Kosulity of directed categories in representation ...homepage.math.uiowa.edu/~fbleher/CGMRT2014/Slides/... · Background Koszul theory of directed graded k-linear categories Type

BackgroundKoszul theory of directed graded k-linear categories

Type A∞ categories

Kosulity of directed categories in representationstability 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

The category FI

I Objects: finite sets.

I Morphisms: injections.

I Equivalently, objects are [n], n ∈ N ∪ {0}I EndC([n]) is precisely Sn.

The category FI

I Equivalently, objects are [n], n ∈ N ∪ {0}

I EndC([n]) is precisely Sn.

The category FI

The category FIq

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

I Morphisms: linear injections.

I Equivalently, objects are Fn, n ∈ N ∪ {0}I EndC([n]) is precisely the general linear group.

The category FIq

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

The category FIq

I Equivalently, objects are Fn, n ∈ N ∪ {0}

I EndC([n]) is precisely the general linear group.

The category FIq

Applications

I These two categories, as well as a lot of variations, areintroduced and studied by CEFN, Putman, Sam, Snowden,Wilson, etc.

I They are used to study representations of a family of groupssimultaneously, in particular the representation stabilitywhen n increases.

I They have many applications in representation theory,algebraic topology, geometry, combinatorics, etc.

Applications

Properties

I Theorem (CEFN): FI is locally Noetherian over any leftNoetherian ring; that is, sub-representations of finitelygenerated representations are still finitely generated.

I Theorem (GL, PS): FIq is locally Noetherian over any leftNoetherian ring.

I Theorem (SS): Every finitely generated projective FI-moduleis also injective over the complex field.

I Many proofs use representations of these particular groups.

Properties

EI categories

I All above categories are examples of locally finite EI categoriesof type A∞, which are small categories such that everyendomorphism is invertible and satisfy:

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

representation theory of categories, and characterize theseproperties using certain conditions independent of particulargroups.

EI categories

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

I objects are indexed by N ∪ {0}, and C(j , s) ◦ C(i , j) = C(i , s).I Therefore, it is natural to consider them from the viewpoint of

EI categories

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

I Therefore, it is natural to consider them from the viewpoint ofrepresentation theory of categories, and characterize theseproperties using certain conditions independent of particulargroups.

EI categories

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

Graded k-linear categories

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

I C(x , y) =⊕

i>0 C(x , y)i and C(y , z)j · C(x , y)i ⊆ C(x , z)i+j ;I For any objects x and y , C(x , y) is finite dimensional;I C(x , y)0 = 0 if x 6= y ;I C(x , x)0 is semisimple;I For each x , there are only finitely many y with C(x , y) 6= 0 orC(y , x) 6= 0;

I C i · C1 = C i+1.

I Let C be a small skeletal k-linear category. We assume:I C(x , y) =

⊕i>0 C(x , y)i and C(y , z)j · C(x , y)i ⊆ C(x , z)i+j ;

I For any objects x and y , C(x , y) is finite dimensional;I C(x , y)0 = 0 if x 6= y ;I C(x , x)0 is semisimple;I For each x , there are only finitely many y with C(x , y) 6= 0 orC(y , x) 6= 0;

I C i · C1 = C i+1.

I For any objects x and y , C(x , y) is finite dimensional;

I C(x , y)0 = 0 if x 6= y ;I C(x , x)0 is semisimple;I For each x , there are only finitely many y with C(x , y) 6= 0 orC(y , x) 6= 0;

I C i · C1 = C i+1.

I For any objects x and y , C(x , y) is finite dimensional;I C(x , y)0 = 0 if x 6= y ;

I C(x , x)0 is semisimple;I For each x , there are only finitely many y with C(x , y) 6= 0 orC(y , x) 6= 0;

I C i · C1 = C i+1.

I For any objects x and y , C(x , y) is finite dimensional;I C(x , y)0 = 0 if x 6= y ;I C(x , x)0 is semisimple;

I For each x , there are only finitely many y with C(x , y) 6= 0 orC(y , x) 6= 0;

I C i · C1 = C i+1.

Directed graded k-linear categories

I The above category C is directed if there is a partial order 6such that x 6 y whenever C(x , y) 6= 0. We assume:

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

I The k-linearization of many categories in representationstability theory satisfy all above assumptions when thecharacteristic of k is 0.

I C(x , x) = C(x , x)0.

I The convex hull of any finite set of objects is a finite category.

Koszul categories

I A graded representation M of C is a homogeneous k-linearfunctor from C to the category of graded vector spaces.

I If dimk Mi

Koszul categories

I If dimk Mi

Koszul categories

I If dimk Mi

Koszul categories

I If dimk Mi

Koszul categories

I If dimk Mi

Koszul duality

I Theorem (G-L): Let C be a directed graded k-linearcategory. We have:

I If C is Koszul, then it is quadratic; its opposite category andYoneda category Y are Koszul. Moreover, Y ∼= (C!)op.

I Koszul duality (for certain module categories and certainderived categories) holds.

Koszul duality

Resuts

I Theorem (G-L): C is Koszul if and only if every finite convexfull subcategory is Koszul.

I D is called an essential subcategory of C if Ob C = ObD,C(x , y) = D(x , y) for x 6= y , and D(x , x) = k1x .

I Theorem (G-L): If C and C′ have the same essentialsubcategory, then one is Koszul if and only if so is the otherone.

Resuts

Definitions

I A directed graded k-linear category C is of type A∞ if objectsare parameterized by non-negative integers, and C(i , j) isconcentrated in degree j − i .

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 6 i .

Definitions

I A directed graded k-linear category C is of type A∞ if objectsare parameterized by non-negative integers, and C(i , j) isconcentrated in degree j − i .

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 6 i .

Main result

I Theorem (G-L): If there is a genetic functor ι : C → C, thenC is a Koszul category.

I The existence of such a fuctor was first observed by CEFN forFI. They call the restriction functor induced by ι a degreeshift functor.

I We described certain combinatorial conditions, whichguarantee the existence of genetic functors, and are prettyeasy to check in practice.

I The k-linearizations of many infinite categories inrepresentation stability theory, such as FI, FIq, FId , FIΓ, OI,OIΓ OId , OS, etc, satisfy these conditions.

Main result