Barycentric invariance and
universal parameters for face rings
Caius Folkerts
Abstract. We prove that barycentric subdivision preserves a face ring as a
module over its universal parameters, for every finite simplicial poset and every
commutative coefficient ring. The comparison is integral, commutes with all
face restrictions, and induces the canonical straightening map on associated
graded; its filtered structure proves the equivariant Tor conjecture of Adams and
Reiner in every characteristic. Equivariance of the comparison itself has a sharp
obstruction; in rank , the universal condition is invertibility of . The construction
lifts comparisons from the boundary of each face, giving relative resolutions and
extending to homogeneous straightening algebras, including arbitrary unions of
Grassmannian Richardson varieties.
Barycentric subdivision preserves the topology of a simplicial poset while changing the
multiplication in its face ring, but the two rings still carry a common system of universal
parameters. Adams and Reiner conjectured that their equivariant Betti numbers over this
parameter ring agree [1], while Murai asked whether the rings themselves are isomorphic
as parameter modules [2].
We construct integral module isomorphisms compatible with restriction to every face,
over every commutative coefficient ring and without a Cohen–Macaulay hypothesis. Their
filtered structure proves the equivariant Tor formula in every characteristic, even though
the comparison itself need not be equivariant. The obstruction to making the
isomorphisms themselves equivariant is also determined exactly; for rank at most , the
universal condition is that be invertible.
Let be a finite simplicial poset with minimum and maximum rank . Thus each interval
is a Boolean lattice. Let be a commutative ring, and write
The generator of the face ring has degree , as does the barycentric vertex , for . The
parameter actions are
On a rank- face, the first action is by the elementary symmetric polynomials in its
vertex variables. The second is by color sums in the order complex, colored by
rank.
A comparison is coherent if it commutes with restriction to every face. Such
comparisons give compatible maps on every lower ideal of and on every kernel
of a restriction. The rank-weight filtration gives a rank- face variable weight .
Straightening lowers this weight, so chain monomials canonically identify with
.
Theorem 1.1.For every finite simplicial poset there are coherent graded -module
isomorphisms
over every commutative ring . They can be constructed over and then
base-changed. They preserve the rank-weight filtration, with the canonical
chain-monomial identification on associated graded. The same comparisons identify
every relative restriction kernel.
Blum-Smith and Marques constructed equivariant comparisons for Cohen–Macaulay
complexes when the group order is invertible, and exhibited modular obstructions [2].
Theorem 1.1 answers the nonequivariant module question in full. Its construction uses
freeness only on individual faces, where the parameters are elementary symmetric
polynomials. Coherence then assembles these local comparisons even when the global face
ring is not free.
Let be a finite group of automorphisms of . For a graded representation we write for
its class in the Grothendieck group . For , let denote the rank-selected order
complex.
Theorem 1.2.There is a finite -stable filtration in each internal degree with
canonical equivariant identification
It is compatible with multiplication on associated graded Tor. For a -stable lower
ideal , the relative version uses . Over a field this gives the corresponding identity in
.
The equivariant Tor formula holds even when an equivariant module isomorphism does
not exist. A single simplex gives the exact obstruction to such isomorphisms in a fixed
rank.
Theorem 1.3.For , coherent graded equivariant parameter-module comparisons
exist over for every simplicial poset of maximum rank at most and every
automorphism group if and only if is a unit in . If it is not a unit, the polynomial
and barycentric rings of the rank-simplex admit no -equivariant parameter-module
isomorphism, even without grading or coherence.
We also determine the integral obstruction; on a rank- simplex, the possible constant
scalars of coherent -equivariant graded parameter-linear maps form exactly the ideal
.
The comparison theorem is proved locally on faces; quotienting by the top parameter
leaves exactly the boundary diagram, where a comparison has already been constructed.
Freeness lifts that map to the face and positive degree makes the lift invertible, while bases
adapted to straightening preserve the canonical leading terms. This boundary-lifting
argument is also the reason the theorem extends to relative modules and survives arbitrary
base change.
The same construction controls parameter changes and their fibers. For the polynomial
face parameters of Section 7, the module depends only on the degree profile,
and its Betti numbers at a character over a field depend only on the highest
active rank. Consequently its nontrivial rank-jump loci are nested coordinate
subspaces.
Local freeness and the boundary quotients also make the lifting argument available for
homogeneous straightening algebras and arbitrary unions of Grassmannian Richardson
varieties.
The factorial obstruction concerns comparisons uniform over all actions; particular
actions can admit integral equivariant lifts without that inversion. Compatible choices on
faces and their fixed-face quotients give such lifts for free actions with cyclic Sylow
subgroups, using integral templates natural for cyclic orders. The relative construction in
Appendix A extends this lifting method to further actions.
2. Boundary lifting and the coherent comparison
2.1. The face diagrams.
For of rank , put
Restrictions kill vertex variables or face generators outside the smaller face. Include the
constant stalk . Then
The constant stalk keeps the components compatible when the complex is disconnected.
Both descriptions follow coefficientwise from the standard bases; weak chains of nonzero
faces, together with the empty monomial. A chain coefficient is detected at its
largest face; compatibility there matches that coefficient in every containing
stalk.
The face-ring relations are
when have common upper bounds, the sum being over their minimal upper bounds,
with ; otherwise the product is zero. On each containing Boolean face this is the ordinary
product identity in its vertex variables. Sorting their exponents gives the chain basis and
proves (1) over and over every .
On a rank- stalk, acts as on and as on . Both are regular, and
Indeed, the common kernel of proper-face restrictions is spanned by the chain
monomials containing the top face. On the barycentric side .
For a two-vertex face, this says
with parameters and . Setting the second parameter to zero identifies both rings with
the boundary ring . The free parameter bases and then lift this boundary identification
by , . This illustrates the general construction; a free basis lifts the comparison already
defined on the boundary, preserving the parameter action without requiring the two
multiplications to agree.
Lemma 2.1 (Boundary lifting).Let be positively graded, let be a finite graded-free
-module, and let be bounded below. Suppose a homogeneous of positive degree
is regular on both modules. Every graded isomorphism lifts to a graded -module
isomorphism .
Proof.Lift the prescribed image of each homogeneous free generator and extend
-linearly. The resulting map is surjective by induction on degree modulo . If an
element of its kernel is , regularity on puts in the kernel too. Repeated division
lowers degree, so boundedness below makes the kernel zero. □
The lemma applies to any finite boundary diagram; on each nonzero stalk require a
finite graded-free source over its local parameter ring, a regular positive boundary
parameter on both sides, and quotients equal to the actual inverse limits of the proper
stalks. Induction constructs coherent comparisons. Every coherent family whose scalar
at the constant stalk is a unit is automatically an isomorphism, by the same
induction.
2.2. Adapted bases.
Recall the rank weights . If incomparable have ranks , and their meet has rank , every
straightening term has other rank , so the weight drops by
Thus the chain bases give a canonical identification
compatible with every restriction. All filtrations are finite in each internal
degree.
Order the atoms of a face by . For , let be its descent set and its first atoms. Set
Neither expression involves the top face.
Lemma 2.2.The form an integral free basis of over its color sums. The form an
adapted free basis of over its elementary symmetric parameters, with leading symbols
.
Proof.Fix a color multidegree with support . Its chain monomials correspond to
chains whose selected ranks are . Sorting each block between successive selected
ranks gives a unique permutation with . Expand the parameter monomials times in
this chain basis, and order permutations lexicographically. The permutation is the
lexicographically least one with its required descent prefixes; its diagonal coefficient
is . The matrix is unitriangular. This proves the first assertion over , without division.
