---
title: IS-Tilting Module in Artin Algebras
url: https://www.emergentmind.com/topics/is-tilting-module
type: topic
---

# IS-Tilting Module in Artin Algebras

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 [1706.00475].

## 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
\[
\operatorname{Gen}(X)=\{M\in \mathrm{mod}\,A \mid X^m\twoheadrightarrow M \text{ for some } m\},
\]
and
\[
\operatorname{Cogen}(X)=\{M\in \mathrm{mod}\,A \mid M\hookrightarrow X^m \text{ for some } m\}.
\]
Writing
\[
Q=\bigoplus_{i=1}^t Q_i
\]
for the direct sum of representatives of all indecomposable projective-injective \(A\)-modules, the ambient category for the IS-tilting module is
\[
\mathcal C_A=(\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q).
\]
Thus \(M\in\mathcal C_A\) means precisely that \(M\) is both generated and cogenerated by projective-injective modules.

The subcategory \(\mathcal C_A\) has strong closure properties. If \(X\in \mathcal C_A\), then there are short exact sequences
\[
0\to N\to Q_0\to X\to 0,\qquad 0\to X\to Q^0\to L\to 0
\]
with \(Q_0,Q^0\) projective-injective. If \(A\) has finite global dimension \(d\), this implies
\[
\operatorname{pd}_A X\le d-1,\qquad \operatorname{id}_A X\le d-1.
\]
Moreover, if \(X\in\mathcal C_A\), then its projective cover \(P(X)\) is injective and its injective envelope \(I(X)\) is projective; hence both are projective-injective. In particular, a projective module in \(\mathcal C_A\) is automatically projective-injective, and an injective module in \(\mathcal C_A\) is automatically projective-injective. Accordingly, \(\mathcal C_A\) contains no purely projective or purely injective objects of those kinds [1706.00475].

## 2. Tilting structure, canonical decomposition, and uniqueness

The relevant tilting notion is classical. A basic \(A\)-module \(T\) is tilting if
\[
\operatorname{pd}_A T\le 1,\qquad \operatorname{Ext}^1_A(T,T)=0,
\]
and there exists an exact sequence
\[
0\to A\to T_0\to T_1\to 0
\qquad\text{with }T_0,T_1\in\operatorname{add}T.
\]
When such a tilting object exists inside \(\mathcal C_A\), it is denoted \(T_{\mathcal C}\).

Its structure is rigid. Let \(\mathcal Q=\operatorname{add}Q\), and let \(\mathcal X\) be the full subcategory of \(\mathcal C_A\) consisting of modules of projective dimension \(1\). If
\[
X=\bigoplus_{i=1}^s X_i
\]
is the direct sum of representatives of indecomposable modules in \(\mathcal X\), then \(Q\oplus X\) is a partial tilting module. More strongly, if a tilting module in \(\mathcal C_A\) exists, then necessarily
\[
T_{\mathcal C}\cong Q\oplus X.
\]
Hence the IS-tilting module is unique.

The decomposition can be made explicit. Let \(P_i\) run through the indecomposable projective non-injective \(A\)-modules. For each \(P_i\), take its injective envelope
\[
0\to P_i\to Q_i\to \Omega^{-1}P_i\to 0
\]
with \(Q_i\) projective-injective. Then
\[
T_{\mathcal C}\cong Q\oplus\left(\bigoplus_i \Omega^{-1}P_i\right).
\]
Thus the non-projective summands of \(T_{\mathcal C}\) are exactly the cosyzygies of the indecomposable projective non-injective modules. Equivalently, they are precisely the indecomposable objects of \(\mathcal C_A\) of projective dimension \(1\).

This description is encoded by the syzygy map
\[
\Omega:\operatorname{ind}\mathcal X\to \operatorname{ind}\mathcal P,
\]
where \(\mathcal P\) is the subcategory of indecomposable projective non-injective modules. The map is injective in general and bijective exactly when a tilting module exists in \(\mathcal C_A\). Numerically, if \(n\) is the number of non-isomorphic simple \(A\)-modules, \(n_{\mathcal Q}\) the number of indecomposable projective-injective modules, and \(n_{\mathcal X}\) the number of indecomposable objects in \(\mathcal X\), then
\[
n_{\mathcal Q}+n_{\mathcal X}\le n,
\]
and
\[
\mathcal C_A \text{ contains a tilting module}\iff n_{\mathcal Q}+n_{\mathcal X}=n.
\]
The uniqueness of \(T_{\mathcal C}\) is therefore both structural and numerical [1706.00475].

