Complete Duality Datum in Morita Rings

• Complete Duality Datum is the structured collection of complete duality pairs on rings A and B, inducing duality on Morita rings with vanishing tensor products.
• It enables lifting duality properties from A-modules and B-modules to Δ-modules via explicit tensor, Tor, and finiteness conditions.
• The framework generalizes triangular matrix ring theory to characterize relative Gorenstein and Ding projective modules in a unified homological setting.

Searching arXiv for the primary paper and closely related terminology. In the setting of Morita rings with vanishing tensor products, a complete duality datum is the package of complete duality pairs on the component rings $A$ and $B$, together with the induced complete duality pair on the Morita ring

$$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$

and the relative Gorenstein and Ding projective theories determined by that induced pair. In the paper “How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings,” this datum is the organizing structure that allows duality-theoretic information on $A$-modules and $B$-modules to be lifted to $\Delta$-modules and then applied to relative homological constructions (Ma et al., 2022).

1. Morita-ring framework and module-theoretic realization

The Morita ring under consideration has underlying abelian group $A\oplus N\oplus M\oplus B$ and multiplication

$$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$

subject to the vanishing conditions $M\otimes_A N=0=N\otimes_B M$. This is the degenerate Morita-context case that contains triangular matrix rings as the special case $N=0$ (Ma et al., 2022).

Left $B$0-modules are described via Green’s equivalence by tuples $B$1, where $B$2, $B$3,

$B$4

and morphisms are pairs commuting with the obvious squares. Under the equivalence with $B$5, the $B$6-action on $B$7 is

$B$8

Right $B$9-modules are similarly described by tuples $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$0 with

$$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$1

This tuple description is essential because every later duality and Gorenstein condition is formulated componentwise.

2. Duality pairs, perfectness, and completeness

For a ring $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$2, a duality pair in the sense used here is a pair $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$3 of module classes such that

$$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$4

and $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$5 is closed under direct summands and finite direct sums. A duality pair is perfect when $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$6 contains the regular module and is closed under arbitrary direct sums and extensions; it is symmetric when both $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$7 and $$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$8 are duality pairs. A complete duality pair, following Gillespie’s terminology, is one that is both symmetric and perfect (Ma et al., 2022).

The paper starts with complete duality pairs

$$\Delta= \begin{pmatrix} A & {}_A N_B\ {}_B M_A & B \end{pmatrix}, \qquad M\otimes_A N=0=N\otimes_B M,$$9

Within this framework, “complete duality datum” is not introduced as a separate formal axiom system; rather, it can be understood as the full package consisting of these two complete duality pairs, the bimodules $A$0 satisfying the vanishing and finiteness hypotheses, and the induced complete duality classes on $A$1. The datum is therefore simultaneously categorical, module-theoretic, and homological.

3. Induced duality classes on $A$2

The induced classes of $A$3-modules are built from the componentwise classes on $A$4 and $A$5.

Class	Side	Defining condition
$A$6	left	$A$7
$A$8	left	$A$9 monic, $B$0
$B$1	right	induced maps $B$2 epi, $B$3

The basic structural theorem states that the following are equivalent: the pairs $B$4 and $B$5 are duality pairs over $B$6 and $B$7; $B$8 is a duality pair over $B$9; and $\Delta$0 is a duality pair over $\Delta$1. If $\Delta$2 and $\Delta$3 are finitely presented, a further variant is equivalent as well. A key computational input is the character-module identity

$\Delta$4

This turns the passage from $\Delta$5 and $\Delta$6 to $\Delta$7 into an explicit character-duality argument rather than a purely abstract existence statement (Ma et al., 2022).

Perfectness and completeness require additional hypotheses. For $\Delta$8, perfectness is equivalent to perfectness of the base pairs provided

$\Delta$9

For the componentwise pair $A\oplus N\oplus M\oplus B$0, one additionally needs $A\oplus N\oplus M\oplus B$1 and $A\oplus N\oplus M\oplus B$2. If $A\oplus N\oplus M\oplus B$3 and $A\oplus N\oplus M\oplus B$4 are finitely generated projective, Corollary 3.8 identifies exactly when these induced pairs are complete. In this sense, the complete duality datum on $A\oplus N\oplus M\oplus B$5 is obtained by transporting symmetry and perfectness from the two component rings through explicit tensor, Tor, and finiteness constraints.

