I'm working in a problem from Hartshorne Algebraic Geometry. But I need a result from Commutative Algebra.

Given a commutative ring $B$ with $1$. For each $b \in B$ define the ring $B_b$ as the localizated ring by the multiplicative set $\{b^n:n\ge 1 \}$.

Given a commutative ring $A$ and elements $f,g \in A$ and some $n_0 \in \mathbb{N}$ I want to know if it's true that $(A_f)_{g/f^{n_0}}\cong A_{fg}$.

I tried to construct an isomorphism but I could not )= If this result is true, and I have an explicit isomorphism, then I'll be able to prove that $(X_f,O_X|_{X_f})\cong (Spec(A_f),O_{Spec(A_f)})$ as locally ringed spaces.

But I need to prove the above result. Thanks


1 Answer 1



Note that $A_{fg}\cong (A_f)_g$. This can be proven by showing that both $A\rightarrow A_{fg}$ and $A\rightarrow A_f \rightarrow (A_f)_g$ are the universal localisation maps wrt $fg$.

Then you need to show that $(A_f)_{g/f^{n_0}}\cong (A_f)_g$ for any $n_0$. This can be done with the same reasoning on the maps $A_f \rightarrow(A_f)_{g/f^{n_0}}$ and $A_f \rightarrow(A_f)_g$.

Both isomorphisms are obtained by appealing to the universal property repeatedly.

Let me know if you still do not get it, ill work this out completely.


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