Lemma 47.22.1. Let A \to B be a local homomorphism of Noetherian local rings. Let \omega _ A^\bullet be a normalized dualizing complex. If A \to B is flat and \mathfrak m_ A B = \mathfrak m_ B, then \omega _ A^\bullet \otimes _ A B is a normalized dualizing complex for B.
47.22 The ubiquity of dualizing complexes
Many Noetherian rings have dualizing complexes.
Proof. It is clear that \omega _ A^\bullet \otimes _ A B is in D^ b_{\textit{Coh}}(B). Let \kappa _ A and \kappa _ B be the residue fields of A and B. By More on Algebra, Lemma 15.99.2 we see that
R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa _ B, \omega _ A^\bullet \otimes _ A B) = R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa _ A, \omega _ A^\bullet ) \otimes _ A B = \kappa _ A[0] \otimes _ A B = \kappa _ B[0]
Thus \omega _ A^\bullet \otimes _ A B has finite injective dimension by More on Algebra, Lemma 15.69.7. Finally, we can use the same arguments to see that
R\mathop{\mathrm{Hom}}\nolimits _ B(\omega _ A^\bullet \otimes _ A B, \omega _ A^\bullet \otimes _ A B) = R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \otimes _ A B = A \otimes _ A B = B
as desired. \square
Lemma 47.22.2. Let A \to B be a flat map of Noetherian rings. Let I \subset A be an ideal such that A/I = B/IB and such that IB is contained in the Jacobson radical of B. Let \omega _ A^\bullet be a dualizing complex. Then \omega _ A^\bullet \otimes _ A B is a dualizing complex for B.
Proof. It is clear that \omega _ A^\bullet \otimes _ A B is in D^ b_{\textit{Coh}}(B). By More on Algebra, Lemma 15.99.2 we see that
R\mathop{\mathrm{Hom}}\nolimits _ B(K \otimes _ A B, \omega _ A^\bullet \otimes _ A B) = R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ) \otimes _ A B
for any K \in D^ b_{\textit{Coh}}(A). For any ideal IB \subset J \subset B there is a unique ideal I \subset J' \subset A such that A/J' \otimes _ A B = B/J. Thus \omega _ A^\bullet \otimes _ A B has finite injective dimension by More on Algebra, Lemma 15.69.6. Finally, we also have
R\mathop{\mathrm{Hom}}\nolimits _ B(\omega _ A^\bullet \otimes _ A B, \omega _ A^\bullet \otimes _ A B) = R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \otimes _ A B = A \otimes _ A B = B
as desired. \square
Lemma 47.22.3. Let A be a Noetherian ring and let I \subset A be an ideal. Let \omega _ A^\bullet be a dualizing complex.
\omega _ A^\bullet \otimes _ A A^ h is a dualizing complex on the henselization (A^ h, I^ h) of the pair (A, I),
\omega _ A^\bullet \otimes _ A A^\wedge is a dualizing complex on the I-adic completion A^\wedge , and
if A is local, then \omega _ A^\bullet \otimes _ A A^ h, resp. \omega _ A^\bullet \otimes _ A A^{sh} is a dualzing complex on the henselization, resp. strict henselization of A.
Proof. Immediate from Lemmas 47.22.1 and 47.22.2. See More on Algebra, Sections 15.11, 15.43, and 15.45 and Algebra, Sections 10.96 and 10.97 for information on completions and henselizations. \square
Lemma 47.22.4. The following types of rings have a dualizing complex:
fields,
Noetherian complete local rings,
\mathbf{Z},
Dedekind domains,
any ring which is obtained from one of the rings above by taking an algebra essentially of finite type, or by taking an ideal-adic completion, or by taking a henselization, or by taking a strict henselization.
Proof. Part (5) follows from Proposition 47.15.11 and Lemma 47.22.3. By Lemma 47.21.3 a regular local ring has a dualizing complex. A complete Noetherian local ring is the quotient of a regular local ring by the Cohen structure theorem (Algebra, Theorem 10.160.8). Let A be a Dedekind domain. Then every ideal I is a finite projective A-module (follows from Algebra, Lemma 10.78.2 and the fact that the local rings of A are discrete valuation ring and hence PIDs). Thus every A-module has finite injective dimension at most 1 by More on Algebra, Lemma 15.69.2. It follows easily that A[0] is a dualizing complex. \square
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