4. Relative Gorenstein projective modules determined by the datum

Given a complete duality pair $A\oplus N\oplus M\oplus B$6 over a ring $A\oplus N\oplus M\oplus B$7, a module is Gorenstein $A\oplus N\oplus M\oplus B$8-projective if it is the $A\oplus N\oplus M\oplus B$9-cycle of an exact complex of projectives that remains exact under $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$0 for every $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$1. The paper applies this relative notion to the induced complete duality pair on $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$2 (Ma et al., 2022).

Under the standing assumptions that $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$3 and $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$4 are finitely generated projective and that $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$5, $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$6 are complete duality pairs, the functors

$$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$7

transport relative Gorenstein projectives from $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$8 and $$\begin{pmatrix} a & n\ m & b \end{pmatrix} \begin{pmatrix} a' & n'\ m' & b' \end{pmatrix} = \begin{pmatrix} aa' & an'+nb'\ ma'+bm' & bb' \end{pmatrix},$$9 to $M\otimes_A N=0=N\otimes_B M$0. Specifically, if $M\otimes_A N=0=N\otimes_B M$1 for all $M\otimes_A N=0=N\otimes_B M$2, then Gorenstein $M\otimes_A N=0=N\otimes_B M$3-projective $M\otimes_A N=0=N\otimes_B M$4-modules yield Gorenstein $M\otimes_A N=0=N\otimes_B M$5-projective $M\otimes_A N=0=N\otimes_B M$6-modules via $M\otimes_A N=0=N\otimes_B M$7. Dually, if $M\otimes_A N=0=N\otimes_B M$8 for all $M\otimes_A N=0=N\otimes_B M$9, then $N=0$0 sends Gorenstein $N=0$1-projectives to relative Gorenstein projectives over $N=0$2.

There are also converse component results. If $N=0$3 and $N=0$4 are projective, if $N=0$5, and if

$N=0$6

then the $N=0$7-component of a Gorenstein $N=0$8-projective $N=0$9-module is Gorenstein $B$00-projective. The dual statement for the $B$01-component uses the hypotheses $B$02 and

$B$03

The complete duality datum therefore governs both construction and detection of relative Gorenstein projectives.

5. Flat–FP-injective specialization and Ding projective modules

A principal application is obtained by taking the complete duality pairs

$B$04

which are complete when the rings are right coherent. In that coherent setting, Ding projective modules coincide with Gorenstein $B$05-projective modules, so the general relative theory specializes directly to Ding projectives (Ma et al., 2022).

The paper also gives a concrete componentwise characterization of FP-injective right $B$06-modules: a right $B$07-module $B$08 is FP-injective if and only if

$B$09

$B$10 is FP-injective as a right $B$11-module, and $B$12 is FP-injective as a right $B$13-module. This identifies the right-hand side of the flat–FP-injective duality datum on $B$14 in explicit tuple-theoretic terms.

If $B$15 is right coherent and $B$16 are finitely generated projective, then $B$17 and $B$18 are right coherent, and the general transfer theorem becomes a transfer theorem for Ding projectives. Under the hypotheses of Corollary 4.8, Ding projective $B$19-modules and $B$20-modules are sent to Ding projective $B$21-modules by $B$22 and $B$23, while suitable finite-injective-dimension assumptions on $B$24 or $B$25 recover the Ding projectivity of the $B$26- or $B$27-components from a Ding projective $B$28-module.

6. Conceptual role and mathematical context

In this paper, the complete duality datum is the homological environment formed by complete duality pairs on $B$29 and $B$30, the induced complete duality pair on $B$31, and the resulting classes of relative Gorenstein and Ding projective $B$32-modules. It packages character-duality, closure under sums and extensions, monomorphism/epimorphism conditions on structure maps, and the finiteness hypotheses on $B$33 and $B$34 needed for those constructions to behave functorially.

The paper positions these results as a generalization from triangular matrix rings to Morita rings with $B$35, extending results of Mao and connecting them to Gillespie’s relative Gorenstein framework. It also shows that flat and FP-injective structures over $B$36 and $B$37 can be assembled into a coherent duality-theoretic framework over $B$38. A plausible implication is that Morita-type constructions can serve as a systematic laboratory for transporting relative homological phenomena between component rings and matrix-like extensions, provided the relevant complete duality datum is available.