Dependencies

If \(X_v\) and \(Y_v\) are families of topological spaces indexed by \(v\in V\), if \(f_v:X_v\to Y_v\) is a continuous map sending the subset \(C_v\subseteq X_v\) into \(D_v\subseteq Y_v\) then there’s an induced continuous map \(\prod '_v(X_v,C_v)\to \prod '_v(Y_v,D_v)\).

If all the \(f_v\) are homeomorphisms identifying \(C_v\) with \(D_v\) then the induced map on restricted products is also a homeomorphism (proof: apply the previous construction to \(f_v\) and their inverses)

In the same setup as definition 8.26 (\(V,W\) index sets, \(f:W\to V\), \(C_v\subseteq X_v\) and \(D_w\subseteq Y_w\), bijections \(b_v:X_v\to \prod _{w:f(w)=v}Y_w\) identifying \(C_v\) with \(\prod _{w:f(w)=v}D_w\)), if all the \(X_v\) and \(Y_w\) are furthermore topological spaces, all the \(C_v\) and \(D_w\) are open, and all the \(b_v\) are homeomorphisms, then the induced map \(\prod '_v(X_v,C_v)\to \prod '_w(Y_w,D_w)\) is also a homeomorphism.

\(\mathbb {\mathbb {A}_L^\infty }\) is homeomorphic to \(\prod _v(B\otimes _AK_v,B\otimes _AA_v)\).

If \(X_v\) and \(Y_v\) are topological spaces with open subspaces \(C_v\) and \(D_v\), then the obvious bijection \(\prod '_v(X_v \times Y_v,C_v\times D_v) \cong \left(\prod '_v(X_v,C_v)\right)\times \left(\prod '_v(Y_v,D_v)\right)\) is a homeomorphism, where the restricted products have the restricted product topology and the binary product has the product topology.

\(B\) is a finitely-presented \(A\)-module.

The \(A\)-bilinear map \(B\times K\to L\) sending \((b,k)\) to \(bk\) is surjective.

If \(M\) is any \(K\)-module then the canonical map \(B\otimes _A M\to L\otimes _K M\) is an isomorphism.

If \(0\not=b\in B\) then there exists \(0\not=a\in A\) such that \(b\) divides the image of \(a\) in \(B\).

This natural map \(L\otimes _K\mathbb {A}_{A,K}^\infty \to \mathbb {A}_{B,L}^\infty \) is an isomorphism.

The induced \(L\)-algebra morphism \(L\otimes _K\mathbb {A}_{A,K}^\infty \to \mathbb {A}_{B,L}^\infty \) is a topological isomorphism.

The natural map \(B\otimes _A\mathbb {A}_K^\infty \to \mathbb {A}_L^\infty \) is a \(B\)-algebra isomorphism.

The natural map \(B\otimes _A\mathbb {A}_K^\infty \to R\) is a \(B\)-algebra isomorphism.

The ring \(R\) introduced above (the restricted product of the \(B\otimes _A K_v\) with respect to the \(B\otimes _A A_v\)) is isomorphic to \(\mathbb {A}_L\).

There’s a natural ring homomorphism \(\mathbb {A}_{A,K}^\infty \to \mathbb {A}_{B,L}^\infty \) lying over \(K\to L\).

The natural map \(B\otimes _AK\to L\) is a \(B\)-algebra isomorphism.

Giving \(L_w\) the \(K_v\)-algebra structure coming from the natural map \(K_v\to L_w\), the \(w\)-adic topology on \(L_w\) is the \(K_v\)-module topology.

The induced \(L\)-algebra homomorphism \(L\otimes _KK_v\to \prod _{w|v}L_w\) is an isomorphism of rings.

The isomorphism \(L\otimes _KK_v\to \prod _{w|v}L_w\) induces an isomorphism \(B\otimes _AA_v\to \prod _{w|v}B_w\) for all \(v\) in the height one spectrum of \(A\).

If we give \(L\otimes _KK_v\) the \(K_v\)-module topology then the \(L\)-algebra isomorphism \(L\otimes _K K_v\cong \prod _{w|v}L_w\) is also a homeomorphism.

There’s a natural ring map \(K_v\to L_w\) extending the map \(K\to L\). It is defined by completing the inclusion \(K\to L\) at the finite places \(v\) and \(w\) (which can be done because the previous lemma shows that the map is uniformly continuous for the \(v\)-adic and \(w\)-adic topologies).

The product of the maps \(K_v\to L_w\) for \(w|v\) is a natural ring map \(K_v\to \prod _{w|v}L_w\) lying over \(K\to L\).

The map \(i_v:K_v\to L_w\) sends the integer ring \(A_v\) into \(B_w\).

There are only finitely many primes \(w\) of \(B\) lying above \(v\).

For \(v\) fixed, the product topology on \(\prod _{w|v}L_w\) is the \(K_v\)-module topology.

