- Source: Derived tensor product
In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is
−
⊗
A
L
−
:
D
(
M
A
)
×
D
(
A
M
)
→
D
(
R
M
)
{\displaystyle -\otimes _{A}^{\textbf {L}}-:D({\mathsf {M}}_{A})\times D({}_{A}{\mathsf {M}})\to D({}_{R}{\mathsf {M}})}
where
M
A
{\displaystyle {\mathsf {M}}_{A}}
and
A
M
{\displaystyle {}_{A}{\mathsf {M}}}
are the categories of right A-modules and left A-modules and D refers to the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor
−
⊗
A
−
:
M
A
×
A
M
→
R
M
{\displaystyle -\otimes _{A}-:{\mathsf {M}}_{A}\times {}_{A}{\mathsf {M}}\to {}_{R}{\mathsf {M}}}
.
Derived tensor product in derived ring theory
If R is an ordinary ring and M, N right and left modules over it, then, regarding them as discrete spectra, one can form the smash product of them:
M
⊗
R
L
N
{\displaystyle M\otimes _{R}^{L}N}
whose i-th homotopy is the i-th Tor:
π
i
(
M
⊗
R
L
N
)
=
Tor
i
R
(
M
,
N
)
{\displaystyle \pi _{i}(M\otimes _{R}^{L}N)=\operatorname {Tor} _{i}^{R}(M,N)}
.
It is called the derived tensor product of M and N. In particular,
π
0
(
M
⊗
R
L
N
)
{\displaystyle \pi _{0}(M\otimes _{R}^{L}N)}
is the usual tensor product of modules M and N over R.
Geometrically, the derived tensor product corresponds to the intersection product (of derived schemes).
Example: Let R be a simplicial commutative ring, Q(R) → R be a cofibrant replacement, and
Ω
Q
(
R
)
1
{\displaystyle \Omega _{Q(R)}^{1}}
be the module of Kähler differentials. Then
L
R
=
Ω
Q
(
R
)
1
⊗
Q
(
R
)
L
R
{\displaystyle \mathbb {L} _{R}=\Omega _{Q(R)}^{1}\otimes _{Q(R)}^{L}R}
is an R-module called the cotangent complex of R. It is functorial in R: each R → S gives rise to
L
R
→
L
S
{\displaystyle \mathbb {L} _{R}\to \mathbb {L} _{S}}
. Then, for each R → S, there is the cofiber sequence of S-modules
L
S
/
R
→
L
R
⊗
R
L
S
→
L
S
.
{\displaystyle \mathbb {L} _{S/R}\to \mathbb {L} _{R}\otimes _{R}^{L}S\to \mathbb {L} _{S}.}
The cofiber
L
S
/
R
{\displaystyle \mathbb {L} _{S/R}}
is called the relative cotangent complex.
See also
derived scheme (derived tensor product gives a derived version of a scheme-theoretic intersection.)
Notes
References
Lurie, J., Spectral Algebraic Geometry (under construction)
Lecture 4 of Part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry
Ch. 2.2. of Toen-Vezzosi's HAG II