Rees Algebras of Modules

• Rees algebras of modules are graded algebras attached to finitely generated modules over Noetherian rings, extending classical ideal theory by encoding blowup behavior, integral dependence, and homological properties.
• They are constructed via versal maps, intrinsic divided-power formulations, and coherent functor frameworks, ensuring torsionless behavior and embedding independence.
• Applications include criteria for linear type, explicit computational methods for defining equations, and geometric interpretations through total blow-ups and Nash transformations.

Rees algebras of modules are graded algebras attached to finitely generated modules over Noetherian rings that extend the classical Rees algebra of an ideal and encode blowup-type, integral-dependence, and homological information. For an ideal $I\subset R$, the classical construction is $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$. For a finitely generated module $M$, the modern theory combines the Eisenbud–Huneke–Ulrich definition via maps to free modules, an intrinsic divided-power description, and a coherent-functor formulation in which the Rees algebra is induced by a canonical natural transformation $t_M\to h^{M^*}$; together these descriptions make independence of embedding, torsionless behavior, and functoriality substantially more transparent (Ståhl, 2014, Ståhl, 2014).

1. Classical definition and module-theoretic generalization

For a finitely generated module $M$ over a Noetherian ring $A$, Eisenbud–Huneke–Ulrich define the Rees algebra by

$$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$

where $g$ ranges over all homomorphisms $M\to E$ with $E$ free. This recovers the ideal case when $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$0, but the passage from ideals to modules is nontrivial because a module need not have a canonical embedding into a free module, and different maps $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$1 can produce different images in $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$2 unless one isolates the correct universal class of maps (Ståhl, 2014).

The relevant class is that of versal maps. A homomorphism $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$3 to a finitely generated free module is versal if every homomorphism from $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$4 to a free module factors through $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$5. For such a map,

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$6

and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$7 is versal if and only if the dual map $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$8 is surjective. Versal maps always exist for finitely generated modules over Noetherian rings, and a versal map factors canonically as $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$9 with $M$0 injective (Ståhl, 2014).

A central structural notion is the torsionless quotient

$M$1

The degree-one part of $M$2 is exactly $M$3, and $M$4. Thus the Rees algebra depends only on the torsionless part of the module, not on any embedded torsion phenomena invisible to maps into free modules. This is one of the sharp distinctions between the ideal case and the general module case (Ståhl, 2014).

Functoriality is partial but precise. If $M$5 is surjective, then $M$6 is surjective. If $M$7 is surjective, then $M$8 is injective. These criteria are especially useful when studying reductions, embeddings, and Bourbaki constructions.

2. Intrinsic divided-power formulation

An intrinsic definition replaces the intersection over all maps to free modules by a single canonical map into a graded dual of a divided power algebra. For a finitely generated module $M$9 over a Noetherian ring $t_M\to h^{M^*}$0, there is a canonical graded algebra homomorphism

$t_M\to h^{M^*}$1

and the Rees algebra is precisely its image: $t_M\to h^{M^*}$2 Here $t_M\to h^{M^*}$3 is the divided power algebra of $t_M\to h^{M^*}$4, and $t_M\to h^{M^*}$5 denotes graded dual (Ståhl, 2014).

This description is intrinsic in the sense that it does not require a choice of embedding $t_M\to h^{M^*}$6. It is also compatible with the versal-map formulation: if $t_M\to h^{M^*}$7 is versal with $t_M\to h^{M^*}$8 finite free, then $t_M\to h^{M^*}$9, and the composite

$M$0

agrees with $M$1. Consequently, the image of the intrinsic map equals the Eisenbud–Huneke–Ulrich Rees algebra (Ståhl, 2014).

The free-module case is particularly simple: if $M$2 is finite free, then $M$3 is an isomorphism. In general, however, $M$4 need not be generated in degree $M$5. This obstruction explains why divided powers provide a correct intrinsic target but not always a convenient replacement for symmetric algebras in explicit computations. The phenomenon is visible already for $M$6 and $M$7, where $M$8, while $M$9 contains higher-degree generators not generated by degree $A$0 elements (Ståhl, 2014).

The divided-power framework also clarifies two recurrent cautions. First, the formula $A$1-torsion}) is reliable in many rank-theoretic contexts, but over rings with zero divisors it is not a universally adequate replacement for the intrinsic definition. Second, $A$2 does not preserve coproducts in general: even in the example above, $A$3 (Ståhl, 2014).

