IS-Tilting Module in Artin Algebras

• IS-Tilting module is a unique classical tilting module in the subcategory defined by projective-injective modules, ensuring both generation and cogeneration in artin algebras.
• Its canonical decomposition expresses the module as a direct sum of projective-injective summands and cosyzygies of non-injective projectives, showcasing a rigid and explicit structure.
• The existence of the IS-tilting module is equivalent to a dominant dimension of at least two, linking it to 1-Auslander-Gorenstein conditions and influencing homological invariants.

An IS-tilting module, in the sense relevant here, is the unique classical tilting module $T_{\mathcal C}$ lying in the subcategory

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$

where $Q$ is the direct sum of representatives of all indecomposable projective-injective $A$-modules. Equivalently, it is a tilting module that is both generated and cogenerated by projective-injective modules. For an artin algebra $A$, the existence of such a module is equivalent to $\operatorname{domdim}A\ge 2$; when it exists, it is unique, has an explicit decomposition by cosyzygies of projective non-injective modules, and need not be cotilting (Nguyen et al., 2017).

1. Definition via the projective-injective subcategory

Let $A$ be an artin algebra and $\mathrm{mod}\,A$ the category of finitely generated left $A$-modules. If $X$ is a module, then

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$0

and

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$1

Writing

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$2

for the direct sum of representatives of all indecomposable projective-injective $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$3-modules, the ambient category for the IS-tilting module is

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$4

Thus $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$5 means precisely that $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$6 is both generated and cogenerated by projective-injective modules.

The subcategory $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$7 has strong closure properties. If $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$8, then there are short exact sequences

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$9

with $Q$0 projective-injective. If $Q$1 has finite global dimension $Q$2, this implies

$Q$3

Moreover, if $Q$4, then its projective cover $Q$5 is injective and its injective envelope $Q$6 is projective; hence both are projective-injective. In particular, a projective module in $Q$7 is automatically projective-injective, and an injective module in $Q$8 is automatically projective-injective. Accordingly, $Q$9 contains no purely projective or purely injective objects of those kinds (Nguyen et al., 2017).

2. Tilting structure, canonical decomposition, and uniqueness

The relevant tilting notion is classical. A basic $A$0-module $A$1 is tilting if

$A$2

and there exists an exact sequence

$A$3

When such a tilting object exists inside $A$4, it is denoted $A$5.

Its structure is rigid. Let $A$6, and let $A$7 be the full subcategory of $A$8 consisting of modules of projective dimension $A$9. If

$A$0

is the direct sum of representatives of indecomposable modules in $A$1, then $A$2 is a partial tilting module. More strongly, if a tilting module in $A$3 exists, then necessarily

$A$4

Hence the IS-tilting module is unique.

The decomposition can be made explicit. Let $A$5 run through the indecomposable projective non-injective $A$6-modules. For each $A$7, take its injective envelope

$A$8

with $A$9 projective-injective. Then

$\operatorname{domdim}A\ge 2$0

Thus the non-projective summands of $\operatorname{domdim}A\ge 2$1 are exactly the cosyzygies of the indecomposable projective non-injective modules. Equivalently, they are precisely the indecomposable objects of $\operatorname{domdim}A\ge 2$2 of projective dimension $\operatorname{domdim}A\ge 2$3.

This description is encoded by the syzygy map

$\operatorname{domdim}A\ge 2$4

where $\operatorname{domdim}A\ge 2$5 is the subcategory of indecomposable projective non-injective modules. The map is injective in general and bijective exactly when a tilting module exists in $\operatorname{domdim}A\ge 2$6. Numerically, if $\operatorname{domdim}A\ge 2$7 is the number of non-isomorphic simple $\operatorname{domdim}A\ge 2$8-modules, $\operatorname{domdim}A\ge 2$9 the number of indecomposable projective-injective modules, and $A$0 the number of indecomposable objects in $A$1, then

$A$2

and

$A$3

The uniqueness of $A$4 is therefore both structural and numerical (Nguyen et al., 2017).

3. Existence and dominant dimension

The controlling invariant is dominant dimension. If

$A$5

is the minimal injective resolution of an $A$6-module $A$7, then

$A$8

In particular, $A$9 means that in the minimal injective resolution of the regular module

$\mathrm{mod}\,A$0

both $\mathrm{mod}\,A$1 and $\mathrm{mod}\,A$2 are projective-injective.

The central characterization is

$\mathrm{mod}\,A$3

Thus the existence of the IS-tilting module is equivalent to dominant dimension at least $\mathrm{mod}\,A$4, with no hypothesis on global dimension.

The forward implication is constructive. If $\mathrm{mod}\,A$5 is an indecomposable projective non-injective module and $\mathrm{mod}\,A$6, then its minimal injective copresentation begins

$\mathrm{mod}\,A$7

with $\mathrm{mod}\,A$8 and $\mathrm{mod}\,A$9 projective-injective. Writing

$A$0

one gets $A$1, $A$2, and $A$3. This gives the required pd-$A$4 summands.

