nLab algebra of functions




Algebra of function on a set

For RR a ring and SS a set, the set of functions SRS \to R (to the underlying set of RR) is itself naturally an associative algebra over RR, where addition and multiplication is given pointwise in SS by addition and multiplication in RR: for f 1,f 2:SRf_1, f_2 \colon S \to R their sum is the function

(f 1+f 2):sf 1(s)+f 2(s)R, (f_1 + f_2) \colon s \mapsto f_1(s) + f_2(s) \in R \,,

their product is the function

(f 1f 2):sf 1(s)f 2(s)R (f_1 \cdot f_2) \colon s \mapsto f_1(s) f_2(s) \in R

and the ring inclusion R[S,R]R \to [S,R] is given by sending rRr \in R to the constant function with value rr.

Algebra of functions on an \infty-stack

More generally, in the context of (∞,1)-topos theory and higher algebra, there is a notion of function algebras on ∞-stacks.


Relation to free modules

If SS is a finite set or else if one restricts to functions that are non-vanishing only for finitely many elements in SS, then the algebra of functions with values in RR also forms the free module over RR generated by SS.

Hadamard product

If SS is a set and RR is a commutative ring, then the pointwise multiplication on the function algebra SRS \to R is the Hadamard product on the function algebra.

Duality between algebra and geometry

Sending spaces to their suitable algebras of functions constitutes a basic duality operation that relates geometry and algebra. For more on this see at Isbell duality.

Last revised on August 2, 2023 at 12:19:02. See the history of this page for a list of all contributions to it.