algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

, a simplicial commutative ring is a

commutative monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...

in the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

simplicial abelian group In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial grou ...

s, or, equivalently, a

simplicial object In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...

in the

category of commutative rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is ...

. If ''A'' is a simplicial commutative ring, then it can be shown that

\pi_0 A

is a ring and

\pi_i A

are

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...

over that ring (in fact,

\pi_* A

is a

graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...

over

\pi_0 A

.) A

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

-counterpart of this notion is a

commutative ring spectrum In the mathematical field of algebraic topology, a commutative ring spectrum, roughly equivalent to a E_\infty-ring spectrum, is a commutative monoid in a goodsymmetric monoidal with respect to smash product and perhaps some other conditions; one ...

Examples

*The ring of

polynomial differential form In algebra, the ring of polynomial differential forms on the standard ''n''-simplex is the differential graded algebra: :\Omega^*_( = \mathbb _0, ..., t_n, dt_0, ..., dt_n(\sum t_i - 1, \sum dt_i). Varying ''n'', it determines the simplicial comm ...

s on simplexes.

Graded ring structure

Let ''A'' be a simplicial commutative ring. Then the ring structure of ''A'' gives

\pi_* A = \oplus_ \pi_i A

the structure of a graded-commutative graded ring as follows. By the

Dold–Kan correspondence In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the ...

\pi_* A

is the homology of the

chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...

corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing

S^1

for the

simplicial circle In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

, let

x:(S^1)^ \to A, \, \, y:(S^1)^ \to A

be two maps. Then the composition :

(S^1)^ \times (S^1)^ \to A \times A \to A

, the second map the multiplication of ''A'', induces

(S^1)^ \wedge (S^1)^ \to A

. This in turn gives an element in

\pi_ A

. We have thus defined the graded multiplication

\pi_i A \times \pi_j A \to \pi_ A

. It is

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

because the smash product is. It is graded-commutative (i.e.,

xy = (-1)^ yx

) since the involution

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

with an action of ''A''), then the similar argument shows that

\pi_* M

has the structure of a graded module over

\pi_* A

(cf.

Module spectrum In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra. The ∞-category of (say right) module spectra is stable A stable is a building in which livestock, especially horses ...

Spec

By definition, the category of affine

derived scheme In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) ...

s is the

opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...

of the category of simplicial commutative rings; an object corresponding to ''A'' will be denoted by

\operatorname A

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
{{algebra-stub Commutative algebra Ring theory Algebraic structures

Examples

Graded ring structure

Spec

See also

References