3. Coherent functors and the canonical map $A$4

A conceptual reformulation places the theory inside the category of coherent functors. Let $A$5 denote the category of finitely generated $A$6-modules, and let $A$7 be the full subcategory of additive covariant functors $A$8 admitting a presentation

$A$9

The tensor functors $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$0 and the representable functors $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$1 are coherent, and kernels, cokernels, and images of morphisms of coherent functors remain coherent (Ståhl, 2014).

Within this category there is an exact contravariant duality $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$2 characterized by $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$3 and $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$4. Applying the evaluation construction to $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$5 produces a canonical morphism $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$6, and dualizing yields the canonical map

$$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$7

Its value on an $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$8-module $$\mathcal R(M)=\operatorname{Sym}(M)\Big/\bigcap_g \ker(\operatorname{Sym}(g)),$$9 is the explicit transformation

$g$0

This is the fundamental map from which the Rees algebra is induced (Ståhl, 2014).

The image

$g$1

is the torsionless quotient functor. It is coherent, satisfies $g$2, preserves surjections, and depends only on $g$3. Moreover, for a homomorphism $g$4, the induced morphism $g$5 is injective if and only if $g$6 is surjective, and $g$7 is surjective if and only if $g$8 is surjective. These criteria parallel the injectivity and surjectivity criteria for Rees algebras and make their functorial origin explicit (Ståhl, 2014).

The coherent-functor framework culminates in a functor

$g$9

such that $M\to E$0 and $M\to E$1, where $M\to E$2 is the largest subring of $M\to E$3 generated in degree $M\to E$4. Under this functor,

$M\to E$5

so the Rees algebra is induced by the canonical coherent-functor morphism $M\to E$6. In this language, versality becomes equivalent to the injectivity of $M\to E$7, or equivalently of $M\to E$8, which gives a canonical explanation of why different versal embeddings yield the same Rees algebra (Ståhl, 2014).

4. Analytic spread, linear type, and Cohen–Macaulayness

For finite modules of positive rank, many later works use

$M\to E$9

as the working definition of the Rees algebra, especially in local or graded contexts where $E$0 is torsion-free and rank-theoretic hypotheses are built into the arguments. In this setting, $E$1 is of linear type if $E$2, and the graded pieces $E$3 play the role of powers of a module. Over a Noetherian local ring $E$4, the analytic spread is

$E$5

and if $E$6 has rank $E$7 and $E$8, then

$E$9

(Costantini, 2018).

The main mechanism for transferring ideal-theoretic results to modules is the generic Bourbaki ideal. Starting from a torsion-free rank-$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$00 module $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$01, one adjoins indeterminates, forms $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$02 generic linear combinations of generators, and obtains an exact sequence

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$03

with $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$04 free of rank $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$05 and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$06 an ideal, the generic Bourbaki ideal of $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$07. Under the standard hypotheses, $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$08 is Cohen–Macaulay if and only if $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$09 is Cohen–Macaulay, $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$10, and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$11. If $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$12 has sufficiently large grade, then $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$13 and the generators of $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$14 form a regular sequence on $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$15 (Costantini, 2018).

These reductions make it possible to formulate module analogues of Johnson–Ulrich and related ideal-theoretic criteria. Over Gorenstein local rings, finite torsion-free orientable modules $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$16 of rank $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$17 and analytic spread $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$18 are of linear type when $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$19 satisfies $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$20 together with suitable Ext-vanishing or depth conditions on the graded pieces $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$21. One sharp criterion states that if

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$22

then $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$23 is of linear type and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$24 is Cohen–Macaulay (Costantini, 2018). A related criterion, formulated through a generic Bourbaki ideal $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$25 of height $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$26, yields Cohen–Macaulayness of $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$27 under the bounds

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$28

(Lin, 2015).

The Bourbaki method also extends to fiber cones. When the deformation $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$29 holds under either an $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$30 hypothesis or local depth conditions at the non-linear-type locus, one obtains analogous deformations for fiber cones and a transfer principle for Cohen–Macaulayness. In particular, if $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$31 has projective dimension $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$32, satisfies $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$33, and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$34 is Cohen–Macaulay, then $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$35 is Cohen–Macaulay (Costantini, 2020).

5. Defining equations and computational methods

The defining ideal of $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$36 is often studied through the quotient map $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$37. In the ideal case, if $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$38 is height two and minimally generated by three homogeneous forms of degree $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$39, then the torsion module

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$40

admits a $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$41-module description. After Fourier transforming the linear equations $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$42 of $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$43 to operators $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$44, one has

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$45

and, equivalently, $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$46 for a suitable de Rham complex. The bigraded pieces $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$47 are characterized exactly by integral roots of Bernstein–Sato type polynomials $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$48: $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$49 This gives a precise algorithmic control of nonlinear equations in terms of Weyl-algebra Gröbner methods, and the same paper outlines an extension of the construction to modules $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$50 (Cid-Ruiz, 2017).

