FI-modules over Noetherian rings

Church, Thomas; Ellenberg, Jordan S.; Farb, Benson; Nagpal, Rohit

doi:10.2140/gt.2014.18.2951

Cited by 153 publications

(157 citation statements)

References 16 publications

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“…We now explain the relation of our definition of central stability given above to the notion of central stability given by Putman and Sam in for complemented categories with a generator, and the relation to the inductive description in the special case of

scriptFI

‐modules as given by Church, Farb, Ellenberg, and Nagpal .…”

Section: Generalitiesmentioning

confidence: 99%

“…A central coefficient system

ρ

defines a

scriptC

‐module

V

over

R

with

V_{n} = ρ_{n}

and such that the standard inclusion

[n] ↪ [n + 1]

induces the map

F_{n}

. If

V

is finitely generated as a

scriptC

‐module over

R

, then by [, Theorem C] it is a centrally stable

scriptC

‐module over

R

.…”

Section: Generalitiesmentioning

confidence: 99%

“…Since the automorphism group of

[n]

in the category

scriptFI

is the symmetric group

S_{n}

, an

scriptFI

‐module

V

over

R

gives rise to a sequence

{V_{n}}

where

V_{n}

is a representation of

S_{n}

. It was shown by Church, Ellenberg, Farb, and Nagpal (see [, ]) that many interesting sequences of representations of symmetric groups arise in this way from finitely generated

scriptFI

‐modules. A principal result they proved is that any finitely generated

scriptFI

‐module over a commutative noetherian ring is noetherian.…”

Section: Introductionmentioning

confidence: 99%

“…To provide the context for our present article, let us briefly describe the way in which is proved in . Suppose that

V

is a finitely generated

scriptFI

‐module over a noetherian ring.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On central stability

Gan

2017

Bull. London Math. Soc.

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The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI‐modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of d‐step central stability, and prove that if the ideal of relations of a category is generated in degrees at most d, then every module presented in finite degrees is d‐step centrally stable.

show abstract

scriptFI

‐modules as given by Church, Farb, Ellenberg, and Nagpal .…”

Section: Generalitiesmentioning

confidence: 99%

“…A central coefficient system

ρ

defines a

scriptC

‐module

V

over

R

with

V_{n} = ρ_{n}

and such that the standard inclusion

[n] ↪ [n + 1]

induces the map

F_{n}

. If

V

is finitely generated as a

scriptC

‐module over

R

, then by [, Theorem C] it is a centrally stable

scriptC

‐module over

R

.…”

Section: Generalitiesmentioning

confidence: 99%

“…Since the automorphism group of

[n]

in the category

scriptFI

is the symmetric group

S_{n}

, an

scriptFI

‐module

V

over

R

gives rise to a sequence

{V_{n}}

where

V_{n}

is a representation of

S_{n}

. It was shown by Church, Ellenberg, Farb, and Nagpal (see [, ]) that many interesting sequences of representations of symmetric groups arise in this way from finitely generated