Dependencies

Let $h:A\to B$ be a homomorphism between commutative rings and suppose $O\subseteq Spec(A)$ is quasi-compact and open. Then $Spec(h{)}^{-1}(O)$ is quasi-compact.

Let $\mathit{A}=(X,A)$ and ${\mathit{A}}^{\mathbf{\prime }}=(X^{\prime },A^{\prime })$ be springs with indices $v$ and $v^{\prime }$ respectively. Let $(f,h)$ be an indexed spring morphism from $(X,A)$ to $(X^{\prime },A^{\prime })$. Suppose $(a,b)\in G({\mathit{A}}^{\mathbf{\prime }},v^{\prime })$ and let $h_1$ be the ring homomorphism constructed in Proposition 5. Then $(f,h_1)$ is an indexed spring morphism from $\mathit{A}[h(a)\mathrm{\#}h(b)],v$ to ${\mathit{A}}^{\mathbf{\prime }}[a\mathrm{\#}b],v^{\prime }$.

Let $\mathit{A}=(X,A)$ and ${\mathit{A}}^{\mathbf{\prime }}=(X^{\prime },A^{\prime })$ be springs with indices $v$ and $v^{\prime }$ respectively. Let $(f,h)$ be an indexed spring morphism from $(X,A)$ to $(X^{\prime },A^{\prime })$. Suppose $(a,b)\in G({\mathit{A}}^{\mathbf{\prime }},v^{\prime })$, ${\mathit{A}}^{\mathbf{\prime }}{}^{\mathbf{\prime }}$ and ${\mathit{A}}^{\mathbf{\prime }}{}^{\mathbf{\prime }}{}^{\mathbf{\prime }}$ extend $\mathit{A}$ and ${\mathit{A}}^{\mathbf{\prime }}[a\mathrm{\#}b]$ respectively, and $h_2:A^{\prime }{}^{\prime }{}^{\prime }\to A^{\prime }{}^{\prime }$ is a ring homomorphism extending $h$. If $(f,h_2)$ is a spring morphism from ${\mathit{A}}^{\mathbf{\prime }}{}^{\mathbf{\prime }}$ to ${\mathit{A}}^{\mathbf{\prime }}{}^{\mathbf{\prime }}{}^{\mathbf{\prime }}$, then $h(a)\mathrm{\#}h(b)=h_2(a\mathrm{\#}b)$, meaning that $h(a)\mathrm{\#}h(b)$ belongs to $A^{\prime }{}^{\prime }$.

Let $\mathit{A}=(X,A)$ and ${\mathit{A}}^{\mathbf{\prime }}=(X^{\prime },A^{\prime })$ be springs with indices $v$ and $v^{\prime }$ respectively. Let $(f,h)$ be an indexed spring morphism from $\mathit{A}$ to ${\mathit{A}}^{\mathbf{\prime }}$. Suppose $(a,b)\in G({\mathit{A}}^{\mathbf{\prime }},v^{\prime })$. Then there exists a unique ring homomorphism $h_1$ from $A^{\prime }[a\mathrm{\#}b]$ to $A[h(a),h(b)]$ that extends $h$ and maps $a\mathrm{\#}b$ to $h(b)\mathrm{\#}h(b)$.

Let $(f,h)$ be a spring morphism from $(X,A)$ to $(X^{\prime },A^{\prime })$. Suppose $x\in X$ and $c\in A^{\prime }$. Then $x\in d(h(c))$ if and only if $f(x)\in d(c)$.

Let $\mathit{A}=(X,A)$ and ${\mathit{A}}^{\mathbf{\prime }}=(X^{\prime },A^{\prime })$ be springs with indices $v$ and $v^{\prime }$ respectively. Let $(f,h)$ be an indexed spring homomorphism from $\mathit{A}$ to ${\mathit{A}}^{\mathbf{\prime }}$. If $(a,b)$ belongs to $G({\mathit{A}}^{\mathbf{\prime }},v^{\prime })$, then (h(a),h(b)) belongs to $G(\mathit{A},v)$.

Let $n$ be a positive integer and let $(X,A)$ be a spring. Suppose $a,b\in A$ satisfy $z(b)\subseteq z(a)$. Then $z(b^n)\subseteq z(a^n)$ and $(a\mathrm{\#}b{)}^n=a^n\mathrm{\#}{b}^n$.