Pedro Tamaro Trinity College Dublin London, November 2018pedro/CITY2018.pdf · Trinity College...

Motivation Associative algebras GHZ-A ADMT dga algebras HTT and trees Main result

Minimal models for monomial algebras

Pedro TamaroffTrinity College Dublin

London November 2018


Resolutions of objects

What we do in the shadows

Generally we want to study an object Y which is linear in naturefor example a module over an algebra

A resolution of Y is loosely speaking a way to unpack Y into asequence of objects which are much more well behaved than Y

We relate these objects by a sequence of boundary maps Theseencode the ldquohomological propertiesrdquo of Y in a non-trivial way


Some examples

We can resolve modules over algebras by free modules Freemodules behave like vector spaces and this makes us happy We cando the same with bimodules

We can resolve algebras by differential graded algebras that are freeas an algebra Free algebras depend only on a vector space and thisalso makes us happy

Notation We will write (TX dX ) for a free algebra with a differentialand call a resolution of an algebra a model


Why do we like models

Resolving an algebra resolves in particular its representations wecan focus on understanding how to obtain models

Models are algebras which are non-linear in nature so they capturemore information than representations which are linear

They have been used to solve open problems like Hanrsquosconjecture [2] and the rational Ganea conjecture [3]












Associative algebras

Associative algebras can be presented by generators V and relations RResolving an algebra involves understanding its syzygyes

The free algebra has no relations

The polynomial algebra has relations R = xy minus yx | x y isin V The exterior algebra has relations R = xy + yx | x y isin V A monomial algebra has relations that are monomials on V likexy2 = 0 or xyx = 0

Notation We will write A = T 〈V |R〉 for an algebra with generators Vand relations R and A for the augmentation ideal
































The plan

Principle of conservation of difficulty We want to resolve an algebraby some object (TX dX ) In general if we can pin down one of X or dXeasily a lot of work will be needed to pin down the other

Show that X can be described combinatorially in terms of Anickchains For example X0 = V X1 = R

Produce a datum that remembers how to compute X from aldquocanonical modelrdquo of A

Show that this datum determines dX and then the desired model(TX dX ) of A

Give explicit formulas for dX on generators


Anick chains combinatorics of syzygyes

If A = T 〈V |R〉 is an associative monomial algebra let us write X0 = Vfor its generators and X1 = R for its relations

Inductively define X2X3 by their bases C2C3 consisting ofmonomials called chains as follows

Definition A monomial m is an i-chain if the following holds

We can write m = m1m2 = mprime1mprime2 so that

m1 is an (i minus 1)-chain and mprime2 is a 1-chain

m2 is a proper right divisor of mprime2 and

no proper left divisor of m can be written so to satisfy the previousconditions


Anick chains useful examples

Example 1 Let us consider k〈t|t2〉 Then C0 = t C1 = t2 and ingeneral Cn = tn+1 for every n isin N0

Example 2 Let us consider k〈x y |x2 xy2〉 Then Cn = xn+1 xny2 foreach n isin N while C0 = x y

Example 3 Let us consider k〈x y z |xy2 y2z〉 Then C0 = x y zC1 = xy2 y2z C2 = xy2z xy3z and Cn = 0 for n gt 3


The bar construction syzygyes are homological

An important construction that can be extracted from an algebra A is thebar construction of A (BA dBA∆) It encodes syzygyes of A

We let BA be the free coalgebra on A so it is a sum kAAotimes2

The operator d is induced by the multiplication of A

dBA[a1| middot middot middot |an] =nminus1sumi=1

(minus1)iminus1[a1| middot middot middot |aiminus1|aiai+1|ai+2| middot middot middot |an]

The operator ∆ BArarr BAotimes BA is deconcatenation

∆[a1| middot middot middot |an] =nsum

i=1

[a1| middot middot middot |ai ]otimes [ai+1| middot middot middot |an]


The differential of BA

One can check that d2BA = 0 is equivalent to the fact A is associative

The operator ∆ is coassociative and makes BA into a dg coalgebra Wewrite TorA for its homology

Theorem (Anick and GreenndashHappelndashZacharia)

Chains give a basis of TorA for A a monomial algebra for eachr isin N0 we have that TorAr+1 Xr

Note There is a shift in the indices between X and TorA We willdenote this by saying that X = sminus1 TorA


What do we have so far

The spaces of chains X produced for us the generators of our model Butwe also need an operator dX How can this be obtained from A

Answer The space X was obtained from (BA∆ dBA) which is a dgcoalgebra It turns out that if we relate X and BA by some linear mapswe can build the new operator dX from this information


Algebraic discrete Morse theory

Algebraic discrete Morse theory developed by by Skoldberg [6] andJollenbeckndashWelker [4] solves our problem It ldquoremembersrdquo how we wentfrom BA to H(BA) = TorA its homology

Geometrically it tells us how to ldquokill contractible complexesrdquo in BA toobtain a model whose generators are H(BA) The picture we want is

TorA BAi

ph 1minus ip = dh + hd pi = 1


and what it can do for you

Theorem (JollenbeckndashWelker)

