Ulrich Ideal in Cohen–Macaulay Rings

• Ulrich ideals are m-primary ideals in Cohen–Macaulay rings defined by I² = QI and the freeness of I/I², encapsulating rigid algebraic behavior.
• They exhibit distinctive homological properties with explicit derived decompositions and rigid minimal free resolutions that connect to generalized Ulrich modules.
• Their classification in one-dimensional semigroup rings and hypersurfaces links to almost Gorenstein conditions and extends to geometric counterparts like Ulrich bundles.

An Ulrich ideal is an $\mathfrak m$-primary ideal $I$ in a Cohen–Macaulay local ring $(A,\mathfrak m)$ that contains a parameter ideal $Q$ as a reduction and satisfies two extremal conditions: $I^2=QI$ and $I/I^2$ is a free $A/I$-module. In this form, Ulrich ideals encode a rigid combination of reduction-theoretic, homological, and multiplicity-theoretic behavior. Equivalently, the associated graded ring $\operatorname{gr}_I(A)$ is Cohen–Macaulay with $a(\operatorname{gr}_I(A))=1-d$, where $d=\dim A$. The theory interacts closely with maximal Cohen–Macaulay modules, syzygies, almost Gorenstein and 2-almost Gorenstein singularities, numerical semigroup rings, hypersurfaces, and the geometry of Ulrich bundles and trace ideals (Goto et al., 2012, Maeda et al., 17 Jul 2025).

1. Definition and numerical characterizations

Let $I$0 be a Cohen–Macaulay local ring of dimension $I$1, and let $I$2 be an $I$3-primary ideal containing a parameter ideal $I$4 as a reduction. The standard definition requires

$I$5

Under infinite residue field, this can be reformulated by requiring that for every minimal reduction $I$6 of $I$7, one has $I$8 and $I$9 free over $(A,\mathfrak m)$0. The basic multiplicity inequality

$(A,\mathfrak m)$1

becomes an equality precisely in the Ulrich case; equivalently, $(A,\mathfrak m)$2 is free over $(A,\mathfrak m)$3 (Goto et al., 2012).

The numerical rigidity of Ulrich ideals is sharpened by the relation

$(A,\mathfrak m)$4

where $(A,\mathfrak m)$5 denotes Cohen–Macaulay type. Hence

$(A,\mathfrak m)$6

In particular, if $(A,\mathfrak m)$7 is Gorenstein, every Ulrich ideal is generated by exactly $(A,\mathfrak m)$8 elements. For one-dimensional Gorenstein rings, every Ulrich ideal is therefore $(A,\mathfrak m)$9-generated (Endo et al., 2021).

In dimension $Q$0, with parameter reduction $Q$1, the definition simplifies to: $Q$2 is $Q$3-primary, $Q$4, and $Q$5 is free over $Q$6. If $Q$7 is $Q$8-generated, then $Q$9 for some $I^2=QI$0, $I^2=QI$1, and $I^2=QI$2. This already exhibits the tight relation between Ulrich ideals and periodic resolutions in low dimension (Endo et al., 2021).

In Gorenstein rings, Ulrich ideals coincide with good ideals satisfying an extremal generator condition. More precisely, a non-parameter ideal $I^2=QI$3 is Ulrich if and only if it is good and $I^2=QI$4, or equivalently if $I^2=QI$5 is Gorenstein. This characterization is especially important in the one-dimensional ADE and hypersurface settings (Goto et al., 2012).

2. Homological structure and generalized Ulrich modules

Ulrich ideals are inseparable from the homological behavior of the quotient $I^2=QI$6 and of its syzygies. A central structural theorem gives an explicit derived decomposition: $I^2=QI$7 where $I^2=QI$8 for $I^2=QI$9, $I/I^2$0, and $I/I^2$1 for $I/I^2$2, with $I/I^2$3. Consequently, every $I/I^2$4 is a free $I/I^2$5-module. This description yields the criterion

$I/I^2$6

and in a G-regular ring every non-parameter Ulrich ideal must satisfy $I/I^2$7 (Goto et al., 2015).

The higher syzygies of $I/I^2$8 are themselves Ulrich objects. If $I/I^2$9 is an Ulrich ideal and not a parameter ideal, then $A/I$0 is an Ulrich module with respect to $A/I$1 for all $A/I$2; conversely, the existence of sufficiently high Ulrich syzygies characterizes Ulrich ideals. The minimal free resolution of $A/I$3 is correspondingly rigid: if $A/I$4, then its Betti numbers satisfy

$A/I$5

and for $A/I$6,

$A/I$7

Moreover, the ideal generated by the entries of every differential in the minimal free resolution is exactly $A/I$8, so $A/I$9 can be recovered from any step of the resolution (Goto et al., 2012).

