algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, a simplicial commutative ring is a

commutative monoid In abstract algebra, 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 . Monoids are semigroups with identity ...

in the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

of simplicial abelian groups, or, equivalently, a

simplicial object In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...

in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be shown that

\pi_0 A

is a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

and

\pi_i A

are modules 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 . The index set is usually the set of nonnegative integers or the set of integers, but ...

over

\pi_0 A

.) A

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

-counterpart of this notion is a

commutative ring spectrum In 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 choice is the category of ...

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

\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 contained in the kernel o ...

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

S^1

for the simplicial circle, 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 that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

because the smash product is. It is

graded-commutative In Abstract algebra, algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'', and , ''y'', ...

(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 simplicial abelian group 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 (topology), spectrum with an action of a ring spectrum; it generalizes a module (mathematics), module in abstract algebra. The ∞-category of (say right) module spectra is stable ∞-category, stable; ...

Spec

By definition, the category of affine derived schemes is the

opposite category In category theory, a branch of mathematics, the opposite category or dual category C^ of a given Category (mathematics), category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal ...

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
Commutative algebra Ring theory Algebraic structures {{commutative-algebra-stub

Examples

Graded ring structure

Spec

See also

References