Transfer each chain monomial to the source. The corresponding map from a
filtered free parameter module has associated graded the first basis isomorphism,
by (3). Finite filtration induction in each degree makes the map an isomorphism,
proving the second assertion and its compatibility with base change. □
Proof of Theorem 1.1. Induct over faces, using the maps on proper faces as the
prescribed boundary map in (2). For each , apply this map to the boundary class and
lift its image by the representative containing no . Extend over . Lemma 2.1 makes
this an isomorphism and ensures coherence. The boundary map has the canonical
leading symbols by induction, and contains no top variable; hence its chosen lift
still has leading symbol . Lemma 2.2 gives a filtered isomorphism with canonical
associated graded. Constructing over and tensoring the maps and their inverses
gives the assertion over every . Inverse limits and restriction kernels preserve these
isomorphisms. □
The correction at the boundary is necessary. The uncorrected assignment
need not commute with face restrictions; the lift above imposes precisely those
compatibilities.
A more separated system of weights makes triangularity automatic for every normalized
coherent comparison.
Proposition 2.3.Every coherent graded comparison normalized to on constants
is automatically triangular for the weights
Its associated graded map is the canonical chain-monomial map.
Proof.These weights also strictly decrease under straightening; with the ranks above,
the decrease is . At fixed internal degree, decreasing weight is equivalent to increasing
the cost of the factors. An adapted generator on a rank- face has cost at most .
Inductively, any discrepancy in its leading symbol vanishes on the boundary and is
therefore divisible by . Such a discrepancy has cost at least , so has smaller weight.
Extend over the parameters and induct on rank. □
3. Choices and equivariant filtrations
Although the comparison need not be unique or equivariant, its choices form an
affine space whose differences are controlled by a nilpotent ideal. This structure
lets us compare equivariant invariants even when no equivariant comparison
exists.
Choose an integral homogeneous free basis of each polynomial stalk over
its elementary symmetric parameters. A prescribed boundary image of has a
unique top-free lift, and every other lift is obtained by adding , where . Thus
the module of coherent degree-zero parameter-linear families has coordinates
The first coordinate is its scalar at ; write for this map. This is a finite free module,
and the coordinates commute with every base change. Its unit-scalar points are precisely
its isomorphisms.
Let be the algebra of degree-zero coherent endomorphisms of the source diagram, and
let be its ideal of maps zero at . A fixed normalized comparison identifies its underlying
module with (4).
The key constraint on a difference is boundary vanishing; once it vanishes on all proper
faces, its image is divisible by the top parameter. Repeated compositions force further
factors of positive degree, while the degrees of a free basis remain bounded, which is why
the ambiguity in the comparison is nilpotent.
Lemma 3.1.The ideal is nilpotent. One may take
Proof.A product of elements vanishes on all proper faces of a rank- face, by
induction. Its component there has image in , and so is for a parameter-linear map
lowering degree by . A free basis has degrees at most . A product of such blocks
lowers degree past every generator, and is zero. □
Normalized comparisons form an affine space of isomorphisms; after choosing
one they are , with inverses given by finite geometric series. Over a field is a
finite-dimensional local algebra with radical . In ranks one and two, every free
generator has degree below the rank, so the normalized coherent comparison is
unique and hence equivariant. In rank three, each face already contributes one free
coefficient.
Proposition 3.2.For a finite group acting coherently on a boundary diagram, the
source and target modules have finite filtrations with identical equivariant successive
quotients. The same holds for restriction kernels and after applying a -linear additive
covariant or contravariant functor to the modules and their group actions. The
actions may be semilinear, permuting equal-degree parameters.
Proof.Transport the target action through a normalized comparison . The two
actions then satisfy
Their ratio is parameter-linear, coherent, and at the constant stalk, even when
both actions are semilinear for the same parameter permutation. Conjugation by
preserves . The filtration of a limit module or restriction kernel is therefore stable
under both actions, which agree on its quotients. After an additive functor use
powers of the image of in its endomorphism algebra; composition and nilpotence are
preserved. For a contravariant additive functor the opposite algebra gives the same
argument. □
For any finite boundary diagram the nilpotence proof is unchanged; replace by the
maximal generator degree at that level and by the degree of the regular boundary
parameter. This observation will also apply to interval diagrams and parameter
twists.
Finally, the affine obstruction to an equivariant coherent choice is additive. Put . For
any normalized comparison, is a cocycle in . Its class
is independent of the choice and vanishes precisely when an equivariant normalized
comparison exists. This concerns coherent maps; an arbitrary isomorphism of global
modules need not preserve face restrictions.
4. Tor and rank-selected cohomology
The comparison reduces the Tor calculation to a properly colored face module.
Let be a finite properly -colored simplicial pair. In the absolute case is the void
complex. Write for the relative face module and let act by the sum of the vertices of color
. We first use the color multigrading .
Lemma 4.1 (Colorful Koszul calculation).The color-multigraded Koszul homology
is supported on squarefree degrees. In degree it is canonically
This is equivariant under color-preserving automorphisms.
Proof.In a strand whose color- degree is at least two, a monomial already contains
the unique vertex of that color in its support. Multiplication by the color sum simply
raises that vertex’s exponent. It is a bijection on the relevant coefficient modules,
and its inverse gives a one-coordinate Koszul contraction.
In a squarefree strand , a basis element is a face with colors , together with the
exterior factor on . The Koszul differential inserts one missing-colored vertex and
sums over its possible cofaces. This is the augmented relative cochain complex of the
rank selection, with degree . The sign change
identifies its differential with the cochain differential. The construction commutes
with every color-preserving automorphism. □
Proof of Theorem 1.2. Give the th Koszul exterior generator weight . The filtered
comparison in Theorem 1.1 gives a filtered chain isomorphism to the barycentric
Koszul complex, whose differential is homogeneous for weight. Thus kernels and
images are direct sums of their weight pieces on the barycentric side. Transporting
this fact back proves strictness; every boundary of a given filtration degree has
a preimage of that degree. Taking homology therefore commutes canonically with
associated graded. Lemma 4.1 computes that homology. The filtration and its leading
symbols are -equivariant, even though the comparison need not be.
The source Koszul complex is a filtered differential graded algebra, with the
barycentric Koszul algebra as associated graded. Leading symbols of products of
cycles are products of their leading symbols, and strictness identifies the leading
boundaries. Consequently the canonical homology identification is multiplicative.
Coherence gives the same strict chain comparison for every relative restriction kernel.
□
The same argument makes the Rees deformation constant as a parameter module; its
Rees module is isomorphic to over the correspondingly homogenised parameter ring. Its
Koszul homology is and commutes with every specialization of . This is a module
assertion; the algebra multiplications need not be isomorphic.
Example 4.2. The filtration can be nonsplit. Over , take the two triangles and
, with the involution interchanging . The face ring is . Set , and eliminate in the
universal-parameter quotient. Its relations become
Its degree-three basis is . The involution fixes the first two and sends the third to
, so its difference from the identity has rank one. On the barycentric side the same
degree is
a three-dimensional trivial representation. The two modules have the same
representation class but are not isomorphic.
4.1. Permutations of the colors.
Suppose also permutes the colors, preserving positive degrees , and give every vertex of
color degree . For , put ; its stabilizer acts by the sign of its permutation of
.
Proposition 4.3.There is an actual -module isomorphism
For a source boundary diagram coherently comparable to , the same formula gives
the class of its Tor representation over a field.
Proof.Repeated-color strands contract as before. In a squarefree strand, write for
the face colors and , with . Use the natural map
It identifies the summand with its oriented cochain line tensored with , naturally
under color permutations. Moving from to changes the shuffle sign by ; the Koszul
sign and the change of the displayed degree factor leave exactly the cochain sign.