## 3. Existence and dominant dimension

The controlling invariant is dominant dimension. If
\[
0\to M\to I_0\to I_1\to I_2\to\cdots
\]
is the minimal injective resolution of an \(A\)-module \(M\), then
\[
\operatorname{domdim}M=\sup\{k\mid I_i \text{ is projective for all }0\le i<k\}.
\]
In particular, \(\operatorname{domdim}A\ge 2\) means that in the minimal injective resolution of the regular module
\[
0\to A\to I_0\to I_1\to I_2\to\cdots,
\]
both \(I_0\) and \(I_1\) are projective-injective.

The central characterization is
\[
\operatorname{domdim}A\ge 2
\iff
\mathcal C_A \text{ contains a tilting module }T_{\mathcal C}
\iff
\mathcal C_A \text{ contains a cotilting module }C_{\mathcal C}.
\]
Thus the existence of the IS-tilting module is equivalent to dominant dimension at least \(2\), with no hypothesis on global dimension.

The forward implication is constructive. If \(P\) is an indecomposable projective non-injective module and \(\operatorname{domdim}A\ge 2\), then its minimal injective copresentation begins
\[
0\to P\to Q\to I
\]
with \(Q\) and \(I\) projective-injective. Writing
\[
0\to P\to Q\to X\to 0,
\]
one gets \(X\in \operatorname{Gen}Q\cap\operatorname{Cogen}Q=\mathcal C_A\), \(\operatorname{pd}X=1\), and \(X\cong \Omega^{-1}P\). This gives the required pd-\(1\) summands.

The converse uses the tilting condition. If \(T_{\mathcal C}\in\mathcal C_A\) is tilting and \(P\) is an indecomposable projective non-injective module, then there is an exact sequence
\[
0\to P\xrightarrow{g} T_0\xrightarrow{f} T_1\to 0
\qquad\text{with }T_0,T_1\in\operatorname{add}T_{\mathcal C}.
\]
Taking \(g\) as a minimal left \(\operatorname{add}T_{\mathcal C}\)-approximation forces \(T_0\) to be projective-injective; since \(T_1\in\mathcal C_A\), its injective envelope is also projective-injective. Hence \(P\) has a minimal injective copresentation whose first two terms are projective-injective, so \(\operatorname{domdim}P\ge 2\), and therefore \(\operatorname{domdim}A\ge 2\) [1706.00475].

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

The existence of an IS-tilting module does not imply cotilting. The precise strengthening is
\[
\mathcal C_A \text{ contains a tilting-cotilting module}
\iff
A \text{ is }1\text{-Auslander-Gorenstein}.
\]
Here
\[
\operatorname{id}A_A\le 2\le \operatorname{domdim}A,
\]
equivalently either \(A\) is selfinjective, or
\[
\operatorname{id}{}_A A=\operatorname{id}A_A=2=\operatorname{domdim}A.
\]
If \(A\) has finite global dimension, then the existence of a tilting-cotilting module in \(\mathcal C_A\) is equivalent to \(A\) being an Auslander algebra.

The dual explicit form is
\[
C_{\mathcal C}\cong Q\oplus\left(\bigoplus_j \Omega I_j\right),
\]
where \(I_j\) ranges over indecomposable injective non-projective modules. The asymmetry is exact: \(T_{\mathcal C}\) is built from cosyzygies of projective non-injectives, while \(C_{\mathcal C}\) is built from syzygies of injective non-projectives.

The endomorphism algebra of the IS-tilting module is also controlled. If
\[
B_{\mathcal C}:=\operatorname{End}_A(T_{\mathcal C})^{op},
\]
then, assuming \(\operatorname{domdim}A\ge 2\),
\[
\operatorname{gldim} B_{\mathcal C}\le \operatorname{gldim}A.
\]
Combined with standard tilting inequalities, this yields
\[
\operatorname{gldim}A-1\le \operatorname{gldim}B_{\mathcal C}\le \operatorname{gldim}A.
\]
More precisely,
\[
\operatorname{gldim} B_{\mathcal C}=\operatorname{gldim}A-1
\iff
\operatorname{pd}_A(\mathcal F(T_{\mathcal C}))<\operatorname{gldim}A,
\]
where \(\mathcal F(T_{\mathcal C})\) is the torsion-free class associated to the tilting module.

The paper also uses \(B_{\mathcal C}\) in connection with the Finitistic Dimension Conjecture. If \(A\) is an artin algebra, \(X\) an Auslander generator-cogenerator, and
\[
\Gamma:=\operatorname{End}_A(X)^{op}
\]
has dominant dimension at least \(2\), then \(\Gamma\) has the special tilting module \(T_{\mathcal C}\). If
\[
\operatorname{repdim}A\le 4
\quad\text{and}\quad
\operatorname{pd}_{\Gamma}(\mathcal F(T_{\mathcal C}))<3,
\]
then
\[
\operatorname{findim}A<\infty.
\]
Accordingly, the IS-tilting module functions not only as a structural object but also as a homological tool [1706.00475].

