Finiteness Conditions on the Ext Algebra of a Monomial Algebra - PowerPoint PPT Presentation

Finiteness Conditions on the Ext Algebra of a Monomial Algebra Ellen Kirkman kirkman@wfu.edu University of Missouri, Columbia, November 23, 2013 ArXiv 1210.3389 J. Pure and Applied Algebra 218 (2014) 52-64 Joint work with Andrew Conner, Jim Kuzmanovich, and W. Frank Moore

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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.

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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?

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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 ) .

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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 m ′ is a minimal Edges of Γ( A ) : m → m ′ ⇔ left annihilator of m . When A is quadratic, Γ( A ) is Ufnarovski’s “relation graph" of A ! = E ( A ) .

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab �

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � G 0 a b c d

� Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � G 0 G 1 a � cda b � ab c d

� � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � G 0 G 1 G 2 a � cda b � cd � ab c d

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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.

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Proposition (Cassidy-Shelton,Phan) The minimal free resolution of A k over A = k � x 1 , . . . , x n � /I has the form � � · · · → A ( − d w ) → A ( − d w ) → A → k → 0 , w ∈ W 2 w ∈ W 1 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 { ǫ 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 ) .

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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).

� � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � a � cda b � cd � ab c d So gldimA = ∞ and GKdim E ( A ) = 1 .

� � � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b � / � ab, ba, b 2 � a b GKdim E ( A ) = ∞ .

� � � � � � � � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � x, y � / � x 2 y, xy 2 , y 3 , x 4 � � x x 3 y 2 x 2 xy y GKdim E ( A ) = 2 . Hence B = k � x, y � / ( x 3 − x 2 y, xy 2 , y 3 ) has GKdim E ( B ) ≤ 2 .

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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.

� � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � a � cda b � cd � ab c 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).

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Yoneda Product α ⋆ β Let p = p 0 · · · p s and q = q 0 · · · q n be admissible walks in Γ( A ) . Then ǫ p ⋆ ǫ 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 ǫ p ⋆ ǫ q = ǫ w where w ∼ q ′ 0 · · · q ′ n p ′ 0 · · · p ′ s .

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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.

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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.

� � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � a � cda b � cd � ab c d All out degrees ≤ 1 ( ab has in-degree 3) cd → ab not admissible E ( A ) is not left or right Noetherian.

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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 .

� � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Example: A = k � a, b, c, d � / � abc, cdab � a � cda b � cd � ab c d 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.

Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) 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.

� � � Motivation Background Using the CPS Graph When is E ( A ) Noetherian? Finite generation of E ( A ) Adding one relation to A we obtain B , where E ( B ) not finitely generated. B = k � a, b, c, d � / � abc, cdab, bcda � � bcd a � cda b � cd � ab c d Now ab → cd is not dense in c → ab → cd → ab → cd · · ·