If \(i:K\to L\) denotes the inclusion then for \(k\in K\) we have \(e\times w(i(k))=v(k)\), where \(e\) is the ramification index of \(w/v\) (recall that valuations here are written additively, unlike in mathlib).

If \(i_v:K_v\to L_w\) denotes the map of the previous definition then for \(x\in K_v\) we have \(e\times w(i(k))=v(k)\), where \(e\) is the ramification index of \(w/v\).

If \(S\) is a finite set of nonzero primes of \(A\) then the natural map \(B\otimes ((\prod _{v\in S}K_v)\times (\prod _{v\notin S}A_v))\to (\prod _{v\in S}(B\otimes _AK_v))\times (\prod _{v\notin S}(B\otimes _AA_v))\) is an isomorphism.

If \(K\to L\) is a ring homomorphism between two number fields then the topology on \(\mathbb {A}_L\) is the \(\mathbb {A}_K\)-module topology, where the module structure comes from the natural map \(\mathbb {A}_K\to \mathbb {A}_L\).

If \(K\to L\) is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) \(L\otimes _K\mathbb {A}_K\cong \mathbb {A}_L\).

The quotient \(\mathbb {A}_K/K\) is compact.

The additive subgroup \(K\) of \(\mathbb {A}_K\) is discrete.

The adeles of a number field are locally compact.

There’s an open subset of \(\mathbb {A}_{K}\) whose intersection with \(K\) is \(\{ 0\} \).

If \(K\to L\) is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) \(L\otimes _KK_\infty \cong L_\infty \).

There is a natural \(L\)-algebra isomorphism \(L \otimes _K K_{\infty } \cong _L L_{\infty }\).

\(L_{\infty }\) has the \(K_{\infty }\)-module topology.

There is a natural \(K_{\infty }\)-linear homeomorphism \(K_{\infty }^{[L:K]} \cong _{K_{\infty }} L_{\infty }\).

Let \(v\) be an infinite place of \(K\). There is a natural \(L\)-algebra homomorphism \(L\otimes _K K_v \to \prod _{w\mid v}L_w\), whose restriction to \(1\otimes _K K_v\) corresponds to the map in 8.36.

For a fixed infinite place \(v\) of \(K\), the map \(L\otimes _K K_v \to \prod _{w\mid v}L_w\) is injective.

For a fixed infinite place \(v\) of \(K\), the map \(L\otimes _K K_v \to \prod _{w\mid v}L_w\) is surjective.

Let \(v\) be an infinite place of \(K\). There is a natural \(L\)-algebra homeomorphism \(L\otimes _K K_v \cong _L \prod _{w\mid v}L_w\), whose restriction to \(1\otimes _K K_v\) corresponds to the map in 8.36.

Let \(S\) be a set of infinite places of \(K\). The image of \(K\) under the embedding \(K\hookrightarrow (K_v)_{v\in S}; k \mapsto (k)_v\) is dense in the product topology.

For a fixed infinite place \(v\) of \(K\), we have \(\text{dim}_{K_v} \prod _{w\mid v} L_w = \text{dim}_{K_v} L\otimes _K K_v\).

If \(w \mid v\) is an infinite place of \(L\) lying above the infinite place \(v\) of \(K\), then \(L_w\) has the \(K_v\)-module topology.

Let \(v\) be an infinite place of \(K\). There is a natural \(K_v\)-linear homeomorphism \(K_v^{[L:K]} \cong _{K_v} \prod _{w\mid v}L_w\).

Let \(v\) be an infinite place of \(K\). There is a continuous \(K\)-algebra homomorphism \(K_v \to \prod _{w\mid v}L_w\), whose restriction to \(K\) corresponds to the global embedding of \(K\) into \((L_w)_w\).

If \(K\) is a number field and \(v\) is a nonzero prime ideal of the integers of \(K\), then the integers of \(K_v\) is a compact open subgroup.

If \(R\) is a commutative ring, if \(M\) is a finitely presented \(R\)-module and if \(N_i\) are a collection of \(R\)-modules, then the canonical map \(M\otimes _R\prod _i N_i\to \prod _i(M\otimes _R N_i)\) is an isomorphism.

The quotient \(\mathbb {A}_{\mathbb {Q}}/\mathbb {Q}\) is compact.

There’s an open subset of \(\mathbb {A}_{\mathbb {Q}}\) whose intersection with \(\mathbb {Q}\) is \(\{ 0\} \).

Let \(V\) and \(W\) be index sets, and let \(f:W\to V\) be a map with finite fibres. Let \(X_v\) be sets, with subsets \(C_v\), let \(Y_w\) be sets with subsets \(D_w\), and say for all \(v\in V\) we’re given a bijection \(X_v\to \prod _{w|f(w)=v}Y_w\), identifying \(C_v\) with \(\prod _{w:f(w)=v}D_w\). Then there’s an induced bijection between the restricted products \(\prod '_v(X_v,C_v)\) and \(\prod '_w(Y_w,D_w)\).