The converse uses the tilting condition. If $A$5 is tilting and $A$6 is an indecomposable projective non-injective module, then there is an exact sequence

$A$7

Taking $A$8 as a minimal left $A$9-approximation forces $X$0 to be projective-injective; since $X$1, its injective envelope is also projective-injective. Hence $X$2 has a minimal injective copresentation whose first two terms are projective-injective, so $X$3, and therefore $X$4 (Nguyen et al., 2017).

4. Cotilting enhancement, Auslander-Gorenstein conditions, and endomorphism algebras

The existence of an IS-tilting module does not imply cotilting. The precise strengthening is

$X$5

Here

$X$6

equivalently either $X$7 is selfinjective, or

$X$8

If $X$9 has finite global dimension, then the existence of a tilting-cotilting module in $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$00 is equivalent to $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$01 being an Auslander algebra.

The dual explicit form is

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$02

where $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$03 ranges over indecomposable injective non-projective modules. The asymmetry is exact: $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$04 is built from cosyzygies of projective non-injectives, while $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$05 is built from syzygies of injective non-projectives.

The endomorphism algebra of the IS-tilting module is also controlled. If

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$06

then, assuming $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$07,

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$08

Combined with standard tilting inequalities, this yields

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$09

More precisely,

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$10

where $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$11 is the torsion-free class associated to the tilting module.

The paper also uses $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$12 in connection with the Finitistic Dimension Conjecture. If $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$13 is an artin algebra, $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$14 an Auslander generator-cogenerator, and

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$15

has dominant dimension at least $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$16, then $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$17 has the special tilting module $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$18. If

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$19

then

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$20

Accordingly, the IS-tilting module functions not only as a structural object but also as a homological tool (Nguyen et al., 2017).

5. Examples, special families, and explicit criteria

The general theory specializes effectively. The paper shows, as special cases, that triangular matrix algebras obtained from Auslander algebras and certain injective modules have such a tilting module, and it gives a detailed treatment of Nakayama algebras.

For Nakayama algebras with admissible sequence $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$21, let $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$22 denote the labels of indecomposable projective non-injective modules and $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$23 the labels of indecomposable projective-injective modules. Then

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$24

When $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$25 exists, it has the explicit form

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$26

where each $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$27 is uniserial with

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$28

and

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$29

This is the Nakayama form of the general decomposition

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$30

The examples also isolate the main homological distinctions. One Nakayama example with dominant dimension at least $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$31 has

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$32

Another example is a $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$33-Auslander-Gorenstein Nakayama algebra with

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$34

yet

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$35

is tilting-cotilting in $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$36. This shows that the existence of a tilting-cotilting module in $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$37 does not force finite global dimension, so the class of $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$38-Auslander-Gorenstein algebras is strictly broader than the class of Auslander algebras. Conversely, the paper also emphasizes examples where $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$39 and $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$40 exists, but the module is not cotilting. The contrast between these cases is exactly the distinction between dominant dimension $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$41 and the stronger $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$42-Auslander-Gorenstein condition (Nguyen et al., 2017).

6. Terminology, scope, and adjacent notions

The paper that identifies the dominant-dimension criterion does not primarily use the label “IS-tilting”; it speaks instead of “tilting modules generated and cogenerated by projective-injective modules” or “tilting modules in $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$43.” In that usage, “IS-tilting module” refers to the unique classical tilting module

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$44

and not to an arbitrary tilting module or to an arbitrary support construction (Nguyen et al., 2017).

A frequent source of ambiguity is the phrase’s proximity to support $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$45-tilting terminology. In “$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$46-tilting theory,” a support $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$47-tilting module is defined by passage to a quotient algebra $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$48, or equivalently by a support $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$49-tilting pair $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$50 with

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$51

That theory does not define an object literally called “IS-tilting module” (Adachi et al., 2012).

The term is also distinct from constructions in which a tilting module is extracted from a two-term tilting complex. For a two-term tilting complex $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$52, the module

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$53

is tilting over the factor algebra

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$54

and its endomorphism algebra is a corresponding factor of $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$55. This is a support-by-annihilator phenomenon rather than a projective-injective-generated/cogenerated one (Abe, 2011).

A related but again different comparison appears in work characterizing $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$56-tilting modules as $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$57-tilting modules over quotient algebras satisfying

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$58

and $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$59-tilting modules as $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$60-tilting modules satisfying

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$61

These criteria concern the relationship between $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$62-tilting and classical $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$63-tilting over annihilator quotients, not the dominant-dimension-controlled module $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$64 in $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$65 (Chen et al., 5 Jan 2025).

Within this terminological landscape, the most precise use of “IS-tilting module” is therefore the one attached to $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$66, projective-injective generation and cogeneration, dominant dimension $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$67, and the canonical decomposition

$$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$68

That object is unique if it exists, may fail to be cotilting, and becomes tilting-cotilting exactly in the $$\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),$$69-Auslander-Gorenstein case.