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Background Koszul theory of directed graded k-linear categories Type A∞ categories

Kosulity of directed categories in representation stability theory

Wee Liang Gan and Liping Li

University of California, Riverside

November 23, 2014

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FI

◮ Objects: finite sets.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FI

◮ Objects: finite sets. ◮ Morphisms: injections.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FI

◮ Objects: finite sets. ◮ Morphisms: injections. ◮ Equivalently, objects are [n], n ∈ N ∪ {0}

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FI

◮ Objects: finite sets. ◮ Morphisms: injections. ◮ Equivalently, objects are [n], n ∈ N ∪ {0} ◮ EndC([n]) is precisely Sn.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FIq

◮ Objects: finite dimensional spaces over a finite field Fq.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FIq

◮ Objects: finite dimensional spaces over a finite field Fq. ◮ Morphisms: linear injections.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FIq

◮ Objects: finite dimensional spaces over a finite field Fq. ◮ Morphisms: linear injections. ◮ Equivalently, objects are Fn, n ∈ N ∪ {0}

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

The category FIq

◮ Objects: finite dimensional spaces over a finite field Fq. ◮ Morphisms: linear injections. ◮ Equivalently, objects are Fn, n ∈ N ∪ {0} ◮ EndC([n]) is precisely the general linear group.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Applications

◮ These two categories, as well as a lot of variations, are

introduced and studied by CEFN, Putman, Sam, Snowden, Wilson, etc.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Applications

◮ These two categories, as well as a lot of variations, are

introduced and studied by CEFN, Putman, Sam, Snowden, Wilson, etc.

◮ They are used to study representations of a family of groups

simultaneously, in particular the representation stability when n increases.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Applications

◮ These two categories, as well as a lot of variations, are

introduced and studied by CEFN, Putman, Sam, Snowden, Wilson, etc.

◮ They are used to study representations of a family of groups

simultaneously, in particular the representation stability when n increases.

◮ They have many applications in representation theory,

algebraic topology, geometry, combinatorics, etc.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Properties

◮ Theorem (CEFN): FI is locally Noetherian over any left

Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Properties

◮ Theorem (CEFN): FI is locally Noetherian over any left

Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated.

◮ Theorem (GL, PS): FIq is locally Noetherian over any left

Noetherian ring.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Properties

◮ Theorem (CEFN): FI is locally Noetherian over any left

Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated.

◮ Theorem (GL, PS): FIq is locally Noetherian over any left

Noetherian ring.

◮ Theorem (SS): Every finitely generated projective FI-module

is also injective over the complex field.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Properties

◮ Theorem (CEFN): FI is locally Noetherian over any left

Noetherian ring; that is, sub-representations of finitely generated representations are still finitely generated.

◮ Theorem (GL, PS): FIq is locally Noetherian over any left

Noetherian ring.

◮ Theorem (SS): Every finitely generated projective FI-module

is also injective over the complex field.

◮ Many proofs use representations of these particular groups.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

EI categories

◮ All above categories are examples of locally finite EI categories

• f type A∞, which are small categories such that every

endomorphism is invertible and satisfy:

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

EI categories

◮ All above categories are examples of locally finite EI categories

• f type A∞, which are small categories such that every

endomorphism is invertible and satisfy:

◮ for every pair x, y ∈ Ob C, |C(x, y)| is finite;

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

EI categories

◮ All above categories are examples of locally finite EI categories

• f type A∞, which are small categories such that every

endomorphism is invertible and satisfy:

◮ for every pair x, y ∈ Ob C, |C(x, y)| is finite; ◮ objects are indexed by N ∪ {0}, and C(j, s) ◦ C(i, j) = C(i, s).

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

EI categories

◮ All above categories are examples of locally finite EI categories

• f type A∞, which are small categories such that every

endomorphism is invertible and satisfy:

◮ for every pair x, y ∈ Ob C, |C(x, y)| is finite; ◮ objects are indexed by N ∪ {0}, and C(j, s) ◦ C(i, j) = C(i, s). ◮ Therefore, it is natural to consider them from the viewpoint of

representation theory of categories, and characterize these properties using certain conditions independent of particular groups.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume:

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume: ◮ C(x, y) = i0 C(x, y)i and C(y, z)j · C(x, y)i ⊆ C(x, z)i+j;

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume: ◮ C(x, y) = i0 C(x, y)i and C(y, z)j · C(x, y)i ⊆ C(x, z)i+j; ◮ For any objects x and y, C(x, y) is finite dimensional;