Group the resulting summands into -orbits.
For the source, let act on Koszul exterior generators by . The transported actions
still differ by the image of , so Proposition 3.2 gives equal representation classes. □
The orientation line matters. For two isolated vertices of distinct colors interchanged by
an involution, both reduced and are sign lines. Their tensor product is trivial, as is the
Tor class of the relation .
5. Resolutions and coefficient diagrams
The colorful Tor formula comes from a resolution whose differential uses only face
incidences; constructing it explicitly will also give minimal resolutions and allow
coefficients that vary from face to face.
5.1. The incidence resolution.
For , let be the free -module on faces of with color set exactly . For , define
Proper coloring gives . The relative complement is upward closed, so this also holds for
relative modules.
Adding a missing color to a face monomial can be expressed either by multiplication by
its parameter or by summing the cofaces obtained by inserting a vertex of that color. Since
these operations agree in , the resolution records their difference, with one interval
coordinate for each color still to be added. Its exactness will be the contraction of these
intervals onto their upper endpoints.
Put
and define
Proposition 5.1.The complex is a finite graded free resolution of over every .
The augmentation sends to the face monomial . The resolution is equivariant under
color-preserving automorphisms.
Proof.Endpoint operations in distinct coordinates commute and their orientation
signs anticommute, giving . In color degree with support , remove the coefficient
monomial from each summand. The strand becomes the cellular chains of a cube;
coordinates in are fixed at , those in vary, and the others are . The coefficient at
that cell is ; the upper endpoint uses and the lower endpoint uses minus the identity.
For , let push to the upper -face. It is the identity if , zero if , and the map if
is absent. Define in the first two cases and otherwise
The boundary formula gives . The commuting contract the cube onto its all-one
vertex , which is the augmentation. The empty strand is the identity on ; negative
strands vanish. Everything is integral and natural in the face incidences. □
For one color and vertices, this resolution is simply
Its single relation says . Cancelling one unit entry gives the free module . The minimal
construction below performs the corresponding cancellations simultaneously in every color
set.
For , these bases are chains with specified rank sets. Composing the augmentation with
the coherent comparison inverse gives the same resolution matrices for and its relative
modules. Its free rank in degree is
An equivariant comparison gives an equivariant augmentation; no such choice is
presumed in modular characteristic.
5.2. Minimal differentials.
Over a field, we can remove the unit entries from this resolution while retaining explicit
differentials. Write , where coface insertion preserves and the parameter part deletes
one element of . On each finite complex choose cohomology splittings with
They preserve . Put
To turn a chosen -cohomology representative into a chain map for , correct it first by ,
then correct the new error by , and continue. Every correction deletes a color, so the
process terminates. This gives
The sum is finite because each lowers .
Proposition 5.2.The complex , augmented through and , is a minimal free
resolution of . Its block from to is a scalar matrix times . The same matrices resolve
the corresponding source face module.
Proof.The identity gives
and hence . Since , the equality gives . The associated graded of for the
finite filtration is the chosen cohomology inclusion, a quasi-isomorphism. Thus is
a quasi-isomorphism. Every term of contains a parameter, making the resolution
minimal. Each step deletes one distinct color, giving the squarefree monomial in the
stated block. □
The matrices require only face incidences and finite cochain splittings, which may be
chosen equivariantly when is invertible. Weighting color by changes the shift in (6) to
.
5.3. Coefficient diagrams.
Let be any contravariant diagram of -modules on . Define
Tensoring the actual stalk isomorphisms and their inverses gives a filtered comparison
natural in . No flatness is needed; after tensoring, the two maps are still mutual inverses on
each stalk, and coherence lets them pass to the inverse limits. Thus the comparison retains
arbitrary local coefficient data, rather than only the constant coefficients of a face
ring.
For , put . Restriction of components gives for . A chain monomial with largest face
has precisely the coefficient module ; the empty monomial has coefficient . For , define an
augmented cochain complex ; its degree term is , and its degree- term is the sum of over
chains with ranks in . Insertion below the largest face uses the identity; insertion of a new
largest face uses the displayed coefficient map. Use the usual alternating signs and the
augmentation .
Proposition 5.3.With , there is a canonical identification
It is equivariant for every compatible action on the diagram.
Proof.The barycentric color decomposition has the coefficient modules just
described. Raising and lowering a repeated-color exponent is the identity on that
coefficient module, so the same Koszul contraction applies. A squarefree strand is
exactly , with the sign change of Lemma 4.1. The filtered comparison and its inverse
give strictness and the canonical associated-graded homology identification. □
6. Straightening algebras and interval unions
The comparison applies whenever chain monomials form a basis, the top parameter cuts
out the boundary, and each stalk is free over its parameter ring. These properties let the
same boundary construction pass to straightening algebras and, through their interval
quotients, to arbitrary unions of Grassmannian Richardson varieties.
6.1. Lower ideals.
The key condition is that straightening an incomparable product produces a strict
upper factor; if one original factor lies outside a lower ideal, the upper factor does too, so
killing the outside generators remains compatible with multiplication. Let be a finite
graded poset with ranks in , such that every maximal chain in has all ranks . Fix positive
degrees . Suppose a commutative graded algebra over a coefficient ring has generators of
degree , with the weak-chain monomials and forming a -basis. Require that every nonzero
standard term in the homogeneous expansion of , for incomparable , contain a factor with
. Assume also that each is Cohen–Macaulay over every field. For a lower ideal , let be the
quotient by generators outside , and put . Include a separate constant stalk ,
so the empty lower ideal has ring . Let act by rank sums on both rings, with
.
Theorem 6.1.Under these hypotheses there are coherent graded -module
isomorphisms for all lower ideals, filtered with canonical chain-monomial symbols.
They identify every relative restriction kernel. If the source is defined integrally,
the maps may be constructed over and then base-changed. For rank-preserving
symmetries over a field, the equivariant Tor formula is
with the analogous relative formula and multiplicative associated graded assertion.
Proof.Choose large enough that . In a straightening relation, homogeneity cancels
the contributions. If , the strict upper factor gives
Thus every straightening strictly lowers weight. An outside factor of a lower ideal
remains outside, because it is replaced by an upper factor. Its standard-monomial
span is therefore an ideal, and the quotient has exactly the chain basis in .
Consequently , compatibly with restrictions and parameter symbols.
We need local freeness on the discrete side. Work first over at a principal ideal
of rank . Its color sums form a system of parameters over every field; for , so their
quotient is spanned by finitely many squarefree chains. Cohen–Macaulayness makes
the parameters a regular sequence. The quotient’s color-multigraded Hilbert series
is the Hilbert series of the chain ring times , independent of the field. Each integral
quotient component therefore has the same dimension over and over every , and has
no torsion.
Lift a color-homogeneous integral basis of this quotient. The lifts may omit
, because the stalk is and . Positive-degree induction gives generation over the
parameter ring; over the Hilbert series gives freeness, and an integral relation would
remain a relation over . Thus the lifts are an integral adapted free basis. Base-change
it to , then transfer its chain expansions to the source. The associated-graded basis
isomorphism proves source freeness by finite filtration induction.
The top variables append the top face to a chain, so are regular, with quotients
equal to the actual proper-stalk boundary limits. Apply the corrected boundary lift
used in Theorem 1.1. The bases omit the top variable, so the lift has canonical
leading symbols. This proves the module and relative statements. The strict filtered
Koszul argument and Lemma 4.1 prove the remaining assertions. □
For a source defined only over one field, it suffices that the principal intervals be
Cohen–Macaulay over that field; ordinary homogeneous-basis lifting replaces the integral
torsion argument. The dual theorem, with a strict lower factor, applies to upper filters
after reversing the poset and its degrees.
