Finiteness Conditions on the Ext Algebra of a Monomial Algebra

Transcription

1 Finiteness Conditions on the Ext Algebra of a Monomial Algebra Ellen Kirkman kirkman@wfu.edu University of Missouri, Columbia, November 23, 2013 ArXiv J. Pure and Applied Algebra 218 (2014) Joint work with Andrew Conner, Jim Kuzmanovich, and W. Frank Moore

2 Commutative graded complete intersections Theorem (Tate,Gulliksen,Bøgvad-Halperin) For a graded Noetherian commutative k-algebra, the following are equivalent: 1 A is a complete intersection 2 Ext A (k, k) is a noetherian k-algebra 3 Ext A (k, k) has finite Gelfand-Kirillov (GK) dimension.

3 Question What is the appropriate notion of complete intersection for a noncommutative algebra? Question For a monomial algebra, when does Ext A (k, k) satisfy a finiteness condition such as in the preceding theorem?

4 Let A be a connected graded noncommutative monomial algebra over a field k: A = k x 1,, x n /I I = m 1,..., m l where the m i are monomials in {x 1,..., x n }. Denote the Ext algebra Ext A (k, k) of A by E(A).

5 The CPS graph Γ(A) of a monomial algebra A: Let G 0 = {x 1,..., x n }, and for i > 0, set G i = minimal left annihilators of elements in G i 1. Vertices of Γ(A) : G i i 0 Edges of Γ(A) : m m m is a minimal left annihilator of m. When A is quadratic, Γ(A) is Ufnarovski s relation graph" of A! = E(A).

6 Example: A = k a, b, c, d / abc, cdab

7 Example: A = k a, b, c, d / abc, cdab G 0 a b c d

8 Example: A = k a, b, c, d / abc, cdab G 0 G 1 a b c cda ab d

9 Example: A = k a, b, c, d / abc, cdab G 0 G 1 G 2 a b c cda ab cd d

10 A walk in Γ(A) is a sequence of vertices v 0 v 1 v 2 where v i v i+1 is an edge. A path is a walk with no repeated edges. A walk in Γ(A) is anchored if it starts in G 0. Denote the set of all anchored walks of length n in Γ(A) by W n. A circuit of length n is a walk v 0 v 1... v n with v 0 = v n and {v i : i n 1} distinct.

11 Proposition (Cassidy-Shelton,Phan) The minimal free resolution of A k over A = k x 1,..., x n /I has the form w W 2 A( d w ) w W 1 A( d w ) A k 0, where for w W n, d w denotes the sum of the degrees of the vertices in the walk, and if w = w m W n for w W n 1 then (e w ) = me w. The graded duals {ɛ w } of the basis elements {e w }, where w is an anchored walk of length i in Γ(A), form a k-basis for Ext i+1 A (k, k).

12 Corollary Let A be a connected graded monomial algebra with CPS graph Γ(A). 1 gldima < if and only if Γ(A) does not contain a circuit. Then gldima is the length of the longest path in Γ(A). 2 GKdim E(A) = if and only if Γ(A) contains distinct circuits that share a common vertex. 3 If GKdim E(A) <, then GKdim E(A) is the maximum number of circuits contained in any walk in Γ(A) (so is an integer).

13 Example: A = k a, b, c, d / abc, cdab a b c cda ab cd d So gldima = and GKdim E(A) = 1.

14 Example: A = k a, b / ab, ba, b 2 a b GKdim E(A) =.

15 Example: A = k x, y / x 2 y, xy 2, y 3, x 4 x 3 x y 2 xy y x 2 GKdim E(A) = 2. Hence B = k x, y /(x 3 x 2 y, xy 2, y 3 ) has GKdim E(B) 2.

16 When is E(A) Noetherian? Walks p 0 p 1... p n and q 0 q 1... q m are equivalent if n = m and p n p n 1... p 0 = q m q m 1... q 0. A walk is admissible if it is equivalent to an anchored walk.

17 Example: A = k a, b, c, d / abc, cdab a b c cda ab cd d ab cd is admissible since it is equivalent to the anchored walk b cda ((cd)(ab) = (cda)b), but cd ab is not admissible. Also ab cd ab is admissible ((ab)(cd)(ab) = ab(cda)b).

18 Yoneda Product α β Let p = p 0 p s and q = q 0 q n be admissible walks in Γ(A). Then ɛ p ɛ q = 0, unless there exists walks p p and q q such that q is anchored and q n p 0 is an edge in Γ(A). Then ɛ p ɛ q = ɛ w where w q 0 q np 0 p s.

19 Theorem (CKKM) Let A be a connected graded monomial algebra. Then E(A) is left (resp. right) noetherian if and only if the following conditions are satisfied: 1 Every vertex of Γ(A) lying on an oriented circuit has out-degree (resp. in-degree) one, and 2 Every edge of every oriented circuit is admissible.

20 Corollary Let A be a connected graded monomial algebra. If A is left or right noetherian, then GKdim A 1. If A is noetherian, then Γ(A) is a disjoint union of cycles and paths.

21 Example: A = k a, b, c, d / abc, cdab a b c d cda ab cd All out degrees 1 (ab has in-degree 3) cd ab not admissible E(A) is not left or right Noetherian.

22 When is E(A) finitely generated? Definition An anchored walk w is decomposable if w = w w where w is an admissible walk of positive length. Definition Let w be an infinite walk in Γ(A). An admissible edge e in w is called dense if w contains an even length admissible extension of e.

23 Example: A = k a, b, c, d / abc, cdab a b c d cda ab cd c ab cd ab cd ab is decomposable since ab cd ab is admissible. Two infinite anchored walks: c ab cd ab b cda ab cd ab ab cd is dense in each, since ab cd ab is admissible.

24 Theorem (CKKM) Let A be a connected graded monomial algebra. Then the following are equivalent: 1 E(A) is a finitely generated algebra 2 Every infinite anchored walk in Γ(A) has finitely many indecomposable prefixes. 3 For every infinite anchored walk p in Γ(A), p contains a dense edge or two admissible edges with lengths of opposite parity.

25 Adding one relation to A we obtain B, where E(B) not finitely generated. B = k a, b, c, d / abc, cdab, bcda a bcd b c d cda ab cd Now ab cd is not dense in c ab cd ab cd