Brenner–Schröer Proj Construction

• Brenner–Schröer Proj is the multigraded extension of Grothendieck’s classical Proj, replacing single-element localization with homogeneous submonoid localization.
• It constructs schemes by gluing affine charts Spec(A_(S)) obtained from degree-zero parts (potions) of localized rings, thereby handling arbitrary finitely generated abelian gradings.
• This framework has practical applications in flag varieties, toric geometry, and Cox rings, and it has been rigorously formalized in proof assistants like Lean4.

The Brenner–Schröer Proj construction is the multigraded extension of Grothendieck’s classical $\operatorname{Proj}$ from $\mathbb N$-graded rings to commutative rings graded by an arbitrary finitely generated abelian group $M$. Its basic innovation is to replace localization at powers of a single homogeneous element by localization at relevant homogeneous submonoids, and to replace the classical rings $(A_f)_0$ by the degree-zero parts $A_{(S)}=(A_S)_0$ of such localizations, called potions. The resulting scheme $\mathrm{Proj}^M(A)$ is obtained by gluing the affine schemes $\operatorname{Spec}(A_{(S)})$; in modern treatments this gluing is taken as the primary definition, while under additional hypotheses one can recover a point-set description analogous to the classical homogeneous-prime picture (Mayeux et al., 18 Sep 2025).

1. Historical position and guiding idea

Brenner–Schröer defined the Proj of a ring graded by an arbitrary finitely generated abelian group, thereby extending the range of projective methods from singly graded situations to genuinely multigraded ones such as $\mathbb Z^r$-graded rings, Cox rings, and multi-projective constructions (Mayeux et al., 18 Sep 2025). In the later formulation emphasized by Mayeux–Riche, and adopted in recent formalization work, the construction is organized around affine gluing rather than around a distinguished subset of a prime spectrum (Mayeux et al., 2023).

The conceptual departure from the classical case is forced by the grading group. For an $\mathbb N$-graded ring $A=\bigoplus_{n\ge 0}A_n$, one localizes at a homogeneous element $\mathbb N$0, takes the degree-zero part $\mathbb N$1, and glues the affine opens $\mathbb N$2. When the grading is by a general finitely generated abelian group $\mathbb N$3, there is no canonical notion of “positive degree,” and a single homogeneous element need not control enough degrees. Brenner–Schröer’s idea is therefore to work with homogeneous submonoids $\mathbb N$4, require a relevance condition on their degrees, form the potion $\mathbb N$5, and glue the corresponding spectra (Mayeux et al., 18 Sep 2025).

This construction sits, in the language of the formalization paper, “at the crossroads of algebraic geometry and Lie theory,” with applications including flag varieties and Springer resolutions (Mayeux et al., 18 Sep 2025). The same general framework also underlies later toric and Cox-ring comparisons, where Brenner–Schröer Proj appears as a universal multigraded projective object attached to a graded ring (Mallick et al., 2022).

2. Algebraic ingredients: gradings, relevance, and potions

Let $\mathbb N$6 be a finitely generated abelian group and let

$\mathbb N$7

A homogeneous element is an element of some $\mathbb N$8; a submonoid $\mathbb N$9 is homogeneous if all its elements are homogeneous (Mayeux et al., 18 Sep 2025).

For a homogeneous submonoid $M$0, one considers the set of occurring degrees

$M$1

its associated degree submonoid $M$2, and the subgroup

$M$3

A further saturation operation is essential: $M$4 denotes the homogeneous submonoid of homogeneous divisors of elements of $M$5. One has canonical isomorphisms $M$6, and similarly for localized graded modules (Mayeux et al., 18 Sep 2025).

Relevance is the multigraded substitute for the classical condition that powers of a positive-degree element eventually shift degrees as needed. A homogeneous submonoid $M$7 is $M$8-relevant if for any $M$9 there exists $(A_f)_0$0 such that $(A_f)_0$1; equivalently, $(A_f)_0$2 is torsion (Mayeux et al., 18 Sep 2025). A homogeneous element or a family of homogeneous elements is called relevant when the submonoid it generates is relevant.

Given a homogeneous submonoid $(A_f)_0$3, the localization $(A_f)_0$4 is canonically $(A_f)_0$5-graded. The potion of $(A_f)_0$6 with respect to $(A_f)_0$7 is

$(A_f)_0$8

For a graded $(A_f)_0$9-module $A_{(S)}=(A_S)_0$0, one likewise sets

$A_{(S)}=(A_S)_0$1

In the classical case $A_{(S)}=(A_S)_0$2, this recovers the usual ring $A_{(S)}=(A_S)_0$3 (Mayeux et al., 18 Sep 2025).

