- Source: Simplicial commutative ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If A is a simplicial commutative ring, then it can be shown that
π
0
A
{\displaystyle \pi _{0}A}
is a ring and
π
i
A
{\displaystyle \pi _{i}A}
are modules over that ring (in fact,
π
∗
A
{\displaystyle \pi _{*}A}
is a graded ring over
π
0
A
{\displaystyle \pi _{0}A}
.)
A topology-counterpart of this notion is a commutative ring spectrum.
Examples
The ring of polynomial differential forms on simplexes.
Graded ring structure
Let A be a simplicial commutative ring. Then the ring structure of A gives
π
∗
A
=
⊕
i
≥
0
π
i
A
{\displaystyle \pi _{*}A=\oplus _{i\geq 0}\pi _{i}A}
the structure of a graded-commutative graded ring as follows.
By the Dold–Kan correspondence,
π
∗
A
{\displaystyle \pi _{*}A}
is the homology of the chain complex corresponding to A; in particular, it is a graded abelian group. Next, to multiply two elements, writing
S
1
{\displaystyle S^{1}}
for the simplicial circle, let
x
:
(
S
1
)
∧
i
→
A
,
y
:
(
S
1
)
∧
j
→
A
{\displaystyle x:(S^{1})^{\wedge i}\to A,\,\,y:(S^{1})^{\wedge j}\to A}
be two maps. Then the composition
(
S
1
)
∧
i
×
(
S
1
)
∧
j
→
A
×
A
→
A
{\displaystyle (S^{1})^{\wedge i}\times (S^{1})^{\wedge j}\to A\times A\to A}
,
the second map the multiplication of A, induces
(
S
1
)
∧
i
∧
(
S
1
)
∧
j
→
A
{\displaystyle (S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A}
. This in turn gives an element in
π
i
+
j
A
{\displaystyle \pi _{i+j}A}
. We have thus defined the graded multiplication
π
i
A
×
π
j
A
→
π
i
+
j
A
{\displaystyle \pi _{i}A\times \pi _{j}A\to \pi _{i+j}A}
. It is associative because the smash product is. It is graded-commutative (i.e.,
x
y
=
(
−
1
)
|
x
|
|
y
|
y
x
{\displaystyle xy=(-1)^{|x||y|}yx}
) since the involution
S
1
∧
S
1
→
S
1
∧
S
1
{\displaystyle S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}}
introduces a minus sign.
If M is a simplicial module over A (that is, M is a simplicial abelian group with an action of A), then the similar argument shows that
π
∗
M
{\displaystyle \pi _{*}M}
has the structure of a graded module over
π
∗
A
{\displaystyle \pi _{*}A}
(cf. Module spectrum).
Spec
By definition, the category of affine derived schemes is the opposite category of the category of simplicial commutative rings; an object corresponding to A will be denoted by
Spec
A
{\displaystyle \operatorname {Spec} A}
.
See also
E_n-ring
References
What is a simplicial commutative ring from the point of view of homotopy theory?
What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?
Reference request - CDGA vs. sAlg in char. 0
A. Mathew, Simplicial commutative rings, I.
B. Toën, Simplicial presheaves and derived algebraic geometry
P. Goerss and K. Schemmerhorn, Model categories and simplicial methods