Quadratic laws with both strict endpoint factors and conserved rank sums admit the
smaller weights and preserve both ordinary degree and rank degree. With only a
strict upper factor, quadratic laws still satisfy the theorem in ordinary degree,
using . These distinctions concern the grading; the boundary construction is the
same.
6.2. Distributive lattices.
Let be a finite poset and its lattice of order ideals. Give degree , including the genuine
generator . The relations
have the required endpoints and rank conservation. On a lower-ideal quotient a relation
is zero if lies outside the quotient. The monomial map
distinguishes chain monomials; the exponents recover the nested ideals, and the
exponent of recovers the empty-ideal factors. The principal order complexes are shellable
by lexicographic order of the linear extensions. The theorem therefore gives coherent
comparisons for every lower ideal in , over every .
6.3. Interval diagrams.
Suppose each standard term in a homogeneous straightening contains both a factor
strictly below and a factor strictly above them. Assume every nonempty closed interval
has Cohen–Macaulay order complex over every field. The intervals keep their original rank
colors; they are not renumbered.
For an interval, a chain vanishes on every proper subinterval exactly when it contains
both endpoints. The product of the two endpoint parameters therefore cuts out the
boundary, taking the place of the single top parameter used for lower ideals. Once the
comparisons are compatible on these boundaries, they glue on arbitrary unions of
intervals.
Theorem 6.2.There are filtered parameter-module comparisons
simultaneously and compatibly on all closed intervals and on the constant stalk.
They have canonical chain-monomial symbols. They also give compatible comparisons
for arbitrary unions of interval order complexes and for all relative restriction kernels.
Proof.An index outside remains outside during straightening; failure to lie above
persists in the strict lower factor, and failure to lie below persists in the strict
upper factor. Thus interval quotients retain precisely their chain bases. The weights
of Theorem 6.1 give their discrete associated graded.
For and , the discrete interval stalk is free over by the same integral
Cohen–Macaulay argument. Its free basis may omit both endpoints, since for the
stalk is . Transfer this basis to the source. Set for and for a singleton. Multiplication
by appends the endpoint or endpoints to a chain and is regular. Its image is exactly
the common kernel of all proper-interval restrictions. Coefficientwise, the quotient is
their inverse limit, with the constant stalk included.
Induct on interval inclusion and apply Lemma 2.1 with . Lift each boundary image
using the chain representative not divisible by both endpoint variables. Because the
free basis omits both endpoints, the lift retains its canonical leading symbol. This
proves the simultaneous interval comparisons.
For a nonempty finite family of intervals, put
The intersection ideal is spanned by chains contained in none of the intervals.
Close the family under subintervals and add the constant stalk. An allowed chain is
detected at the interval between its endpoints, so both rings are the inverse limits
of these diagrams. The coherent interval maps give the comparison, compatibly with
subunions and relative kernels. □
The complex is the union of the order complexes, not in general the order complex on
the union of their vertices; the latter can introduce mixed chains. The constant stalk
records the common origin of the affine cones, and the strict Koszul and colorful
arguments give all the absolute and relative Tor formulas for these unions. If quadratic
relations conserve rank sums, the comparisons preserve ordinary degree and rank degree
simultaneously.
6.4. Grassmannian applications.
Let be the componentwise ordered lattice of -subsets of , with
The integral Plücker algebra has a standard basis of products of comparable maximal
minors [3, Theorem 11.1]. For , the relation
already exhibits both endpoint conditions; each term on the right has one factor strictly
below and one strictly above them. The rank sums agree as well, since . In the
straightening of incomparable , the lower factor has . Column-content conservation gives .
For every prefix , subtracting
from the content equality yields , so . Incomparability makes all endpoint inequalities
strict. Summing the column indices proves conservation of rank sums. Every interval in is
distributive and shellable.
Corollary 6.3.The coordinate algebra of every nonempty union of Grassmannian
Richardson varieties is coherently isomorphic, over the global rank-sum parameters,
to the face ring of the union of its interval order complexes. The comparisons are
integral and preserve both ordinary Plücker degree and rank degree. They commute
with every subunion restriction and give the absolute and relative colorful Tor
formulas.
Proof.The Richardson variety with allowed indices has coordinate ideal [4,
Proposition 3.4.2]. This is also immediate from the sum of the Schubert and opposite
Schubert coordinate ideals and the common standard basis. Intersecting the ideals
defines the scheme union, so Theorem 6.2 applies. □
Schubert unions are the special case with common minimal lower endpoint. With a
fixed lower bound , one may instead use ; the quadratic rank conservation survives this
constant shift. No common endpoint is needed in Corollary 6.3.
Order-reversing symmetries are allowed in ordinary degree. More generally, suppose a
symmetry transports intervals and their straightening laws and permutes colors while
preserving the chosen degrees. Its boundary elements are transported, so the semilinear
argument of Proposition 3.2 and the orientation formula of Proposition 4.3 apply even
when the straightening weights are not invariant. For example, reversing the
ambient coordinates of a Grassmannian sends to and rank to . All minors acquire
the same sign, removed by a common scalar change; unsigned index reversal is
therefore an integral algebra symmetry. Every invariant Richardson union and
relative subunion has the ordinary-graded equivariant Tor formula including this
reversal.
7. Parameter changes and their fibers
The boundary construction also separates the degree of a parameter from its lower
coefficients. Since a polynomial vanishing at the origin has the same support as the
variable it replaces, a Newton basis change can be made simultaneously on every face. This
compares polynomial parameters and lets us follow the parameter module away from the
origin by computing its Tor at arbitrary characters.
7.1. Polynomial face parameters.
Fix a profile of positive integers . For each face of rank , choose
Let act on the two face rings by
For pure powers the action is graded with ; general polynomials give filtered
modules.
Theorem 7.1.The source and barycentric polynomial-parameter modules are
coherently isomorphic over every , compatibly with all relative kernels. Their
parameter-module isomorphism type depends only on . In particular, lower
coefficients and unit leading coefficients introduce no new module types. The same
assertion holds for the lower-ideal straightening algebras of Theorem 6.1.
Proof.On the barycentric ring use the support-preserving Newton basis change
Triangularity with units on the diagonal gives an inverse by finite degree induction,
while preservation of each exact chain support makes the map commute with restrictions.
It intertwines the power action with the polynomial action. To check this, if color is
already present, all other vertices of that color annihilate the monomial; if absent, both
sides sum over exactly the compatible vertices of that color. The condition ensures
support preservation.
For the source, use on a rank- face, with or . For the power action the target has
adapted basis
with no top variable. Apply (8) in the lower colors, leaving unchanged, and transfer the
resulting chain expansions to the source. Their initial forms give precisely the same free
module over the weighted-power parameters. Positive filtration induction proves that they
are an adapted free -basis.
The actual boundary quotients are obtained by setting . Lift each prescribed boundary
image by its top-free representative and extend -linearly. The lift is filtered with canonical
leading symbols. Its associated-graded map is the basis isomorphism already
constructed, so finite induction on filtration degree makes the lift an isomorphism,
including for nonhomogeneous . Commutation with gives commutation with the
actual top parameter . Induction glues the comparison. For general straightening
algebras use their adapted local bases, which likewise omit the top variable.
□
The pure-power colorful formula has a useful relative form. For , put
An empty union is the void complex.
Proposition 7.2.For the power action on a properly colored relative face module,
for , and all other strands vanish. For the source face ring this computes canonical
associated-graded Tor. In the rank grading its internal degree is .