A recurring misconception is that multigraded Proj should still be governed by a single irrelevant ideal in the same elementary way as in the $A_{(S)}=(A_S)_0$4-graded case. Later work shows that relevance in the general $A_{(S)}=(A_S)_0$5-graded setting is subtler: for a homogeneous element $A_{(S)}=(A_S)_0$6, one defines $A_{(S)}=(A_S)_0$7 as the subgroup generated by the degrees of homogeneous units in $A_{(S)}=(A_S)_0$8, calls $A_{(S)}=(A_S)_0$9 relevant when $\mathrm{Proj}^M(A)$0 has finite index in $\mathrm{Proj}^M(A)$1, and lets the irrelevant ideal $\mathrm{Proj}^M(A)$2 be generated by all relevant elements (Goebler, 13 Feb 2026). This formulation is compatible with the Brenner–Schröer viewpoint but makes explicit that periodicity of localizations replaces positivity of degree.

3. Gluing affine charts and the scheme $\mathrm{Proj}^M(A)$3

The central technical input is what one source calls the “magic of potions.” If $\mathrm{Proj}^M(A)$4 and $\mathrm{Proj}^M(A)$5 are homogeneous submonoids, there is a canonical ring homomorphism

$\mathrm{Proj}^M(A)$6

When $\mathrm{Proj}^M(A)$7 is relevant and $\mathrm{Proj}^M(A)$8 is finitely generated, $\mathrm{Proj}^M(A)$9 is canonically a localization of $\operatorname{Spec}(A_{(S)})$0 at explicit degree-zero elements constructed from generators of $\operatorname{Spec}(A_{(S)})$1 together with degree-correcting elements from $\operatorname{Spec}(A_{(S)})$2. Consequently,

$\operatorname{Spec}(A_{(S)})$3

is an open immersion (Mayeux et al., 18 Sep 2025).

This yields the affine overlap calculus. For finitely generated relevant homogeneous submonoids $\operatorname{Spec}(A_{(S)})$4, set

$\operatorname{Spec}(A_{(S)})$5

Then $\operatorname{Spec}(A_{(S)})$6 is canonically an open subscheme of both $\operatorname{Spec}(A_{(S)})$7 and $\operatorname{Spec}(A_{(S)})$8, one has $\operatorname{Spec}(A_{(S)})$9 and $\mathbb Z^r$0, and triple intersections satisfy the compatibility needed for gluing (Mayeux et al., 18 Sep 2025).

Let $\mathbb Z^r$1 denote the set of relevant homogeneous submonoids of $\mathbb Z^r$2 that are finitely generated as submonoids of $\mathbb Z^r$3. Gluing the family $\mathbb Z^r$4 along the open subschemes $\mathbb Z^r$5 produces a scheme

$\mathbb Z^r$6

equipped with open immersions $\mathbb Z^r$7 whose images cover the scheme (Mayeux et al., 18 Sep 2025). In this presentation the underlying topological space is literally the quotient obtained by gluing the affine spectra $\mathbb Z^r$8.

The structure sheaf is the sheaf obtained by gluing the affine structure sheaves on the $\mathbb Z^r$9. For a graded module $\mathbb N$0, there is a unique quasi-coherent $\mathbb N$1-module $\mathbb N$2 characterized by

$\mathbb N$3

for every $\mathbb N$4 (Mayeux et al., 18 Sep 2025). This is the direct multigraded analogue of the classical correspondence $\mathbb N$5 on $\mathbb N$6.

The construction is quasi-separated: intersections of basic affine opens are covered by affine opens of the form $\mathbb N$7 coming from affine open immersions (Mayeux et al., 18 Sep 2025). In the original Brenner–Schröer quotient picture, the same scheme may also be described as the quotient of $\mathbb N$8 by the diagonalizable group attached to the grading lattice; Mayeux–Riche show that this quotient description coincides with the potion-gluing construction (Mayeux et al., 2023).

4. Topological description, $\mathbb N$9-prime ideals, and the role of factoriality

In the gluing formulation, multigraded Proj is not defined as a subset of a prime spectrum. That omission is deliberate. For general multigraded rings, the classical description by homogeneous prime ideals not containing an irrelevant ideal does not extend directly. A later topological analysis identifies the correct replacement, under additional hypotheses, as a spectrum of $A=\bigoplus_{n\ge 0}A_n$0-prime ideals rather than ordinary prime ideals (Goebler, 13 Feb 2026).

