Partition of Unity for a commutative ring

Question

The Partition of Unity is an important theorem in geometry. Has this theorem any interpretation for a commutative ring with respect to the Zariski topology?

Answer 1

When one talks about partition of unity for a commutative ring, one means a sequence $f_1,\ldots,f_n$ of elements of the ring such that $(f_1,\ldots,f_n)$ is the unit ideal

Answer 2

Yes, it has. Try to solve the following problems and you will see how it naturally appears.

• Show that if $U_{f_i}$ cover $Spec(A)$ and $g, h \in A$ are such that $g = h \in A_{f_i}$ for all $f_i$, then $g=h$. Thus $g \in A$ is determined by its restrictions to an open cover.
• Now let $g_i \in A_{f_i}$ be such that $g_i = g_j \in A_{f_i, f_j}$ for all $i, j$. Show that there exists $g \in A$ such that $g = g_i \in A_{f_i}$. Such $g$ is unique.

Answer 3

I don't know the story for general commutative rings, but for affine $k$-algebras (or affine algebraic varieties, equivalently) I think of Nullstellensatz as exactly the analogue of partition of unity.

The relevant version is, if $f_1, \cdots, f_n$ are regular functions on the affine algebraic variety $X$ with no common zeroes, then there exists $g_1, \cdots, g_n$ such that $\sum f_i g_i = 1$. Which is to say $(f_1, \cdots, f_n)$ is the unital ideal.

To see why this is analogous to the partition of unity, suppose you look at $U_i = X - \mathcal{Z}(f_i)$. By hypothesis these cover $X$. $h_i = f_i g_i$ are then "the bump functions" on $U_i$ which sum to unity.

Like partition of unity, you can use Nullstellensatz to patch local things to global ones. Indeed, that the structure sheaf of an affine algebraic variety actually satisfies the gluing axiom is a corollary of Nullstellensatz.