inclusion of a smooth subvariety, then this sequence is a short exact sequence.[3] This suggests that the cotangent complex of the inclusion of a smooth variety is equivalent to the conormal sheaf shifted by one term.

Early work on cotangent complexes

The cotangent complex dates back at least to SGA 6 VIII 2, where Pierre Berthelot gave a definition when f is a smoothable morphism, meaning there is a scheme V and morphisms i : X → V and h : V → Y such that f = hi, i is a closed immersion, and h is a smooth morphism. (For example, all projective morphisms are smoothable, since V can be taken to be a projective bundle over Y.) In this case, he defines the cotangent complex of f as an object in the derived category of coherent sheaves X as follows:

$L_{0}^{X/Y}=i^{*}\Omega _{V/Y},$
If J is the ideal of X in V, then $L_{1}^{X/Y}=J/J^{2}=i^{*}J,$
$L_{i}^{X/Y}=0$ for all other i,
The differential $L_{1}^{X/Y}\to L_{0}^{X/Y}$ is the pullback along i of the inclusion of J in the structure sheaf ${\mathcal {O}}_{V}$ of V followed by the universal derivation $d:{\mathcal {O}}_{V}\to \Omega _{V/Y}.$
All other differentials are zero.

Berthelot proves that this definition is independent of the choice of V[4] and that for a smoothable complete intersection morphism, this complex is perfect.[5] Furthermore, he proves that if g : Y → Z is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an exact triangle

\mathbf {L} f^{*}L_{\bullet }^{Y/Z}\to L_{\bullet }^{X/Z}\to L_{\bullet }^{X/Y}\to \mathbf {L} f^{*}L_{\bullet }^{Y/Z}[1].

The definition of the cotangent complex

The correct definition of the cotangent complex begins in the homotopical setting. Quillen and André worked with the simplicial commutative rings, while Illusie worked with simplicial ringed topoi. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that $A$ and $B$ are simplicial rings and that $B$ is an $A$ -algebra. Choose a resolution $r:P^{\bullet }\to B$ of $B$ by simplicial free $A$ -algebras. Such a resolution of $B$ can be constructed by using the free commutative $A$ -algebra functor which takes a set $S$ and yields the free $A$ -algebra $A[S]$ . For an $A$ -algebra $B$ , this comes with a natural augmentation map $\eta _{B}:A[B]\to B$ which maps a formal sum of elements of $B$ to an element of $B$ via the rule

$a_{1}[b_{1}]+\cdots +a_{n}[b_{n}]\mapsto a_{1}\cdot b_{1}+\cdots a_{n}\cdot b_{n}$

Iterating this construction gives a simplicial algebra

$\cdots \to A[A[A[B]]]\to A[A[B]]\to A[B]\to B$

where the horizontal maps come from composing the augmentation maps for the various choices. For example, there are two augmentation maps $A[A[B]]\to A[B]$ via the rules

${\begin{aligned}a_{i}[a_{i,1}[b_{i,1}]+\cdots +a_{i,n_{i}}[b_{i,n_{i}}]]&\mapsto a_{i}a_{i,1}[b_{i,1}]+\cdots +a_{i}a_{i,n_{i}}[b_{i,n_{i}}]\\&\mapsto a_{i,1}[a_{i}\cdot b_{i,1}]+\cdots +a_{i,n_{i}}[a_{i}\cdot b_{i,n_{i}}]\end{aligned}}$

which can be adapted to each of the free $A$ -algebras $A[\cdots A[A[B]]$ .

Applying the Kähler differential functor to $P^{\bullet }$ produces a simplicial $B$ -module. The total complex of this simplicial object is the cotangent complex L^B/A. The morphism r induces a morphism from the cotangent complex to Ω_B/A called the augmentation map. In the homotopy category of simplicial A-algebras (or of simplicial ringed topoi), this construction amounts to taking the left derived functor of the Kähler differential functor.

Given a commutative square as follows:

there is a morphism of cotangent complexes $L^{B/A}\otimes _{B}D\to L^{D/C}$ which respects the augmentation maps. This map is constructed by choosing a free simplicial C-algebra resolution of D, say $s:Q^{\bullet }\to D.$ Because $P^{\bullet }$ is a free object, the composite hr can be lifted to a morphism $P^{\bullet }\to Q^{\bullet }.$ Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms $A\to B\to C,$ this produces the sequence

L^{B/A}\otimes _{B}C\to L^{C/A}\to L^{C/B}.