Let $A=\bigoplus_{n\ge 0}A_n$1 be a $A=\bigoplus_{n\ge 0}A_n$2-graded integral domain, with $A=\bigoplus_{n\ge 0}A_n$3 a finitely generated abelian group. A graded ideal $A=\bigoplus_{n\ge 0}A_n$4 is $A=\bigoplus_{n\ge 0}A_n$5-prime if

$A=\bigoplus_{n\ge 0}A_n$6

for all homogeneous $A=\bigoplus_{n\ge 0}A_n$7. The ring is factorially $A=\bigoplus_{n\ge 0}A_n$8-graded if every nonzero nonunit homogeneous element is a product of $A=\bigoplus_{n\ge 0}A_n$9-prime elements (Goebler, 13 Feb 2026). These notions are weaker than their ordinary ring-theoretic counterparts; the paper’s example $\mathbb N$00 with $\mathbb N$01-grading shows that a $\mathbb N$02-prime element need not be prime in the usual sense.

For relevant homogeneous $\mathbb N$03, one has the localizing map

$\mathbb N$04

The decisive result is that, for factorially $\mathbb N$05-graded integral domains, $\mathbb N$06 is a homeomorphism for every relevant $\mathbb N$07 (Goebler, 13 Feb 2026). This restores the classical local picture once one replaces homogeneous prime ideals by $\mathbb N$08-prime ideals.

Under the assumptions that $\mathbb N$09 is factorially and effectively $\mathbb N$10-graded and $\mathbb N$11 is a conical ring, the multihomogeneous $\mathbb N$12-prime spectrum

$\mathbb N$13

admits a scheme structure for which every relevant chart $\mathbb N$14 is isomorphic to $\mathbb N$15, and the resulting scheme is canonically isomorphic to the Brenner–Schröer construction $\mathbb N$16 (Goebler, 13 Feb 2026). In this sense the point-set description exists, but only after strengthening the hypotheses and modifying the notion of primality.

The same framework yields a multigraded Nullstellensatz. For factorially $\mathbb N$17-graded conical rings, radical homogeneous ideals in the relevant part correspond to closed subsets of $\mathbb N$18, and closure is computed by vanishing of the appropriate homogeneous functions (Goebler, 13 Feb 2026). This places the Brenner–Schröer construction much closer to the classical $\mathbb N$19 once the correct prime-like objects have been identified.

5. Structural theorems, twists, and non-separatedness

The Brenner–Schröer Proj construction retains several formal properties of classical Proj. A homomorphism $\mathbb N$20 of $\mathbb N$21-graded rings induces a canonical morphism

$\mathbb N$22

obtained by gluing the affine maps $\mathbb N$23 (Mayeux et al., 18 Sep 2025). If $\mathbb N$24 is surjective, then $\mathbb N$25 is a closed immersion (Mayeux et al., 18 Sep 2025).

The construction is also compatible with tensor products and products. If $\mathbb N$26 is $\mathbb N$27-graded and $\mathbb N$28 is $\mathbb N$29-graded over a commutative ring $\mathbb N$30, then

$\mathbb N$31

(Mayeux et al., 18 Sep 2025). A related base-change statement is established in the Mayeux–Riche treatment (Mayeux et al., 2023).

Twisting sheaves are indexed by the grading group itself. For $\mathbb N$32, the degree shift $\mathbb N$33 is defined by $\mathbb N$34, and one sets

$\mathbb N$35

If $\mathbb N$36 is noetherian, then $\mathbb N$37 is coherent, and $\mathbb N$38 is coherent for every finitely generated graded $\mathbb N$39-module $\mathbb N$40 (Mayeux et al., 18 Sep 2025). Under a covering condition by maximally relevant submonoids, these twisting sheaves are invertible, and if $\mathbb N$41 is quasi-compact the functor $\mathbb N$42 induces an equivalence between graded modules modulo negligible modules and quasi-coherent sheaves on $\mathbb N$43 (Mayeux et al., 18 Sep 2025).

Later work refines the invertibility problem in terms of local degree groups. For $\mathbb N$44 and $\mathbb N$45, the twist $\mathbb N$46 is invertible if and only if, for every relevant $\mathbb N$47, the localization $\mathbb N$48 contains a unit of degree $\mathbb N$49; equivalently, $\mathbb N$50 lies in the intersection of the subgroups $\mathbb N$51 attached to generators of the irrelevant ideal (Goebler, 13 Feb 2026). This makes precise the difference between the full divisor-class grading and the subgroup of degrees that actually yield line bundles.

Separatedness is the major point where multigraded Proj diverges sharply from the classical $\mathbb N$52-graded theory. In the $\mathbb N$53-graded setting, $\mathbb N$54 is separated if and only if the multiplication maps

$\mathbb N$55

are surjective for relevant pairs $\mathbb N$56 (Goebler, 13 Feb 2026). Weak pairs detect the failure of this surjectivity, and linear dependencies in the degree group explain when such failures occur. If every linear dependency among relevant degrees is of length $\mathbb N$57, then $\mathbb N$58 is separated; if there exists a non-trivial irreducible dependency of higher length, then $\mathbb N$59 is not separated (Goebler, 13 Feb 2026).

