Contents

Definition

Algebra of function on a set

For $RR$ a ring and $SS$ a set, the set of functions $S\to RS\; \backslash to\; R$ (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 $f_1,f_2:S\to Rf\_1,\; f\_2\; \backslash colon\; S\; \backslash to\; R$ their sum is the function

$$(f_1+f_2):s\mapsto f_1(s)+f_2(s)\in R,(f\_1\; +\; f\_2)\; \backslash colon\; s\; \backslash mapsto\; f\_1(s)\; +\; f\_2(s)\; \backslash in\; R\; \backslash ,,$$

their product is the function

$$(f_1\cdot f_2):s\mapsto f_1(s)f_2(s)\in R(f\_1\; \backslash cdot\; f\_2)\; \backslash colon\; s\; \backslash mapsto\; f\_1(s)\; f\_2(s)\; \backslash in\; R$$

and the ring inclusion $R\to [S,R]R\; \backslash to\; [S,R]$ is given by sending $r\in Rr\; \backslash in\; R$ to the constant function with value $rr$.

Algebra of functions on an $\mathrm{\infty }\backslash infty$-stack

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

Properties

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 $S\to RS\; \backslash to\; R$ 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.

• pullback of functions
• composition ring