The theory extends naturally from ideals to modules relative to an $\operatorname{gr}_I(A)$0-primary ideal $\operatorname{gr}_I(A)$1. A finitely generated module $\operatorname{gr}_I(A)$2 is Ulrich with respect to $\operatorname{gr}_I(A)$3 if it is maximal Cohen–Macaulay, $\operatorname{gr}_I(A)$4 for a parameter reduction $\operatorname{gr}_I(A)$5, and $\operatorname{gr}_I(A)$6 is free over $\operatorname{gr}_I(A)$7. If $\operatorname{gr}_I(A)$8 is an Ulrich ideal and not a parameter ideal, then $\operatorname{gr}_I(A)$9 is Ulrich with respect to $a(\operatorname{gr}_I(A))=1-d$0 for all $a(\operatorname{gr}_I(A))=1-d$1. Under suitable Ext-vanishing, the Hom functor preserves the generalized Ulrich property, and in the Gorenstein case horizontal linkage carries sufficiently high syzygies of Ulrich ideals to Ulrich modules again (Miranda-Neto et al., 2022).

This suggests a broader interpretation: Ulrich ideals act as input data for a stable homological package consisting of derived decompositions, periodic or asymptotically periodic resolutions, and linkage-closed classes of generalized Ulrich modules.

3. One-dimensional classifications: semigroup rings and hypersurfaces

One of the most developed parts of the theory is the explicit classification of Ulrich ideals in one-dimensional rings. In numerical semigroup rings $a(\operatorname{gr}_I(A))=1-d$2, the classification reduces to semigroup arithmetic. For Gorenstein $a(\operatorname{gr}_I(A))=1-d$3, a monomial ideal $a(\operatorname{gr}_I(A))=1-d$4 is Ulrich precisely when $a(\operatorname{gr}_I(A))=1-d$5 with $a(\operatorname{gr}_I(A))=1-d$6 satisfying $a(\operatorname{gr}_I(A))=1-d$7, $a(\operatorname{gr}_I(A))=1-d$8, the enlarged semigroup $a(\operatorname{gr}_I(A))=1-d$9 symmetric, and $d=\dim A$0. In the two-generated case $d=\dim A$1, non-parameter monomial Ulrich ideals exist if and only if at least one of $d=\dim A$2 is even (Goto et al., 2012).

A particularly detailed classification is available for the semigroup rings $d=\dim A$3 and $d=\dim A$4. In the Gorenstein ring $d=\dim A$5, every Ulrich ideal has the form

$d=\dim A$6

and the parameters $d=\dim A$7 are uniquely determined. Thus the classification depends on $d=\dim A$8: if $d=\dim A$9, every such pair occurs; otherwise $I$00. In the non-Gorenstein ring $I$01, the valuation analysis leaves only the pairs $I$02 and $I$03, producing two $I$04-parameter families of Ulrich ideals; in this ring there are no Ulrich ideals generated only by monomials in $I$05 (Endo et al., 2021).

For hypersurfaces $I$06 with $I$07 regular local of dimension $I$08, Ulrich ideals admit an equational description. If

$I$09

then $I$10 is Ulrich if and only if $I$11 form a system of parameters in $I$12 and there exist $I$13 and a unit $I$14 such that

$I$15

This criterion yields complete classifications in several one-dimensional hypersurface families. For $I$16, all Ulrich ideals are

$I$17

For $I$18, they are

$I$19

For $I$20, decomposable Ulrich ideals are exactly $I$21, and for $I$22 there are additional indecomposable families such as

$I$23

and, when $I$24 is odd, ideals of the form

$I$25

(Isobe, 2019).

These explicit classifications show that in dimension $I$26, Ulrich ideals are governed by a mixture of valuation constraints, conductor structure, and quadratic or matrix-factorization identities.

4. Almost Gorenstein, 2-AGL, and surface singularities

Ulrich ideals become especially rigid in almost Gorenstein contexts. If $I$27 is a one-dimensional almost Gorenstein but non-Gorenstein local ring admitting a canonical module, then every non-parameter Ulrich $I$28-primary ideal is necessarily the maximal ideal: $I$29 More generally, in higher dimension, annihilator conditions arising from the almost Gorenstein exact sequence

$I$30

force strong restrictions on any Ulrich ideal with $I$31 (Goto et al., 2015).

The class of $I$32-almost Gorenstein local rings refines this picture. In a one-dimensional $I$33-AGL ring $I$34 with canonical fractional ideal $I$35, conductor $I$36, and minimal multiplicity, the set of Ulrich ideals is completely determined: $I$37 according as $I$38 is not free or is free over $I$39. If $I$40 is not free over $I$41, then $I$42 is G-regular and contains no two-generated Ulrich ideals. If two-generated Ulrich ideals do exist, then they force $I$43 to be free over $I$44 and constrain the conductor sharply (Goto et al., 2019).

