In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the cotangent complex is a common generalisation of the
cotangent sheaf,
normal bundle and
virtual tangent bundle of a map of geometric spaces such as
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s or
schemes. If
is a
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
of geometric or algebraic objects, the corresponding cotangent complex
can be thought of as a universal "linearization" of it, which serves to control the
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...
of
.
It is constructed as an object in a certain
derived category of
sheaves on
using the methods of
homotopical algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a com ...
.
Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s,
Michel André and
Daniel Quillen independently came up with the correct definition for a morphism of
commutative rings, using
simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian)
left derived functor of
Kähler differentials.
Luc Illusie then globalized this definition to the general situation of a morphism of
ringed topoi, thereby incorporating morphisms of
ringed spaces,
schemes, and
algebraic spaces into the theory.
Motivation
Suppose that
and
are
algebraic varieties and that
is a morphism between them. The cotangent complex of
is a more universal version of the relative
Kähler differentials . The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If
is another variety, and if
is another morphism, then there is an exact sequence
:
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
and
imperfection modules. Most of these were motivated by
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...
.
This sequence is exact on the left if the morphism
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
:
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 In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if
*(i) it is locally of finite presentation
*(ii) it is flat, and
*(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular.
(iii) means ...
. If ''f'' is the inclusion of a smooth subvariety, then this sequence is a short exact sequence. 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
Cotangent complexes appeared in multiple and partially incompatible versions of increasing generality in the early 1960s. The first instance of the related homology functors in the restricted context of
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s appeared in Cartier (1956).
Alexander Grothendieck then developed an early version of cotangent complexes in 1961 for his general
Riemann-Roch theorem in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
in order to have a theory of
virtual tangent bundles. This is the version described by
Pierre Berthelot
Pierre Berthelot (; born 1943) is a mathematician at the University of Rennes. He developed crystalline cohomology and rigid cohomology.
Publications
*Berthelot, Pierre ''Cohomologie cristalline des schémas de caractéristique p>0.'' Lecture ...
in SGA 6, Exposé VIII. It only applies when ''f'' is a ''smoothable'' morphism (one that factors into a closed immersion followed by a smooth morphism). In this case, the cotangent complex of ''f'' as an object in the
derived category of
coherent sheaves on ''X'' is given as follows:
*
*If ''J'' is the ideal of ''X'' in ''V'', then
*
for all other ''i.''
*The differential
is the pullback along ''i'' of the inclusion of ''J'' in the structure sheaf
of ''V'' followed by the universal derivation
*All other differentials are zero.
This definition is independent of the choice of ''V,'' and for a smoothable complete intersection morphism, this complex is perfect. Furthermore, if is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an
exact triangle In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy categ ...
:
In 1963 Grothendieck developed a more general construction that removes the restriction to smoothable morphisms (which also works in contexts other than algebraic geometry). However, like the theory of 1961, this produced a cotangent complex of length 2 only, corresponding to the truncation
of the full complex which was not yet known at the time. This approach was published later in Grothendieck (1968). At the same time in the early 1960s, largely similar theories were independently introduced for commutative rings (corresponding to the "local" case of
affine scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
s in algebraic geometry) by
Gerstenhaber and
Lichtenbaum and
Schlessinger , he, ×©×œ×–×™× ×’×¨), Slesinger, Slezak, ÅšlÄ™zak
; Similar surnames:
* Schleußinger ( Schleussinger, Schleusinger, from Schleusingen)
, footnotes
Schlessinger is a German language, German surname meaning "from Silesia" (German: ''Schlesien ...
. Their theories extended to cotangent complexes of length 3, thus capturing more information.
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 more generally with simplicial ringed
topoi, thus covering "global" theory on various types of geometric spaces. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that
and
are
simplicial rings and that
is an
-algebra. Choose a resolution
of
by simplicial free
-algebras. Such a resolution of
can be constructed by using the free commutative
-algebra functor which takes a set
and yields the free
-algebra