The Stacks project

Situation 47.13.6. Let $R \to A$ be a ring map. We will give an alternative construction of $R\mathop{\mathrm{Hom}}\nolimits (A, -)$ which will stand us in good stead later in this chapter. Namely, suppose we have a differential graded algebra $(E, d)$ over $R$ and a quasi-isomorphism $E \to A$ where we view $A$ as a differential graded algebra over $R$ with zero differential. Then we have commutative diagrams

\[ \vcenter { \xymatrix{ D(E, \text{d}) \ar[rd] & & D(A) \ar[ll] \ar[ld] \\ & D(R) } } \quad \text{and}\quad \vcenter { \xymatrix{ D(E, \text{d}) \ar[rr]_{- \otimes _ E^\mathbf {L} A} & & D(A) \\ & D(R) \ar[lu]^{- \otimes _ R^\mathbf {L} E} \ar[ru]_{- \otimes _ R^\mathbf {L} A} } } \]

where the horizontal arrows are equivalences of categories (Differential Graded Algebra, Lemma 22.37.1). It is clear that the first diagram commutes. The second diagram commutes because the first one does and our functors are their left adjoints (Differential Graded Algebra, Example 22.33.6) or because we have $E \otimes ^\mathbf {L}_ E A = E \otimes _ E A$ and we can use Differential Graded Algebra, Lemma 22.34.1.

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