There is a connecting homomorphism,

L^{C/B}\to \left(L^{B/A}\otimes _{B}C\right)[1],

which turns this sequence into an exact triangle.

The cotangent complex can also be defined in any combinatorial model category M. Suppose that $f:A\to B$ is a morphism in M. The cotangent complex $L^{f}$ (or $L^{B/A}$ ) is an object in the category of spectra in $M_{B//B}$ . A pair of composable morphisms, $f:A\to B$ and $g:B\to C$ induces an exact triangle in the homotopy category,

L^{B/A}\otimes _{B}C\to L^{C/A}\to L^{C/B}\to \left(L^{B/A}\otimes _{B}C\right)[1].

Cotangent complexes in deformation theory

Setup

One of the first direct applications of the cotangent complex is in deformation theory. For example, if we have a scheme $f:X\to S$ and a square-zero infinitesimal thickening $S\to S'$ , that is a morphism of schemes where the kernel

${\mathcal {I}}={\text{ker}}\{{\mathcal {O}}_{S'}\to {\mathcal {O}}_{S}\}$

has the property its square is the zero sheaf, so

${\mathcal {I}}^{2}=0$

one of the fundamental questions in deformation theory is to construct the set of $X'$ fitting into cartesian squares of the form

$\left\{{\begin{matrix}X&\to &X'\\\downarrow &&\downarrow \\S&\to &S'\end{matrix}}\right\}$

A couple examples to keep in mind is extending schemes defined over $\mathbb {Z} /p$ to $\mathbb {Z} /p^{2}$ , or schemes defined over a field $k$ of characteristic $0$ to the ring $k[\varepsilon ]$ where $\varepsilon ^{2}=0$ . The cotangent complex $\mathbf {L} _{X/S}^{\bullet }$ then controls the information related to this problem. We can reformulate it as considering the set of extensions of the commutative diagram

${\begin{matrix}0&\to &{\mathcal {G}}&\to &{\mathcal {O}}_{X'}&\to &{\mathcal {O}}_{X}&\to &0\\&&\uparrow &&\uparrow &&\uparrow \\0&\to &{\mathcal {I}}&\to &{\mathcal {O}}_{S'}&\to &{\mathcal {O}}_{S}&\to &0\end{matrix}}$

which is a homological problem. Then, the set of such diagrams whose kernel is ${\mathcal {G}}$ is isomorphic to the abelian group

${\text{Ext}}^{1}(\mathbf {L} _{X/S}^{\bullet },{\mathcal {G}})$

showing the cotangent complex controls the set of deformations available.[1] Furthermore, from the other direction, if there is a short exact sequence

${\begin{matrix}0&\to &{\mathcal {G}}&\to &{\mathcal {O}}_{X'}&\to &{\mathcal {O}}_{X}&\to &0\end{matrix}}$

there exists a corresponding element

$\xi \in {\text{Ext}}^{2}(\mathbf {L} _{X/S}^{\bullet },{\mathcal {G}})$

whose vanishing implies it is a solution to the deformation problem given above. Furthermore, the group

${\text{Ext}}^{0}(\mathbf {L} _{X/S}^{\bullet },{\mathcal {G}})$

controls the set of automorphisms for any fixed solution to the deformation problem.

Some important implications

One of the most geometrically important properties of the cotangent complex is the fact given a morphism of $S$ -schemes

$f:X\to Y$

we can form the relative cotangent complex $\mathbf {L} _{X/Y}^{\bullet }$ as the cone of

$f^{*}\mathbf {L} _{Y/S}^{\bullet }\to \mathbf {L} _{X/S}^{\bullet }$

fitting into a distinguished triangle

$f^{*}\mathbf {L} _{Y/S}^{\bullet }\to \mathbf {L} _{X/S}^{\bullet }\to \mathbf {L} _{X/Y}^{\bullet }\xrightarrow {+1}$

This is one of the pillars for cotangent complexes because it implies the deformations of the morphism $f$ of $S$ -schemes is controlled by this complex. In particular, $\mathbf {L} _{X/Y}^{\bullet }$ controls deformations of $f$ as a fixed morphism in ${\text{Hom}}_{S}(X,Y)$ , deformations of $X$ which can extend $f$ , meaning there is a morphism $f':X'\to S$ which factors through the projection map $X'\to X$ composed with $f$ , and deformations of $Y$ defined similarly. This is a powerful technique and is foundational to Gromov-Witten theory (see below), which studies morphisms from algebraic curves of a fixed genus and fixed number of punctures to a scheme $X$ .