The standard example is $\mathbb N$60 with $\mathbb N$61 and

$\mathbb N$62

Here $\mathbb N$63 is a dependency of length $\mathbb N$64, the map $\mathbb N$65 fails to be surjective, and $\mathbb N$66 is not separated; geometrically, the source compares this to a “double origin” phenomenon (Goebler, 13 Feb 2026).

6. Geometric realizations and comparisons with toric Proj-like constructions

A principal application is the realization of flag varieties as multigraded Proj schemes. Let $\mathbb N$67 be algebraically closed, $\mathbb N$68 a connected reductive group, $\mathbb N$69 a maximal torus, and $\mathbb N$70 a Borel subgroup. If

$\mathbb N$71

then $\mathbb N$72 is naturally graded by the character group $\mathbb N$73, and one has

$\mathbb N$74

(Mayeux et al., 18 Sep 2025). This realizes the flag variety without choosing a single dominant weight. More generally, if $\mathbb N$75 is a finite-dimensional $\mathbb N$76-module and $\mathbb N$77 is a $\mathbb N$78-stable subspace, then the associated bundle $\mathbb N$79 is likewise realized as a multigraded Proj; in the case $\mathbb N$80 and $\mathbb N$81, this gives the Springer resolution (Mayeux et al., 18 Sep 2025).

The same multigraded formalism encompasses products and multi-projective geometry. The product formula above is the multigraded analogue of the familiar description of $\mathbb N$82 by a $\mathbb N$83-graded ring (Mayeux et al., 18 Sep 2025). This suggests, in the language of the sources, a natural role for Cox rings and toric constructions, although some of those implications are described as standard context rather than developed in detail.

In the toric setting, Brenner–Schröer Proj has been compared with Perling’s toric Proj. For the Cox-type ring $\mathbb N$84 attached to a toric variety $\mathbb N$85, there is always a canonical torus-equivariant open embedding

$\mathbb N$86

from Perling’s toric Proj into the Brenner–Schröer multihomogeneous Proj (Mallick et al., 2022). When $\mathbb N$87 is simplicial, $\mathbb N$88 is an isomorphism if and only if $\mathbb N$89 is simplicially complete (Mallick et al., 2022). This comparison makes precise that Brenner–Schröer Proj is generally larger: it uses all relevant homogeneous elements, not only those singled out by the toric fan.

A common misunderstanding is therefore to treat Brenner–Schröer Proj as merely a toric reconstruction device. The toric comparison shows instead that it is a general multigraded projective construction; Perling’s toric Proj is a fan-sensitive subconstruction that agrees with it only under a specific combinatorial condition (Mallick et al., 2022).

7. Formalization and current extensions

The multi-graded Brenner–Schröer Proj construction has been formalized in Lean4. One formalization presents the multi-graded Proj construction for rings graded by finitely generated abelian groups and emphasizes the gluing of affine schemes built from potions (Mayeux et al., 18 Sep 2025). A subsequent development extends this to a detailed Lean4 formalization of Brenner–Schröer Proj schemes together with algebraic dilatations (Mayeux et al., 31 May 2026).

In the formalization, homogeneous localization is implemented as a quotient type built from triples $\mathbb N$90 with numerator and denominator homogeneous of the same degree. Mathematically, this recovers the degree-zero fraction subring $\mathbb N$91, while making the universal-property aspect explicit (Mayeux et al., 18 Sep 2025). The “magic of potions” is encoded through a structure PotionGen, which packages generators of a submonoid together with exponents and degree-correcting data used to exhibit $\mathbb N$92 as a localization of $\mathbb N$93 (Mayeux et al., 18 Sep 2025).

The later Lean4 treatment introduces GoodPotionIngredient for finitely generated relevant homogeneous submonoids and constructs the scheme by Scheme.GlueData, using the affines $\mathbb N$94 and the overlaps $\mathbb N$95 (Mayeux et al., 31 May 2026). It also proves functoriality for graded ring homomorphisms, open-immersion properties for overlaps, and invariance under enlarging the covering family (Mayeux et al., 31 May 2026).

This formalized perspective suggests a methodological point rather than a new theorem: the Brenner–Schröer construction is especially well suited to mechanization because it reduces the geometry to graded localizations, degree-zero parts, and explicit gluing morphisms (Mayeux et al., 18 Sep 2025). A plausible implication is that multigraded algebraic geometry, particularly in contexts such as Cox rings, toric geometry, and dilatation-type constructions, becomes more tractable in proof assistants when formulated in the Brenner–Schröer language rather than through prime-ideal descriptions alone.