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 ExtA(k, k) is a noetherian k-algebra 3 ExtA(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 ExtA(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 = kx1, · · · , xn/I I = m1, . . . , mℓ where the mi are monomials in {x1, . . . , xn}. Denote the Ext algebra ExtA(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 G0 = {x1, . . . , xn}, and for i > 0, set Gi = minimal left annihilators of elements in Gi−1. Vertices of Γ(A) :

i≥0

Gi 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).

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

Example: A = ka, b, c, d/abc, cdab

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

Example: A = ka, b, c, d/abc, cdab G0 a b c d

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

Example: A = ka, b, c, d/abc, cdab G0 G1 a b

cda

• c

ab

d

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

Example: A = ka, b, c, d/abc, cdab G0 G1 G2 a b

cda

• c

ab cd

• 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 v0v1v2 · · · where vi → vi+1 is an edge. A path is a walk with no repeated edges. A walk in Γ(A) is anchored if it starts in G0. Denote the set of all anchored walks of length n in Γ(A) by Wn. A circuit of length n is a walk v0v1 . . . vn with v0 = vn and {vi : 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 Ak over A = kx1, . . . , xn/I has the form · · · →

• w∈W2

A(−dw) →

• w∈W1

A(−dw) → A → k → 0, where for w ∈ Wn, dw denotes the sum of the degrees of the vertices in the walk, and if w = w′m ∈ Wn for w′ ∈ Wn−1 then ∂(ew) = mew′. The graded duals {ǫw} of the basis elements {ew}, where w is an anchored walk of length i in Γ(A), form a k-basis for Exti+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

• f 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 = ka, b, c, d/abc, cdab a b

cda

• c

ab cd

• 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 = ka, b/ab, ba, b2 a

• b
• GKdim E(A) = ∞.

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

Example: A = kx, y/x2y, xy2, y3, x4 x3

x

• y2
• xy
• y
• x2
• GKdim E(A) = 2.

Hence B = kx, y/(x3 − x2y, xy2, y3) 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 p0p1 . . . pn and q0q1 . . . qm are equivalent if n = m and pnpn−1 . . . p0 = qmqm−1 . . . q0. 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 = ka, b, c, d/abc, cdab a b

cda

• c

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).

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

Yoneda Product α ⋆ β Let p = p0 · · · ps and q = q0 · · · qn be admissible walks in Γ(A). Then ǫp ⋆ ǫq = 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

ǫp ⋆ ǫq = ǫw where w ∼ q′

0 · · · q′ np′ 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

• nly 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 = ka, b, c, d/abc, cdab a b

cda

• c

ab cd

• 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 = ka, b, c, d/abc, cdab a b

cda

• c

ab cd

• 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 = ka, b, c, d/abc, cdab, bcda a

bcd

• b

cda

• c

ab cd

• d

Now ab → cd is not dense in c → ab → cd → ab → cd · · ·