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Dualizing biduality for finite algebras over a regular base

Statement

Assume AC and DC. Let $R$ be a regular Noetherian ring of finite dimension $d$ , and let $B \neq 0$ be a module-finite $R$ -algebra. For any integer $s$ , $D_{B} = R {Hom}_{R} (B, R [s])$ is a dualizing complex over $B$ . For every $M \in D_{f i n}^{b} (B)$ the canonical evaluation $M \to R {Hom}_{B} (R {Hom}_{B} (M, D_{B}), D_{B})$ is an isomorphism; both duals are bounded with finite cohomology. These constructions commute with localization on $R$ ; localization on $B$ preserves the dualizing conditions and coherent bidual evaluation.

Facts & Assumptions

Given: A regular Noetherian ring $R$ of finite dimension $d$ , a nonzero module-finite $R$ -algebra $B$ , an integer $s$ , and $M \in D_{f i n}^{b} (B)$ .

[F1]

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $F$ all of whose members are nonempty, there exists a function $g$ with domain $F$ satisfying $g (S) \in S$ for all $S \in F$ . (The Axiom of Choice)

[F2]

def-dependent-choice. Let $X$ be a set and let $R \subseteq X \times X$ be a binary relation on $X$ . Call $R$ entire on $X$ when $for every x \in X there is y \in X with x R y .$ The Axiom of Dependent Choice, written $D C$ , is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain)

[F3]

def-dualizing-complex-on-projective-cm-scheme. A dualizing complex on a Noetherian scheme $X$ is an object $D_{X} \in D_{C o h}^{b} (X)$ such that, locally on affine open neighborhoods $U = Spec B$ , its corresponding complex has finite injective dimension over $B$ and the homothety map $B \to R {Hom}_{B} (D_{X} ∣_{U}, D_{X} ∣_{U})$ is an isomorphism. (def-dualizing-complex-on-projective-cm-scheme)

[F4]

lem-finite-closed-immersion-derived-coinduction-adjunction. Assume AC. For a finite homomorphism $A \to B$ of Noetherian rings and $G \in D^{+} (A)$ , the complex $f^{!} G = R {Hom}_{A} (B, G)$ has its natural $B$ -action and is right adjoint to restriction of scalars. (Derived adjunction for finite rings and closed immersions)

[F5]

thm-localisation-and-polynomial-extension-of-regular-rings. Assume the Axiom of Choice (The Axiom of Choice). Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular. Regularity can equivalently be tested at maximal ideals. For every nonzero such ring, $gldim R = \dim R$ , allowing infinity. (localisation and polynomial extension of regular rings)

[F6]

lem-global-dimension-is-detected-on-cyclic-modules. For a unital ring $R$ , its left global dimension equals $\sup_{I} {pd}_{R} (R / I)$ over all left ideals $I$ , and equals the supremum of the injective dimensions of all left modules. The equalities allow infinity; in the commutative Noetherian case the cyclic modules are finite. (global dimension is detected on cyclic modules)

Proof

1.1F5F6given

Every finite $R$ -module has a bounded resolution by finite projective modules: $R$ has finite global dimension by [F5], the resolution can be terminated at that bound, and the syzygies of a finite module over the Noetherian ring $R$ are again finite. Similarly every bounded complex of finite $R$ -modules has a bounded complex of finite projective modules, and the finite global dimension is detected on cyclic modules by [F6].

2.1F5step 1.1

The free module $R$ of rank one has injective dimension at most $d = \dim R$ over $R$ , because the global dimension of $R$ bounds both the projective and the injective dimensions of its modules; hence the shift $R [s]$ has a bounded injective resolution of length at most $d$ , with injective terms that are $R$ -modules.

3.1F4step 2.1

Applying $R {Hom}_{R} (B, -)$ to a bounded injective resolution of $R [s]$ gives a bounded complex $D_{B} = R {Hom}_{R} (B, R [s])$ of $B$ -modules whose terms are injective over $B$ by the finite-ring coinduction adjunction [F4], so $D_{B}$ has finite injective dimension over $B$ .

4.1F4F6step 1.1step 3.1

Since $B$ is a finite $R$ -module, a bounded resolution of $B$ by finite projective $R$ -modules computes $R {Hom}_{R} (B, R [s])$ , so $D_{B}$ has bounded cohomology with finite $R$ -modules, hence finite $B$ -modules, in each degree.

5.1F4step 3.1givenstep 4.1

For $M \in D_{f i n}^{b} (B)$ , coinduction along the finite map $R \to B$ gives $R {Hom}_{B} (M, D_{B}) = R {Hom}_{R} (M, R [s])$ after forgetting the $B$ -structure; applying the same identity to the $B$ -module $R {Hom}_{B} (M, D_{B})$ and substituting yields $R {Hom}_{B} (R {Hom}_{B} (M, D_{B}), D_{B}) ≅ R {Hom}_{R} (R {Hom}_{R} (M, R [s]), R [s])$ .

6.1F6step 5.1

The right-hand side is the double dual of $M$ computed from a bounded resolution of $M$ by finite projective $R$ -modules: a finite projective module is canonically isomorphic to its double dual, so termwise double duality of the resolution is a quasi-isomorphism, and the evaluation at $1 \in B$ is the actual bidual map of $B$ -complexes; forgetting scalars reflects quasi-isomorphisms, so the canonical evaluation $M \to R {Hom}_{B} (R {Hom}_{B} (M, D_{B}), D_{B})$ is an isomorphism and both duals are bounded with finite cohomology.

7.1F4F5F6step 6.1

All constructions commute with localization on $R$ : finite projective resolutions and $R Hom$ localize, and bounded injective resolutions localize by the ideal test for injectivity together with localization of Hom from finitely presented ideals; localization on $B$ preserves the dualizing conditions, the finite cohomology and the coherent bidual evaluation, computed from a degreewise finite free $B$ -resolution.

8.1F1F2step 1.1step 7.1∎

The Axiom of Choice is inherited from the resolution suppliers and the Axiom of Dependent Choice is retained from the Ext and resolution data; no further choice is made.

Remarks

This is the finite-algebra case of the quotient statement used elsewhere on this page; the proof does not assume that $B$ is a quotient of $R$ by an ideal, only that $R \to B$ is finite.
The shift $s$ is carried through all identifications; biduality is shift-invariant because the two shifts cancel.

Depends on

Used by

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Resolution of Surfaces, 54.8.8 and 54.11.6: exact imports replaced by the local argument (standard reference, not scraped)