Proof.If , the color- factor is already present on both sides of its Koszul
multiplication; raising its exponent by is a bijection and contracts that strand. If ,
the exterior generator cannot occur, and every face in the strand must contain color
. If , that color may instead occur in the exterior factor. Thus the surviving strand is
exactly the augmented relative cochain complex displayed above, with degree . The
sign calculation is unchanged. Theorem 7.1 and strictness give the source assertion.
□
For arbitrary polynomial parameters, compose the comparison with . Its associated
graded identifies Tor with the same power-profile cohomology groups, as filtration
quotients, without an internal-grading assertion. Equal weights group summands; one may
separate them by choosing positive straightening weights outside the finitely many
equality hyperplanes for these . If , every leading identification is equivariant. The
coefficient-diagram construction of Section 5.3 also extends; it tensors actual coherent
maps and requires no additional flatness.
7.2. Active colors and triangular symmetry.
We now work over a field . For , let be the character of with acting as .
These Tor groups are ungraded when . We first use the ordinary color-sum action
.
For a properly colored complex , inverting a color sum separates the mutually
annihilating vertices of that color into distinct factors. Doing this for all colors in
selects a face with those colors; the unused vertices form its link. Explicitly,
The orthogonal idempotents are . Indeed, for a color- vertex; the idempotents sum to .
In the factor the chosen active-color variable equals , the others vanish, and the remaining
monomials are exactly the link faces. The formula is equivariant under color-preserving
automorphisms.
Writing and , regularity of the active Laurent parameters and Lemma 4.1 give
There is no homological shift from the active parameters; their Koszul complex has
homology only in degree zero.
Upper-triangular changes of parameters will reduce an arbitrary character to its highest
active rank; if and for , adding multiples of makes all earlier coordinates equal to , and
scaling makes . Thus it is enough to understand the initial active colors . We must first
show that these changes preserve the parameter module and keep track of equivariant
representation classes.
Lemma 7.3.Suppose finite parameter modules have nonnegative filtrations, and
a filtered isomorphism with filtered inverse has equivariant associated graded. Then
every additive -linear functor gives objects with common finite equivariant filtrations
and identical successive quotient actions.
Proof.Let be the algebra of parameter-linear endomorphisms of lowering filtration
by at least one. Choose finite generators of filtration at most . A product of such
maps kills the generators, hence . The transported action differs from the source by
. Use powers of , or of its image after the functor, as in Proposition 3.2. □
Proposition 7.4.For the power action on , every invertible upper-triangular
change
gives a coherently isomorphic ungraded parameter module. Over a field,
finite-dimensional additive-functor outputs have the same equivariant representation
classes for rank-preserving symmetries.
Proof.Diagonal units are removed by multiplying a chain monomial of color
exponents by . This map is invertible, equivariant, and support-preserving. Assume
henceforth that the diagonal is .
Put and give a color- vertex auxiliary degree . On a rank- stalk use , where . The
source action is and the target is ; all have degree . Both boundary quotients set .
The top-free adapted power basis used in Theorem 7.1 therefore permits the same
corrected boundary lift. It commutes with and hence with the top parameter .
Give a vertex of color weight . Then strictly decreases with , so each twisted
parameter has its original leading form. The corrected lift has the canonical leading
symbol; its boundary does by induction, and a remaining discrepancy would contain
the top variable, excluded by the choice of lift. It is therefore a filtered isomorphism
with filtered inverse.
The global power module is generated by chain monomials whose present color-
exponents are between and , together with . Thus it is finite over . Lemma 7.3
proves the representation assertion. □
For , this also gives the ordinary-graded triangular symmetry of and, by the coherent
comparison, an ungraded triangular symmetry of . The original rank grading is generally
not preserved by .
7.3. Only the highest active rank.
For the polynomial face parameters of Section 7, assume and let , with at the origin.
For a rank- face , put and let be the set of full flags in . For use , its empty flag, and
.
The ordinary-parameter case explains the contribution of each face
Localization at the initial active colors selects a full flag ending at , leaving only faces
above available for the colorful Koszul calculation in . For higher powers, colors of
intermediate exponent are required to occur in the face; as in Proposition 7.2, this
replaces cohomology by relative cohomology.
For a vector on the colors with , define as before and put
Theorem 7.5.For every character and every field,
In particular,
These dimensions and classes depend on only through , and vanish for .
Proof.The comparison, its equivariant leading symbols, and Lemma 7.3 compare
Tor at every character. The Newton basis change is an actual equivariant
isomorphism, so we may use pure powers. Characters with the same highest active
rank are conjugate under the invertible upper triangular group. By Proposition 7.4,
it therefore suffices to take for and otherwise.
The localization (9) splits the ring into factors indexed by full flags below rank-
faces. Their links are exactly , since no further face can be inserted below . Each
factor has the form
The active Koszul complex has only degree-zero homology
of dimension , with trivial stabilizer action. No separability is needed; the
successive monic polynomials are nonzerodivisors with nonzero constant terms.
Proposition 7.2 computes the remaining colors as . There are flags in the Boolean
interval. Finally and reduced cohomology vanishes below degree . □
The same ordinary-parameter argument applies to the locally Cohen–Macaulay
straightening posets of Theorem 6.1, replacing by the number of full flags below . For a
relative subposet , add to when ; if , keep the absolute pair. Coherence and the same
localization prove this relative formula.
The class statement need not be an actual modular isomorphism. For a two-vertex
simplex over , its barycentric ring is , with , and interchanged. At the fiber is with
trivial action; at it is with , the regular two-dimensional representation. Both have
highest active rank two and class .
7.4. Fitting ideals and rank jumps.
The Fitting ideals of the source parameter module equal those of its barycentric power
module. They are monomial ideals in the specified parameters; a color-multigraded
presentation has minors homogeneous for , and each such homogeneous polynomial is a
scalar times one monomial. Proposition 7.4 makes these ideals invariant under every
invertible upper-triangular change, over the universal coefficient ring of such a
matrix. Thus the invariance holds in every characteristic. In characteristic zero it
implies reversed strong stability; if , then for . Apply and take the coefficient of
.
Every nonempty proper ungraded Betti rank-jump locus is a tail coordinate subspace
Indeed, Theorem 7.5 makes each Betti number constant on the corresponding
highest-active stratum. Its upper level sets are closed, since a finite free resolution
expresses fiber homology dimension as a free rank minus the ranks of two adjacent
matrices. The closures of these strata are the displayed nested subspaces, so any such
closed union is one of them. The argument holds over an algebraic closure and descends;
monomiality also permits testing at coordinate zero-one points. Consequently every
Fitting ideal has radical a tail-coordinate ideal, the zero ideal, or the unit ideal.
All these statements hold for relative kernels and for the polynomial actions
above.
8. Equivariant boundary lifting
An equivariant boundary map need not admit an equivariant lift when the stabilizer order
is not invertible. Prescribing its image on suitable fixed-face quotients removes the part
that cannot be averaged, leaving a correction in permutation modules with small
stabilizers.
Because boundary lifting makes every normalized coherent map invertible, we need only
construct equivariant maps with scalar on constants. Corrections and averages can
therefore be performed additively, with invertibility supplied by the boundary
construction.
Lemma 8.1 (Parameter-linear lifting).Let a finite group act on a positively graded
polynomial ring trivially, on a bounded-below graded-free -module , and on an
equivariant surjection of graded -modules. Suppose this surjection has an equivariant
graded -linear section, and its kernel is degreewise injective with respect to -split
inclusions of -modules. Every equivariant graded -linear map lifts equivariantly to .