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume: ◮ C(x, y) = i0 C(x, y)i and C(y, z)j · C(x, y)i ⊆ C(x, z)i+j; ◮ For any objects x and y, C(x, y) is finite dimensional; ◮ C(x, y)0 = 0 if x = y;

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume: ◮ C(x, y) = i0 C(x, y)i and C(y, z)j · C(x, y)i ⊆ C(x, z)i+j; ◮ For any objects x and y, C(x, y) is finite dimensional; ◮ C(x, y)0 = 0 if x = y; ◮ C(x, x)0 is semisimple;

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume: ◮ C(x, y) = i0 C(x, y)i and C(y, z)j · C(x, y)i ⊆ C(x, z)i+j; ◮ For any objects x and y, C(x, y) is finite dimensional; ◮ C(x, y)0 = 0 if x = y; ◮ C(x, x)0 is semisimple; ◮ For each x, there are only finitely many y with C(x, y) = 0 or

C(y, x) = 0;

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Graded k-linear categories

◮ Let C be a small skeletal k-linear category. We assume: ◮ C(x, y) = i0 C(x, y)i and C(y, z)j · C(x, y)i ⊆ C(x, z)i+j; ◮ For any objects x and y, C(x, y) is finite dimensional; ◮ C(x, y)0 = 0 if x = y; ◮ C(x, x)0 is semisimple; ◮ For each x, there are only finitely many y with C(x, y) = 0 or

C(y, x) = 0;

◮ Ci · C1 = Ci+1.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Directed graded k-linear categories

◮ The above category C is directed if there is a partial order

such that x y whenever C(x, y) = 0. We assume:

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Directed graded k-linear categories

◮ The above category C is directed if there is a partial order

such that x y whenever C(x, y) = 0. We assume:

◮ C(x, x) = C(x, x)0.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Directed graded k-linear categories

◮ The above category C is directed if there is a partial order

such that x y whenever C(x, y) = 0. We assume:

◮ C(x, x) = C(x, x)0. ◮ The convex hull of any finite set of objects is a finite category.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Directed graded k-linear categories

◮ The above category C is directed if there is a partial order

such that x y whenever C(x, y) = 0. We assume:

◮ C(x, x) = C(x, x)0. ◮ The convex hull of any finite set of objects is a finite category. ◮ The k-linearization of many categories in representation

stability theory satisfy all above assumptions when the characteristic of k is 0.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul categories

◮ A graded representation M of C is a homogeneous k-linear

functor from C to the category of graded vector spaces.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul categories

◮ A graded representation M of C is a homogeneous k-linear

functor from C to the category of graded vector spaces.

◮ If dimk Mi < ∞ for every i, and Mi = 0 when i << 0, then M

has a projective cover. We always consider these representations.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul categories

◮ A graded representation M of C is a homogeneous k-linear

functor from C to the category of graded vector spaces.

◮ If dimk Mi < ∞ for every i, and Mi = 0 when i << 0, then M

has a projective cover. We always consider these representations.

◮ M is Koszul if it has a linear projective resolution.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul categories

◮ A graded representation M of C is a homogeneous k-linear

functor from C to the category of graded vector spaces.

◮ If dimk Mi < ∞ for every i, and Mi = 0 when i << 0, then M

has a projective cover. We always consider these representations.

◮ M is Koszul if it has a linear projective resolution. ◮ C is a Koszul category if C(x, x) is a Koszul module for every

• bject x.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul categories

◮ A graded representation M of C is a homogeneous k-linear

functor from C to the category of graded vector spaces.

◮ If dimk Mi < ∞ for every i, and Mi = 0 when i << 0, then M

has a projective cover. We always consider these representations.

◮ M is Koszul if it has a linear projective resolution. ◮ C is a Koszul category if C(x, x) is a Koszul module for every

• bject x.

◮ A Koszul theory of non-negatively graded k-linear categories

has been established by Mazorchuk, Ovsienko, and Stroppel. But they assumed that C(x, x) ∼ = k.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul duality

◮ Theorem (G-L): Let C be a directed graded k-linear

• category. We have:

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul duality

◮ Theorem (G-L): Let C be a directed graded k-linear

• category. We have:

◮ If C is Koszul, then it is quadratic; its opposite category and

Yoneda category Y are Koszul. Moreover, Y ∼ = (C!)op.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Koszul duality

◮ Theorem (G-L): Let C be a directed graded k-linear

• category. We have:

◮ If C is Koszul, then it is quadratic; its opposite category and

Yoneda category Y are Koszul. Moreover, Y ∼ = (C!)op.

