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• Tags
  • #AG/schemes
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  • #todo/add-references
• Links:
  • coalgebra
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algebra over a ring

Definition: an \(R{\hbox{-}}\)algebra is a unital ring \(A\) with a ring morphism \(R\xrightarrow{f} A\) with \(1_R \mapsto 1_A\) such that \(\operatorname{im}f \subseteq Z(A)\).

Equivalently, \(A\) is an \(R{\hbox{-}}\)module with a compatible ring structure, i.e. a product \((a_1, a_2) \mapsto a_1 . a_2\) such that \(r(a_1.a_2) = (ra_1).a_2 = a_1.(ra_2)\),

An \(A\)-algebra \(B\) is the same data as a ring map \(A \rightarrow B\). Ways to remember which direction this should go:

• $\mathsf{CRing}\cong \mathsf{CRing}{{{\mathbf{Z}}/}} \cong {}{{\mathbf{Z}}} \mathsf{Alg} $ where \({\mathbf{Z}}\to {\mathbf{R}}\) is determined by \(1\mapsto 1_R\).
• \(M \coloneqq\operatorname{Mat}_{n\times n}(R)\) is an \(R{\hbox{-}}\)algebra where \(f: R\to M\) by \(r\mapsto \operatorname{diag}(r,r,c\dots, r) \in Z(M)\).
• Field extensions \(L/K\) admit ring morphisms \(K\to L\) making \(L\) a \(K{\hbox{-}}\)algebra.