In dimension $I$45, the recent theory of rational surface singularities gives a different kind of rigidity. For a two-dimensional rational triple point $I$46, the canonical trace ideal

$I$47

is an Ulrich ideal. More precisely, there exist a minimal system of generators $I$48 of $I$49 and an integer $I$50 such that

$I$51

this ideal is Ulrich, and every Ulrich ideal contains it: $I$52 Hence

$I$53

For two-dimensional quotient singularities $I$54 with multiplicity $I$55, the maximal ideal is the unique Ulrich ideal: $I$56 and in fact for any two-dimensional quotient singularity one has $I$57 (Maeda et al., 17 Jul 2025).

These results show two recurrent mechanisms. In one dimension, the conductor and the canonical fractional ideal dictate the size of $I$58. In two-dimensional rational singularities, the canonical trace ideal plays the analogous role of a minimal Ulrich ideal.

5. Geometric counterparts: bundles, sheaves, and projective embeddings

Ulrich ideals admit a geometric counterpart in Ulrich bundles and Ulrich sheaves. On a projective variety $I$59, an Ulrich bundle is characterized by linear resolution, maximal generation, and complete vanishing

$I$60

together with the extremal Hilbert polynomial. Through the standard dictionary between coherent sheaves on $I$61 and graded $I$62-modules, Ulrich bundles are geometric incarnations of Ulrich modules.

For the Veronese surface $I$63, every Ulrich bundle $I$64 of rank $I$65 fits into an exact sequence

$I$66

For $I$67, there are no Ulrich line bundles on $I$68, whereas for $I$69 there is a unique rank-$I$70 Ulrich bundle and every Ulrich bundle is a direct sum of copies of it (Coskun et al., 2016).

The weaker notion of a $I$71-Ulrich sheaf asks only that the restriction to a smooth one-dimensional linear section be Ulrich. Every normal ACM variety admits a reflexive $I$72-Ulrich sheaf, and in dimension $I$73 the pushforward of a $I$74-Ulrich sheaf under a general finite linear projection is an instanton sheaf on $I$75. This suggests a systematic way of producing Ulrich behavior on curves even when global Ulrich bundles are unavailable (Kulkarni et al., 2015).

The tangent bundle supplies a particularly rigid test case. The only polarized projective manifolds whose tangent bundle is Ulrich are the twisted cubic

$I$76

and the Veronese surface

$I$77

while the cotangent bundle is never Ulrich (Benedetti et al., 2021). A complementary rigidity theorem states that if $I$78 is a smooth complete intersection of dimension at least $I$79, then a vector bundle on $I$80 restricts to an Ulrich bundle on $I$81 only in the trivial case $I$82 and the ambient bundle trivial. For arbitrary $I$83, a characterization is available under the small-positivity condition $I$84 on the extending bundle (Sarkar, 16 Jun 2026).

A plausible implication is that the scarcity of Ulrich ideals in many local settings mirrors the scarcity of rank-one or extendable Ulrich objects on the projective side.

6. Asymptotic, categorical, and related extensions

Several recent directions study structures that are not themselves Ulrich ideals but are governed by the same extremal philosophy. One such direction is asymptotic. A sequence $I$85 of modules is lim Ulrich if it is lim Cohen–Macaulay and

$I$86

weakly lim Ulrich relaxes the Cohen–Macaulay condition to weakly lim Cohen–Macaulay. In a Cohen–Macaulay ring, a constant sequence given by an actual Ulrich module is lim Ulrich. Standard graded domains over an infinite F-finite field of characteristic $I$87, localized at the homogeneous maximal ideal, admit weakly lim Ulrich sequences, and the existence of such a sequence implies Lech’s conjecture for flat local extensions of the base domain (Ma, 2020).

A second direction concerns ideals with partially linear resolutions. If $I$88 is an $I$89-primary ideal generated in degree $I$90 and its resolution is virtually linear for $I$91 steps, then

$I$92

and

$I$93

This is a weaker form of the Eisenbud–Huneke–Ulrich conjecture for a more general class of ideals, and it shows that sufficiently linear powers eventually coincide with powers of the maximal ideal (Yang, 12 Jun 2026).

A third categorical direction is the theory of Ulrich-split rings. A local Cohen–Macaulay ring is Ulrich-split if every short exact sequence of Ulrich modules splits. In minimal multiplicity, this is equivalent to

$I$94

and, over $I$95, two-dimensional Ulrich-split rings that are normal and of minimal multiplicity are precisely cyclic quotient singularities with at most two indecomposable Ulrich modules up to isomorphism (Dao et al., 2022).

Taken together, these developments indicate that the classical notion of Ulrich ideal has become a reference point for several broader theories: exact-category rigidity, asymptotic linearity, linkage-stable generalized Ulrich modules, and the geometry of extremal vector bundles. The common theme is not merely maximal generation, but the persistence of linear or near-linear structure across reductions, syzygies, powers, and geometric realizations.