Properties of the cotangent complex

Flat base change

Suppose that B and C are A-algebras such that $\operatorname {Tor} _{q}^{A}(B,C)=0$ for all q > 0. Then there are quasi-isomorphisms[6]

{\begin{aligned}L^{B\otimes _{A}C/C}&\cong C\otimes _{A}L^{B/A}\\L^{B\otimes _{A}C/A}&\cong \left(L^{B/A}\otimes _{A}C\right)\oplus \left(B\otimes _{A}L^{C/A}\right)\end{aligned}}

If C is a flat A-algebra, then the condition that $\operatorname {Tor} _{q}^{A}(B,C)$ vanishes for q > 0 is automatic. The first formula then proves that the construction of the cotangent complex is local on the base in the flat topology.

Vanishing properties

Let f : A → B. Then:[7][8]

If B is a localization of A, then $L_{B/A}\simeq 0$ .
If f is an étale morphism, then $L_{B/A}\simeq 0$ .
If f is a smooth morphism, then $L_{B/A}$ is quasi-isomorphic to $\Omega _{B/A}$ . In particular, it has projective dimension zero.
If f is a local complete intersection morphism, then $L_{B/A}$ has projective dimension at most one.
If A is Noetherian, $B=A/I$ , and $I$ is generated by a regular sequence, then $I/I^{2}$ is a projective module and $L_{B/A}$ is quasi-isomorphic to $I/I^{2}[1].$

Finite amplitude

It was a conjecture of Quillen[9] who asked if the cotangent complex of a finite-type $A$ -algebra $B$ has finite amplitude, meaning it's cohomology is concentrated in finitely many degrees, then $B$ is necessarily a local complete intersection $A$ -algebra. This turned out to be true as proved by Avramov.[9]

Examples

Smooth schemes

Let $X\in \operatorname {Sch} /S$ be smooth. Then the cotangent complex is $\Omega _{X/S}$ . In Berthelot's framework, this is clear by taking $V=X$ . In general, étale locally on $S,X$ is a finite dimensional affine space and the morphism $X\to S$ is projection, so we may reduce to the situation where $S=\operatorname {Spec} (A)$ and $X=\operatorname {Spec} (A[x_{1},\ldots ,x_{n}]).$ We can take the resolution of $\operatorname {Spec} (A[x_{1},\ldots ,x_{n}])$ to be the identity map, and then it is clear that the cotangent complex is the same as the Kähler differentials.

Closed embeddings in smooth schemes

Let $i:X\to Y$ be a closed embedding of smooth schemes in ${\text{Sch}}/S$ . Using the exact triangle corresponding to the morphisms $X\to Y\to S$ , we may determine the cotangent complex $\mathbf {L} _{X/Y}$ . To do this, note that by the previous example, the cotangent complexes $\mathbf {L} _{X/S}$ and $\mathbf {L} _{Y/S}$ consist of the Kähler differentials $\Omega _{X/S}$ and $\Omega _{Y/S}$ in the zeroth degree, respectively, and are zero in all other degrees. The exact triangle implies that $\mathbf {L} _{X/Y}$ is nonzero only in the first degree, and in that degree, it is the kernel of the map $i^{*}\mathbf {L} _{X/S}\to \mathbf {L} _{Y/S}.$ This kernel is the conormal bundle, and the exact sequence is the conormal exact sequence, so in the first degree, $\mathbf {L} _{X/Y}$ is the conormal bundle $C_{X/Y}$ .

Local complete intersection

More generally, a local complete intersection morphism $X\to Y$ with a smooth target has a cotangent complex perfect in amplitude $[-1,0].$ This is given by the complex

$I/I^{2}\to \Omega _{Y}|_{X}.$

For example, the cotangent complex of the twisted cubic $X$ in $\mathbb {P} ^{3}$ is given by the complex

${\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2){\xrightarrow {s}}\Omega _{\mathbb {P} ^{3}}|_{X}.$

Cotangent complexes in Gromov-Witten theory

In Gromov–Witten theory mathematicians study the enumerative geometric invariants of n-pointed curves on spaces. In general, there are algebraic stacks

${\overline {\mathcal {M}}}_{g,n}(X,\beta )$

which are the moduli spaces of maps

$\pi :C\to X$

from genus $g$ curves with $n$ punctures to a fixed target. Since enumerative geometry studies the generic behavior of such maps, the deformation theory controlling these kinds of problems requires the deformation of the curve $C$ , the map $\pi$ , and the target space $X$ . Fortunately, all of this deformation theoretic information can be tracked by the cotangent complex $\mathbf {L} _{C/X}^{\bullet }$ . Using the distinguished triangle