Proof.The section gives a preliminary graded lift . Suppose the lift has been made
parameter-linear in degrees below . Its values on
are prescribed by . This is well-defined; the first syzygies of the variables on the
-free module are the Koszul syzygies, and the already imposed linearity in lower
degrees annihilates them. The difference takes values in the kernel. Since is free over
, the inclusion is -split. Extend the difference equivariantly and add the extension
to . Induction on degree proves the assertion. □
The required relative injectivity has an elementary source. If and is a unit in ,
then is injective with respect to -split inclusions. By coinduction, extending
a map to amounts to extending a -equivariant map to . Extend -linearly and
average over . In particular, sums of regular modules have this property over every
.
We shall also use a controlled form of the lemma. Let be a face of rank , put , and let
kill a group-invariant collection of face variables, retaining the top variable. Assume that
the retained chain monomials form a face subcomplex. Suppose an equivariant boundary
map and an equivariant -linear map have been prescribed and agree on their
common boundary. Lift the boundary by the top-free section, obtaining . The
difference is divisible by the top variable in . Include it by the span of retained
chain monomials and add it to , obtaining with . This section is only required to
be -linear. In degree the correction in the preceding proof now has the form
Indeed, it vanishes on the boundary, and regularity of on puts the quotient in . Thus
degreewise relative injectivity of gives a boundary-compatible lift satisfying . A
normalized lift is an isomorphism by Lemma 2.1.
Corollary 8.2.If the order of the effective vertex-permutation group of every face
stabilizer is a unit in , there is a coherent graded equivariant comparison over .
Proof.Induct over face orbits. At a representative face, lift its prescribed boundary
and average over the effective stabilizer. Its kernel fixes the entire Boolean face
interval and hence both stalks, so every local map is already equivariant under that
kernel. Averaging preserves the boundary; boundary lifting makes the resulting map
an isomorphism. Transport it over the orbit and continue. □
8.1. An integral orbit retraction.
To prescribe the fixed-face quotient, we need a map on the polynomial side that
combines each free orbit into one variable and sends the elementary symmetric parameters
to those of the orbit set. The following substitution achieves this integrally; roots of unity
cancel the intermediate coefficients within each orbit, and an integral coefficient projection
removes the auxiliary roots.
Let , with prime, act freely on a finite set , and choose a generator. Write its
orbits as , with cyclic labels , . Put and choose a primitive th root . Define a
map over by substituting , retaining only terms whose exponent in every is
divisible by , and replacing by . Apply coefficientwise the integral functional
to obtain
This is an integral map; the roots of unity only verify its properties. For a single orbit
of size two, for example,
The cancellation that kills the first elementary parameter occurs without division by
the orbit size, while the orbit product is retained.
Lemma 8.3.The map retracts the orbit-product embedding . It commutes with
restrictions to unions of -orbits and with every vertex permutation centralizing . For
elementary symmetric polynomials,
Proof.The identity proves (12). The retained exponent classes are preserved by
the substituted symmetric polynomials, so the coefficient projection commutes with
their multiplication. Also maps to . A permutation centralizing permutes the orbits
and shifts their cyclic labels. A shift multiplies a retained term by . This proves
both equivariance and independence of the starting point in each orbit. Restriction
is immediate. □
The construction depends on a generator, but cyclic rotations and induced cyclic
orders preserve its positive choice, providing exactly the compatibility needed
below.
9. Cyclic orders and the factorial obstruction
An ordered comparison can be symmetrized by summing over all linear orders. To obtain
the sharper factor , we construct comparisons that need only a cyclic order. The
difficulty is to make them compatible with restriction to every subset, including
when a rotation group has noninvertible order. The orbit retraction handles
precisely those fixed subsets; the remaining chains have stabilizers whose orders are
invertible.
Theorem 9.1 (Integral cyclic templates).For each there are normalized graded
parameter-module isomorphisms
for all finite cyclically ordered sets . They are natural under cyclic-order-preserving
bijections and commute with restriction to every subset with its induced cyclic order.
Proof.We first construct the system over , for each prime . Induct on , beginning
with the identity at the empty set. The existing maps on smaller cyclic sets give a
prescribed boundary map equivariant for the rotation group .
If , lift this map and average over . Suppose , and let be the subgroup of order
, generated by rotation through places. Its orbit set has its quotient cyclic order.
Write
for the projection retaining exactly the -fixed subsets; degrees in the target are
multiplied by . Impose, in addition, the identity
Here is the retraction of Lemma 8.3 for the positive generator. Both sides
are parameter-linear after is sent to zero for and to the quotient parameter
.
The prescription agrees with the boundary. To see this precisely, identify with . A
-invariant subset of size is the union of the orbits indexed by an -subset of . In
its induced cyclic order, acts by rotation through places, its positive order-
generator. Its orbit order is the induced order on that -subset. Thus both the
retraction and the smaller template restrict to the terms in (13) already prescribed
there.
Apply the controlled lift (11) with . Every chain monomial in contains a subset not
fixed by . Its stabilizer in has order prime to ; a subgroup of order divisible by contains ,
and ranks force a stabilizer of a chain to fix every member of the chain. Hence each graded
kernel piece is a sum of modules with a unit. The relative injectivity above supplies the
correction. This constructs the -local system, and boundary lifting makes all its maps
isomorphisms.
It remains to make the system integral. For each , specify images of one integral Artin
basis of over . These images lie in finitely many finite free graded pieces. Rotation
invariance, subset compatibility and normalization are a finite system of integer linear
equations. Solvability over every implies solvability over ; the class of the inhomogeneous
term in the finitely generated cokernel of the coefficient matrix vanishes in every
localization, and therefore vanishes. □
A compatible cyclic order on a simplicial poset means a cyclic order on each face
inducing that on every subface. The templates assemble into a coherent comparison
natural for order-preserving isomorphisms, and more generally for nondegenerate
simplicial-poset maps preserving these orders. Such a map is an isomorphism on each
Boolean face interval; its pullback on face generators sums over their preimages. The local
comparisons commute with these relabelings and restrictions, so the assertion follows on
inverse limits. In particular, an orientation-preserving action on an oriented triangulated
surface admits an integral equivariant comparison; use the oriented cyclic order on each
triangle, and the unique orders on edges and vertices. The resulting equivariant
isomorphisms also identify their Tor modules.
Corollary 9.2.If is a unit in , there are coherent equivariant comparisons for
every simplicial poset of maximum rank at most and every automorphism group.
Proof.For a nonempty -set, average over its cyclic orders. If has elements,
each cyclic order on is induced by exactly orders on . Thus these symmetric
averages remain coherent. They are normalized, so are isomorphisms, and assemble
independently of face labels. □
We next determine the scalar obstruction, including its prime-power multiplicities. A
suitable parameter fiber turns the polynomial ring into the regular permutation
representation on root orderings, while the barycentric fiber retains an invariant
evaluation. An invariant functional on a regular representation takes the same value on
every coordinate, forcing a factorial divisor.
Lemma 9.3.Suppose is a unit in a commutative ring . If is an -equivariant
graded parameter-linear map with , then .
Proof.Specialize and . The source fiber is the splitting algebra of ; the integral
Artin basis makes it free of rank over , with basis containing . In particular there is
a -linear retraction with .
Let be the roots in . They and all their pairwise differences are units; the
discriminant identity gives
Give the trivial group action and extend coefficients to it. Successive Chinese
remaindering now splits the source fiber as , with acting regularly on the ordered
root assignments. The target fiber has an invariant evaluation to , obtained by setting
all proper-face variables to zero and the top variable to . Composition with gives
an invariant functional taking to . It has the same value on every coordinate
idempotent, whence . Apply to conclude. □
For the rank- simplex, abbreviate and write for its comparison obstruction.
Theorem 9.4.For the rank-simplex and every commutative ring ,
The obstruction class in (5) satisfies . For every simplicial poset of maximum
rank at most and every automorphism group, annihilates its coherent comparison
obstruction.