For modules of projective dimension $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$51 over hypersurface rings $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$52, explicit defining equations can be obtained via generic Bourbaki ideals and modified Jacobian duals. If $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$53 is linearly presented, satisfies $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$54, has rank $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$55, and is minimally generated by $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$56 homogeneous elements of the same degree, then after lifting the presentation matrix to $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$57 and iterating the modified Jacobian dual construction $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$58 times, the defining ideal of $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$59 is

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$60

In this setting $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$61 is Cohen–Macaulay if and only if $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$62, and is almost Cohen–Macaulay otherwise. The special fiber is a hypersurface of degree $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$63, so $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$64 and the relation type is $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$65 (Weaver, 2021).

A different method uses duality inside the symmetric algebra when $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$66 is a complete intersection. For modules $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$67 of projective dimension $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$68, the ideal $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$69 defining $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$70 inside $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$71 satisfies

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$72

and the top component is determinantal: $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$73 Lower components are recovered from dualized Koszul strands via the Kim–Mukundan complex. In the case $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$74 with $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$75 quadrics and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$76, the tangent algebra $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$77 has $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$78 generated by one equation of bidegree $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$79, $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$80 minimally generated by four equations of bidegree $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$81, and, under a height hypothesis, $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$82 minimally generated by one equation of bidegree $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$83 (Weaver, 2024).

Linearly presented modules with weak residual conditions admit still more explicit descriptions. Over $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$84, if a projective-dimension-one module $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$85 satisfies $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$86 but not $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$87, and the presentation matrix has rank $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$88 modulo $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$89, then two cases arise. In the column case,

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$90

and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$91 is a Cohen–Macaulay domain of fiber type with $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$92. In the row case,

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$93

and $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$94 is again a Cohen–Macaulay domain, but not of fiber type when $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$95; here $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$96 (Costantini et al., 2024). Under the more specific hypothesis $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$97 but not $$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$98, one obtains the formula

$$\mathcal R(I)=\bigoplus_{n\ge 0} I^n t^n\subset R[t]$$99

both for linearly presented perfect height-two ideals and, via generic Bourbaki ideals, for linearly presented modules of projective dimension one (Costantini et al., 2023).

6. Geometric interpretations and later developments

The projective spectrum of the Rees algebra of a coherent sheaf provides a geometric incarnation of the theory. For a coherent sheaf $M$00 on a locally Noetherian scheme $M$01, one defines

$M$02

This scheme, called the total blow-up, carries a universal quotient $M$03. If $M$04 is locally free on an open $M$05, then for any $M$06 with $M$07 schematically dense, morphisms $M$08 correspond exactly to surjections $M$09 with $M$10 invertible. For ideal sheaves this recovers the classical blow-up. The same framework identifies the Nash transformation with the total blow-up of the determinant, and characterizes certain birational projective morphisms as $M$11 for a very ample line bundle $M$12 (Ståhl, 2015).

This geometric perspective also clarifies a persistent subtlety: base change for module Rees algebras is not fully formal. Under suitable injective flat extensions making the pulled-back module locally free, $M$13 is injective and the base-changed Rees algebra maps onto the Rees algebra of the base-changed module. Without such hypotheses, there may be no canonical morphism $M$14 (Ståhl, 2015).

More recently, Rees algebras of modules have entered local algebra through normalization questions. For a Noetherian local ring $M$15, a submodule $M$16 of a finite free module defines

$M$17

If $M$18 denotes the integral closure of $M$19 in $M$20, then $M$21 is analytically unramified if and only if $M$22 is finitely generated as an $M$23-module for every such $M$24; equivalently, it suffices that this hold for one kernel $M$25 of a surjection $M$26 with $M$27 a nonzero finite-length module. This extends Rees’s classical ideal-theoretic characterization of analytic unramifiedness to the module setting (Kodiyalam et al., 17 Jul 2025).

Taken together, these developments place Rees algebras of modules at the intersection of blowup algebras, residual intersection theory, homological criteria for linear type and Cohen–Macaulayness, explicit determinantal and $M$28-module equations, and geometric constructions such as total blow-ups and tangent algebras. The subject retains the classical ideal-theoretic core while acquiring genuinely module-theoretic phenomena—torsionless quotients, dependence on coherent-functor images, Bourbaki reduction, and nontrivial behavior under coproducts and base change—that have no exact analogue in the ideal case.