◮ Koszul duality (for certain module categories and certain

derived categories) holds.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Resuts

◮ Theorem (G-L): C is Koszul if and only if every finite convex

full subcategory is Koszul.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Resuts

◮ Theorem (G-L): C is Koszul if and only if every finite convex

full subcategory is Koszul.

◮ D is called an essential subcategory of C if Ob C = Ob D,

C(x, y) = D(x, y) for x = y, and D(x, x) = k1x.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Resuts

◮ Theorem (G-L): C is Koszul if and only if every finite convex

full subcategory is Koszul.

◮ D is called an essential subcategory of C if Ob C = Ob D,

C(x, y) = D(x, y) for x = y, and D(x, x) = k1x.

◮ Theorem (G-L): If C and C′ have the same essential

subcategory, then one is Koszul if and only if so is the other

• ne.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Definitions

◮ A directed graded k-linear category C is of type A∞ if objects

are parameterized by non-negative integers, and C(i, j) is concentrated in degree j − i.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Definitions

◮ A directed graded k-linear category C is of type A∞ if objects

are parameterized by non-negative integers, and C(i, j) is concentrated in degree j − i.

◮ A faithful k-linear functor ι : C → C is genetic if ι(i) = i + 1,

and the pullback of C(i, −) via ι (the restricted representation) is a projective C-module generated in positions i.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Theorem (G-L): If there is a genetic functor ι : C → C, then

C is a Koszul category.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Theorem (G-L): If there is a genetic functor ι : C → C, then

C is a Koszul category.

◮ The existence of such a fuctor was first observed by CEFN for

• FI. They call the restriction functor induced by ι a degree

shift functor.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Theorem (G-L): If there is a genetic functor ι : C → C, then

C is a Koszul category.

◮ The existence of such a fuctor was first observed by CEFN for

• FI. They call the restriction functor induced by ι a degree

shift functor.

◮ We described certain combinatorial conditions, which

guarantee the existence of genetic functors, and are pretty easy to check in practice.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Theorem (G-L): If there is a genetic functor ι : C → C, then

C is a Koszul category.

◮ The existence of such a fuctor was first observed by CEFN for

• FI. They call the restriction functor induced by ι a degree

shift functor.

◮ We described certain combinatorial conditions, which

guarantee the existence of genetic functors, and are pretty easy to check in practice.

◮ The k-linearizations of many infinite categories in

representation stability theory, such as FI, FIq, FId, FIΓ, OI, OIΓ OId, OS, etc, satisfy these conditions.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Let ρ : C → FI be an arbitrary functor. We constructed

explicitly a category Ctw, called the twisted category of C.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Let ρ : C → FI be an arbitrary functor. We constructed

explicitly a category Ctw, called the twisted category of C.

◮ Theorem (G-L): Let C be one of FIΓ, FId, OIΓ, OId. Then

• ne has:

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Let ρ : C → FI be an arbitrary functor. We constructed

explicitly a category Ctw, called the twisted category of C.

◮ Theorem (G-L): Let C be one of FIΓ, FId, OIΓ, OId. Then

• ne has:

◮ The Yoneda category Y is isomorphic to Ctw.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Let ρ : C → FI be an arbitrary functor. We constructed

explicitly a category Ctw, called the twisted category of C.

◮ Theorem (G-L): Let C be one of FIΓ, FId, OIΓ, OId. Then

• ne has:

◮ The Yoneda category Y is isomorphic to Ctw. ◮ The category of Y-modules is equivalent to the category of

C-modules.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory

Background Koszul theory of directed graded k-linear categories Type A∞ categories

Main result

◮ Let ρ : C → FI be an arbitrary functor. We constructed

explicitly a category Ctw, called the twisted category of C.

◮ Theorem (G-L): Let C be one of FIΓ, FId, OIΓ, OId. Then

• ne has:

◮ The Yoneda category Y is isomorphic to Ctw. ◮ The category of Y-modules is equivalent to the category of

C-modules.

◮ The bounded derived category of finite dimensional graded

C-modules is self-dual.

Wee Liang Gan and Liping Li Kosulity of directed categories in representation stability theory