Proof.Put ; the case is immediate. A coherent family of scalar restricts to an
-equivariant map of the same scalar on every rank- face. Apply Lemma 9.3 with over
, and with over . It follows that belongs to after both localizations. Some powers
and therefore belong to . The two powers are coprime, so a Bezout combination
gives . This argument applies without finiteness or reducedness assumptions on .
For the opposite inclusion, use the integral cyclic templates and set on a rank-
face
The coefficient is integral. Counting induced cyclic orders as in Corollary 9.2
shows that is coherent; summing over all orders makes it equivariant for every
automorphism. It has scalar . Base change and scalar multiplication give all of .
Finally, the exact sequence has a surjective last arrow by the nonequivariant
comparison theorem. Its connecting map sends to and has kernel . This proves
the claimed cyclic module and, by the universal family , the universal annihilator
assertion. □
The scalar obstruction concerns coherent graded maps. To rule out arbitrary
isomorphisms, we instead choose a fiber whose two group representations have different
projectivity properties. A fixed root assignment gives a trivial summand on the polynomial
side; on the barycentric side, sufficiently frequent active ranks force the flag components
into free orbits.
Proposition 9.5 (Fixed-vertex obstruction).Let have characteristic . If an
order-permutation of the vertices of a simplex has both fixed and moved
vertices, its polynomial and barycentric rings are not isomorphic as equivariant
parameter-modules, even without grading or coherence.
Proof.Suppose there are moved orbits and fixed vertices. Over , specialize the
parameters to the elementary symmetric coefficients of the list with copies of each
and one copy of each . Its generating polynomial is
Algebraic independence implies that the nonzero coefficient ranks are
Terms with different cannot cancel, since their degrees in the differ. Every
successive gap from through to is less than .
In the barycentric fiber, the elements at each rank are orthogonal idempotents
summing to . Their products decompose the fiber into flag components of rank set
. The cyclic group acts freely on these flags. Indeed, in a fixed flag every successive
difference is a union of vertex orbits, and its size is less than ; it could therefore contain
only fixed vertices, never reaching the full set. The components are consequently
permuted in free orbits, so the barycentric fiber is a free -module.
Evaluation at the displayed root assignment is an equivariant map from the
polynomial fiber to , split by constants. Thus its representation contains the trivial
module as a direct summand. That module is not projective over ; a finite projective
over this local ring is free, whereas its dimension is . The polynomial fiber cannot
be free. An equivariant parameter-module isomorphism would survive specialization
and contradict this distinction. □
Completion of Theorem 1.3. The scalar statement is Theorem 9.4, and
Corollary 9.2 gives the asserted comparisons when is a unit. Otherwise some prime
is a nonunit in . Reduce a putative ungraded isomorphism modulo a maximal ideal
containing , and restrict to a -cycle fixing the other vertices. Proposition 9.5 gives a
contradiction. Thus the universal field range is characteristic zero or . □
10. Cyclic face stabilizers
Call an action locally semiregular if, for every nonempty face , the effective image of its
stabilizer on the vertices of acts freely. Equivalently, all vertex stabilizers inside are the
kernel of that vertex action. This kernel fixes the entire Boolean interval below
.
Theorem 10.1.A locally semiregular action of a cyclic -group on a simplicial poset
admits an integral coherent equivariant comparison.
Proof.Induct on the order of the group . The trivial case is Theorem 1.1. Let have
order . On a -fixed face, either fixes every vertex or it acts freely; the local condition
makes its intersection with every vertex stabilizer the same kernel.
Let consist of the first kind of fixed face and the minimum. It is a downset, and
acts locally semiregularly on it. Let consist of the second kind and the minimum,
dividing ranks by . This is a simplicial poset whose faces below a fixed face are the
unions of its -orbits. The -action on its atoms is free. Indeed, a vertex in such an
orbit has trivial stabilizer in , since every nontrivial subgroup contains . If stabilizes
its -orbit, then for some , forcing . Induction therefore gives comparisons and on and .
Construct the original maps in increasing face rank. On use . A face not fixed by
has trivial stabilizer, so lift its boundary and transport the lift around its orbit. At
, let retain the -fixed face variables and prescribe
This agrees with the boundary; every proper positive -fixed subface of lies in ,
and Lemma 8.3 commutes with its restriction. The kernel of has a chain basis with
trivial stabilizers in . A nontrivial stabilizer would contain and fix each member
of the chain, contrary to membership in the kernel. Thus (11) has values in sums
of regular -modules and extends. Transport these maps over their face orbits. The
resulting family is coherent and normalized, hence consists of isomorphisms. □
Theorem 10.2.Let have characteristic , and suppose a Sylow -subgroup of is
cyclic. A coherent graded equivariant comparison exists if and only if, for every face
, the effective action of a Sylow -subgroup of on its vertices is free.
If every Sylow subgroup of is cyclic, the corresponding criterion over an arbitrary
ring is that every prime for which this local condition fails be a unit in .
Proof.For sufficiency over , a Sylow -subgroup of acts locally semiregularly; is
contained in a Sylow -subgroup of , and semiregularity modulo the kernel passes
to subgroups. Apply Theorem 10.1 and take its coset trace to , divided by . The
resulting family is coherent and normalized.
For necessity, restrict a coherent comparison to the offending face and pass to
the effective cyclic -group. If this faithful action is not free, its unique subgroup of
order fixes a vertex and also moves a vertex. Proposition 9.5 contradicts the stalk
isomorphism.
For the ring assertion, call a prime dividing bad if the local condition fails. A
bad nonunit prime belongs to a maximal ideal of , and reduction to its residue
field proves necessity. For sufficiency, trace an ordinary integral comparison from the
trivial subgroup; its scalar is . For every good prime , trace the integral comparison
equivariant under its cyclic Sylow subgroup ; its scalar is . The greatest common
divisor of these integers is
A Bezout combination gives an integral equivariant coherent family of scalar .
Divide by its unit image in . Boundary lifting again makes the normalized family an
isomorphism. □
In particular, a free vertex action of a group with cyclic Sylow subgroups admits an
integral equivariant comparison. For a faithful action on a full simplex, the field criterion
simplifies; if its Sylow -subgroup is cyclic, an equivariant parameter-module isomorphism
exists exactly when that subgroup acts freely. An isomorphism can then be graded and
coherent; failure rules out even an ungraded isomorphism, by Proposition 9.5. When all
Sylow subgroups are cyclic, the same statement over requires precisely that the primes of
nonfree Sylow actions be units.
Appendix A. Relative equivariant lifting
The preceding construction chooses its own compatible boundary maps; extending an
arbitrary equivariant boundary instead requires information about all subgroup-fixed
quotients. Throughout this section is a field of characteristic .
A.1. Cofixed polynomial sources.
Let a cyclic -group of order act freely on a vertex set of size . Every -fixed
subset is a union of -element orbits, so its rank is divisible by . On the barycentric
ring of these fixed faces, the other parameters act by zero and acts trivially. A
-equivariant parameter-linear map from the polynomial ring to this fixed-face
quotient must therefore factor through the following coinvariant module. Put
The subscript denotes coinvariants for the linear group action. The parameters act by
elementary symmetric functions, and the integral Artin basis makes finite graded-free over
. Proving the same freeness after taking coinvariants will allow the lifting lemma to apply
on fixed-face quotients.
Lemma A.1.Over a ring of finite global dimension, a doubly infinite complex of
projective modules with projective homology has projective cycles, boundaries and
differential cokernels.
Proof.Write for its cycles, boundaries and homology. The sequence splits. From
we obtain, for ,
Iterate beyond the global dimension to obtain . Thus and are projective. The
differential cokernel is an extension of by , so it too is projective. □
Theorem A.2.For every free cyclic -group action, both and are finite graded-free
over .
