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
is a ring and
are modules over that ring (in fact,
is a graded ring over
A topology-counterpart of this notion is a commutative ring spectrum.
Let A be a simplicial commutative ring.
Then the ring structure of A gives
the structure of a graded-commutative graded ring as follows.
is the homology of the chain complex corresponding to A; in particular, it is a graded abelian group.
Next, to multiply two elements, writing
for the simplicial circle, let
Then the composition the second map the multiplication of A, induces
This in turn gives an element in
We have thus defined the graded multiplication
It is associative because the smash product is.
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
has the structure of a graded module over
Module spectrum).
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
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