## 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 \(c=(c_1,\dots,c_n)\), let \(\mathcal P_c\) denote the labels of indecomposable projective non-injective modules and \(\mathcal Q_c\) the labels of indecomposable projective-injective modules. Then
\[
\mathcal C_A \text{ contains a tilting module}
\iff
\mathcal P_c \subseteq \{\, j-c_j \mid j\in \mathcal Q_c\,\}.
\]
When \(T_{\mathcal C}\) exists, it has the explicit form
\[
T_{\mathcal C}\cong \bigoplus_{j\in \mathcal Q_c} P_j \;\oplus\; \bigoplus_{i\in \mathcal P_c} T_i,
\]
where each \(T_i\) is uniserial with
\[
\operatorname{soc} T_i = S_{i+1},\qquad |T_i|=\delta(i),
\]
and
\[
\delta(i):=\min\{k\in \mathbb N \mid i+k\in \mathcal Q_c\}.
\]
This is the Nakayama form of the general decomposition
\[
T_{\mathcal C}\cong Q\oplus \bigoplus_i \Omega^{-1}P_i.
\]

The examples also isolate the main homological distinctions. One Nakayama example with dominant dimension at least \(2\) has
\[
T_{\mathcal C}= {}^{3}_{2}\oplus {}^{3}_{1}.
\]
Another example is a \(1\)-Auslander-Gorenstein Nakayama algebra with
\[
\operatorname{gldim}A=\infty,
\]
yet
\[
T_{\mathcal C}=P_1\oplus P_2\oplus P_4\oplus P_5\oplus S_4
\]
is tilting-cotilting in \(\mathcal C_A\). This shows that the existence of a tilting-cotilting module in \(\mathcal C_A\) does not force finite global dimension, so the class of \(1\)-Auslander-Gorenstein algebras is strictly broader than the class of Auslander algebras. Conversely, the paper also emphasizes examples where \(\operatorname{domdim}A=2\) and \(T_{\mathcal C}\) exists, but the module is not cotilting. The contrast between these cases is exactly the distinction between dominant dimension \(\ge 2\) and the stronger \(1\)-Auslander-Gorenstein condition [1706.00475].

## 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 \(\mathcal C_A\).” In that usage, “IS-tilting module” refers to the unique classical tilting module
\[
T_{\mathcal C}\in (\operatorname{Gen}Q)\cap(\operatorname{Cogen}Q),
\]
and not to an arbitrary tilting module or to an arbitrary support construction [1706.00475].

A frequent source of ambiguity is the phrase’s proximity to support \(\tau\)-tilting terminology. In “\(\tau\)-tilting theory,” a support \(\tau\)-tilting module is defined by passage to a quotient algebra \(\Lambda/\langle e\rangle\), or equivalently by a support \(\tau\)-tilting pair \((M,P)\) with
\[
\operatorname{Hom}_{\Lambda}(M,\tau M)=0,\qquad \operatorname{Hom}_{\Lambda}(P,M)=0,\qquad |M|+|P|=|\Lambda|.
\]
That theory does not define an object literally called “IS-tilting module” [1210.1036].

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 \(T^\bullet\), the module
\[
H^0(T^\bullet)
\]
is tilting over the factor algebra
\[
A/\operatorname{ann}_A(H^0(T^\bullet)),
\]
and its endomorphism algebra is a corresponding factor of \(\operatorname{End}_{\mathcal D(A)}(T^\bullet)\). This is a support-by-annihilator phenomenon rather than a projective-injective-generated/cogenerated one [1104.0627].

A related but again different comparison appears in work characterizing \(\tau\)-tilting modules as \(1\)-tilting modules over quotient algebras satisfying
\[
\operatorname{Ann}(T)\otimes_A T=0,
\]
and \(1\)-tilting modules as \(\tau\)-tilting modules satisfying
\[
{\rm Tor}_1^A(\operatorname{Ann}(T),T)=0.
\]
These criteria concern the relationship between \(\tau\)-tilting and classical \(1\)-tilting over annihilator quotients, not the dominant-dimension-controlled module \(T_{\mathcal C}\) in \(\mathcal C_A\) [2501.02466].

Within this terminological landscape, the most precise use of “IS-tilting module” is therefore the one attached to \(\mathcal C_A\), projective-injective generation and cogeneration, dominant dimension \(\ge 2\), and the canonical decomposition
\[
T_{\mathcal C}\cong Q\oplus \bigoplus_i \Omega^{-1}P_i.
\]
That object is unique if it exists, may fail to be cotilting, and becomes tilting-cotilting exactly in the \(1\)-Auslander-Gorenstein case.

Source: https://www.emergentmind.com/topics/is-tilting-module