Proof.Induct on . For this is the ordinary Artin basis theorem. Let have order and
put . There is a characteristic- orbit map
Identify the variables in each orbit with , retain terms whose orbit exponents
are divisible by , and replace by . It is -equivariant and satisfies (12), since .
The orbit-product embedding is an equivariant -linear section. The kernel has
permutation basis
Every basis stabilizer is trivial; a nontrivial subgroup of contains . Thus each
internal-degree piece of is free over .
Apply the complete two-periodic Tate complex for to the bounded Koszul complex
. Its Koszul homology is , because the indicated parameters form a regular sequence.
Replacing by through leaves total Tate cohomology unchanged; the kernel Koszul
complex is bounded and has degreewise free -terms, whose Tate cohomology vanishes.
Both assertions also follow from the two spectral sequences. The bounded Koszul
direction makes them valid for the complete Tate complex, separately in each internal
degree.
On , the parameters with act as zero. The others form the regular sequence with
. Hence the replacement Koszul complex is equivariantly quasi-isomorphic to
with zero differential on the exterior factor. All these are modules over , with
degrees in multiplied by . By induction is projective over , and is free. The two
exact sequences for , where generates , then split successively over ; they also make
projective.
The -action on factors through . Its norm is times the quotient norm and is
zero. Thus the complete Tate complex of alternates with zero, and its cohomology
alternates the invariant and coinvariant modules just proved projective. The exterior
decomposition shows that every Tate cohomology module of is projective over .
The complete Tate complex of has finite free -terms. Since the polynomial ring
has finite global dimension, Lemma A.1 makes its differential cokernel projective.
The two splitting sequences for also make its invariant module projective.
Finally, a finite graded projective over a positively graded polynomial ring with
degree-zero part is graded-free. Lift a homogeneous basis modulo the positive-degree
ideal, split the resulting graded free surjection, and apply graded Nakayama to its
kernel. This proves both assertions. □
A.2. The subgroup-fixed lifting criterion.
Proposition A.3.Let a finite -group act freely on the vertices of a rank-simplex.
For every subgroup , suppose
is graded-free over . Then every coherent -equivariant graded comparison
prescribed on the proper faces extends equivariantly to the simplex.
Proof.Let be the barycentric ring on the -fixed faces, a Boolean poset on the -orbits
of vertices with degrees multiplied by . Construct maps in decreasing subgroup order,
on conjugacy representatives and then by transport. Require normalizer equivariance,
agreement with the prescribed boundary, and compatibility with all larger-subgroup
fixed quotients.
At the stage, take the union, inside the order complex of its fixed faces,
of the proper-face boundary and the -fixed complexes for all . Intersections are
fixed complexes for the generated subgroups, so the maps already constructed glue
coefficientwise to its face ring . This map is -invariant, and parameters of rank not
divisible by act as zero. It therefore factors through , equivariantly for .
The kernel of has a basis of chain monomials containing the top face and having
stabilizer exactly in . Indeed, the boundary removes chains omitting the top, while
the other fixed complexes remove exactly those with a larger stabilizer. Consequently
acts freely on this basis. Each graded kernel piece is a sum of regular modules, hence
injective. The standard-monomial section is graded and normalizer-equivariant.
Apply Lemma 8.1 to the free -source . Normalizer equivariance makes transport
well-defined; the imposed larger-subgroup compatibilities handle the intersections
with conjugate subgroups.
At this gives an -equivariant parameter-linear map extending the entire prescribed
boundary. Lemma 2.1 makes it an isomorphism. □
Corollary A.4.For a free cyclic -group action, every prescribed coherent
equivariant graded boundary comparison extends to the simplex.
Proof.Apply Theorem A.2 to every subgroup and then Proposition A.3. □
Lemma A.5.If is a power of and , then
is graded-free over .
Proof.Make the finite faithfully flat base change to . Translation of every variable
by is -equivariant. Since in for , the elementary translation formula identifies the
base-changed ring with
Coinvariants commute with base change, so the resulting module is free over .
Faithful flatness makes the original finite module torsion-free over the PID , hence
projective and graded-free. □
Theorem A.6.Let be a finite -group all of whose proper subgroups are cyclic,
acting regularly on its own vertices. Every prescribed coherent -equivariant graded
boundary comparison extends to the simplex. This applies to in characteristic and
to in characteristic two.
Proof.For each proper subgroup use Theorem A.2; for itself use Lemma A.5, since
there is one vertex orbit. Proposition A.3 applies. □
Corollary A.7.Let act on a simplicial poset over . Suppose that, on each face ,
the effective image of a Sylow -subgroup of is trivial, cyclic and free on its vertices,
or a -group with all proper subgroups cyclic acting regularly on those vertices. Then
a coherent graded equivariant comparison exists. When all these effective images are
cyclic, the free-or-trivial condition is also necessary.
Proof.Induct over face orbits. The previously chosen maps prescribe a boundary
invariant under . Pass to the effective image of a Sylow subgroup; its kernel acts
trivially on both stalks. Corollary A.4, Theorem A.6, or ordinary lifting extends this
same boundary. Take the coset trace to and divide by the Sylow index. The average
still extends the prescribed boundary, so is an isomorphism; transport it over the
face orbit. Necessity in the cyclic case is Proposition 9.5 on the offending stalk. □
Thus the fixed-quotient construction extends the earlier criterion from cyclic Sylow
subgroups to actions controlled only by their effective stabilizers on individual
faces.
A.3. Integral relative lifting.
For a prime-order effective action, the relative modular theorem lifts integrally by a
lattice argument.
Lemma A.8.Let a group of order act on a nonempty affine torsor under a finite
free abelian group . If has an -fixed point, then has an -fixed point.
Proof.Lift the fixed point to , and write with . In the rational affine space,
The right side lies in , and the left side is invariant. □
Theorem A.9.For a free -action on the vertices of a simplex, every prescribed
integral coherent equivariant graded boundary comparison extends to an integral
equivariant comparison on the simplex.
Proof.Choose an integral Artin basis of over . Ordinary boundary lifting makes
the set of extensions nonempty. It is a torsor under the finite free lattice
These coordinates commute with reduction modulo ; lift the boundary image of
each basis element by its top-free representative, and choose its arbitrary correction
in . Thus is exactly the space of extensions of the reduced boundary. Corollary A.4
supplies a fixed point there. Lemma A.8, with , gives an integral fixed point. Every
such boundary lift is an isomorphism by Lemma 2.1. □
Theorem A.10.Suppose that, for every face and every prime , the effective image
of a Sylow -subgroup of is trivial or has order and acts freely on the vertices of
. Then there is an integral coherent graded -equivariant comparison, and hence one
over every commutative coefficient ring. It identifies all subcomplex restrictions and
relative kernels.
Proof.Induct on face orbits and let . For each Sylow subgroup , Theorem A.9, or
ordinary lifting in the trivial effective case, gives a -equivariant lift of the same
prescribed boundary. Its coset trace to has boundary times that map, where . The
integers have greatest common divisor one. Choose ; then
is an integral -equivariant lift of the prescribed boundary. For use an ordinary
lift. Boundary lifting makes each map an isomorphism. Transport over face orbits
and pass to inverse limits. Coherence identifies restrictions and their kernels, and
tensoring the maps and their inverses gives arbitrary base change. □
Within the class whose effective local Sylow images all have order or , the condition in
Theorem A.10 is exact; a nonfree effective order- action is obstructed after reduction
modulo by Proposition 9.5. In particular the theorem applies when every effective face
stabilizer has squarefree order and each of its Sylow subgroups acts freely on the face
vertices.
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