$\pi ^{*}\mathbf {L} _{X}^{\bullet }\to \mathbf {L} _{C}^{\bullet }\to \mathbf {L} _{C/X}^{\bullet }\to$

associated to the composition of morphisms

$C\xrightarrow {\pi } X\rightarrow {\text{Spec}}(\mathbb {C} )$

the cotangent complex can be computed in many situations. In fact, for a complex manifold $X$ , its cotangent complex is given by $\Omega _{X}^{1}$ , and a smooth $n$ -punctured curve $C$ , this is given by $\Omega _{C}^{1}(p_{1}+\cdots +p_{n})$ . From general theory of triangulated categories, the cotangent complex $\mathbf {L} _{C/X}^{\bullet }$ is quasi-isomorphic to the cone

${\text{Cone}}(\pi ^{*}\mathbf {L} _{X}^{\bullet }\to \mathbf {L} _{C}^{\bullet })\simeq {\text{Cone}}(\pi ^{*}\Omega _{X}^{1}\to \Omega _{C}^{1}(p_{1}+\cdots +p_{n}))$

Notes

"Section 90.21 (08UX): Deformations of ringed spaces and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02.
"Section 90.23 (08V3): Deformations of ringed topoi and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02.
Grothendieck 1967, Proposition 17.2.5
Berthelot 1966, VIII Proposition 2.2
Berthelot 1966, VIII Proposition 2.4
Quillen 1970, Theorem 5.3
Quillen 1970, Theorem 5.4
Quillen 1970, Corollary 6.14
Avramov, Luchezar L. (1999-08-31). "Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology". arXiv:math/9909192.

References

Applications

https://mathoverflow.net/questions/372128/what-is-the-cotangent-complex-good-for

Generalizations

References

André, M. (1974), Homologie des Algèbres Commutatives, Grundlehren der mathematischen Wissenschaften, vol. 206, Springer-Verlag
Berthelot, Pierre (1971), Grothendieck, Alexandre; Illusie, Luc (eds.), Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French), Berlin; New York: Springer-Verlag, xii+700
Grothendieck, Alexandre; Dieudonné, Jean (1967), "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie", Publications Mathématiques de l'IHÉS, 32: 5–361, doi:10.1007/BF02732123, ISSN 1618-1913, S2CID 189794756
Grothendieck, Alexandre (January 7, 1969), Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Mathematics 79 (in French), Berlin, New York: Springer-Verlag, ISBN 978-3-540-04248-8
Harrison, D. K. (1962), "Commutative algebras and cohomology", Transactions of the American Mathematical Society, American Mathematical Society, 104 (2): 191–204, doi:10.2307/1993575, JSTOR 1993575
Illusie, Luc (2009) [1971], Complexe Cotangent et Déformations I, Lecture Notes in Mathematics 239 (in French), Berlin, New York: Springer-Verlag, ISBN 978-3-540-05686-7
Lichtenbaum; Schlessinger (1967), "The cotangent complex of a morphism", Transactions of the American Mathematical Society, 128: 41–70, doi:10.1090/s0002-9947-1967-0209339-1
Quillen, Daniel (1970), On the (co-)homology of commutative rings, Proc. Symp. Pure Mat., vol. XVII, American Mathematical Society

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[:0-1] "Section 90.21 (08UX): Deformations of ringed spaces and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02.

[2] "Section 90.23 (08V3): Deformations of ringed topoi and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02.

[3] Grothendieck 1967, Proposition 17.2.5

[4] Berthelot 1966, VIII Proposition 2.2

[5] Berthelot 1966, VIII Proposition 2.4

[6] Quillen 1970, Theorem 5.3

[7] Quillen 1970, Theorem 5.4

[8] Quillen 1970, Corollary 6.14

[:1-9] Avramov, Luchezar L. (1999-08-31). "Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology". arXiv:math/9909192.

Cotangent complex

Motivation

Early work on cotangent complexes

The definition of the cotangent complex

Cotangent complexes in deformation theory

Setup

Some important implications

Properties of the cotangent complex

Flat base change

Vanishing properties

Finite amplitude

Examples

Smooth schemes

Closed embeddings in smooth schemes

Local complete intersection

Cotangent complexes in Gromov-Witten theory

See also

Notes

References

Applications

Generalizations

References