If B is a complex of free modules and M is a Morse matching thereis a complex BM and maps

i BM rarr B p B rarr BM

such that pi = 1BM and ip is homotopic to the identity of B

In [4] the authors give a Morse matching M on BA so that (BA)M hasbasis the set of Anick chains This gives us the diagram we wanted


Interlude dg algebras

We can enhance associative algebras by introducing a grading and anoperator on them We care only about free algebras say TX

Grading We will assume X is a direct sum of Xi of subspaces Wecall Xi the ith graded piece of X

Differential We assume that we have an operator d TX rarr TX ofdegree minus1 such that d2 = 0 which satisfies the Leibniz rule

d(xy) = dx middot y + (minus1)|x|x middot dy

This rule means d is determined completely by its action on X























What is a model of an algebra

Given some dg algebra (TX d) the 0th homology H0(TX ) is a anassociative algebra We want to exhibit associative algebras in the formH0(TX ) where we additionally require that H+(TX ) = 0

Explicitly given an algebra T 〈V |R〉 we want to give

a graded vector space X

a map f TX rarr A and

a differential d on TX so that

the induced map Hf H(TX d)rarr A is an isomorphism


so what do we do with the retraction

Relevant data For our purposes it suffices to understand here

The homotopy h BA minusrarr BA afforded by ADMT

Its behaviour with respect to ∆ BA minusrarr BAotimes BA

How these two operators act on TorA

Indeed the following result describes the differential in the minimalmodel with generators TorA in terms of these two operators


Transferring the structure to chains

Homotopy Transfer Theorem

Let C be a dga coalgebra (ex C = BA) and consider a retraction(i p h) to a complex p C rarr C prime as before Then there is aretraction (j q k) between dg algebras q T (sminus1C )rarr T (sminus1C )

This gives us what we want Indeed T (sminus1BA) is a (very big) model ofA but now we have a new model TX where X = sminus1 TorA is as smallas can be


How to pin down the differential Trees

For a given planar binary tree T let ∆T be the operator obtained bydecorating the root of T by i (inclusion TorA sube BA) its leaves by p(projection BArarr TorA) its internal edges by h and its vertices by ∆These define maps from X to TX

Theorem (Markl [5] and many others before)

We have that d |sX =sum

ngt2 ∆n where for each n isin N0 ∆n =sumT (minus1)ϑ(T )∆T as T ranges through all planar binary trees with

n leaves


Some trees go away The comb remains

The explicit description of the differential in TX was simplified with thefollowing vanishing and exchange rules

Proposition (T)

If h is the homotopy h BA minusrarr BA and ∆ is the (deconcatenation)of BA then

∆h = (h otimes 1)∆ mod (TorAotimesBA)

Moreover h vanishes on TorA and h2 = 0 All this implies that∆T = 0 unless T is a right comb


The last of the trees and its action

Decompositions If γ is an r -chain in TorA a decomposition of γ is asequence (γ1 γn) of chains of length r1 rn such that

r1 + middot middot middot+ rn = r minus 1 (correct degree)

γ = γ1 γn (as monomials)

We say the decomposition has length n and call it an n-decomposition

Example The 2-chain xy2z has two 2-decompositions (x y2z) and(xy2 z) and no other decompositions The 1-chain xy2 has a single3-decomposition (x y y) and no other decompositions

Theorem (T)

If T is the right comb and γ is any chain then ∆T (γ) is a signedsum through all decompositions of γ








Theorem (T)









Theorem (T)



Description of the model

The end result is the following description of minimal models ofmonomial quiver algebras

Theorem (T)

For each monomial algebra A there is a minimal model(TX dX ) minusrarr A For a chain γ isin X

dX (γ) =sumngt2

(minus1)(n+12 )+|γ1|minus1γ1 middot middot middot γn

where the sum ranges through all possible decompositions of γ


A non-monomial example

The super Jordan plane is the associative algebra

k〈x y |x2 x2y minus yx2 minus xyx〉

Its minimal model has two generators xn yn in degree n for each n isin N0

corresponding to the ambiguities xn+1 and y2xn Its differential vanisheson degree 0 and for n isin N0 is as follows

dyn+1 = y2xn minus xny2 minus

sums+t=n

xsyxt minussum

s+t=ntgt1

(xsyt minus (minus1)tytxs)

dxn+1 =sum

s+t=n

(minus1)sxsxt


Application the Gerstenhaber bracket

If TX rarr A is a model of an algebra A one can compute Hochschildcohomology of A through derivations of TX

Fact The Gerstenhaber bracket in Hochschild cohomology is just the Liebracket on derivations So one can compute it easily if one can computeHHlowast(A) through a model Of course computing models is quite a task

One can also compute Hochschild homology and cyclic homology andother gadgets associated to these invariants


Thank you


References

[1] David J Anick On the homology of associative algebras Trans Amer Math Soc296 (1986) no 2 641ndash659 MR846601

[2] Luchezar L Avramov and Micheline Vigue-Poirrier Hochschild homology criteriafor smoothness Internat Math Res Notices 1 (1992) 17ndash25 DOI101155S1073792892000035 MR1149001

[3] Kathryn P Hess A proof of Ganearsquos conjecture for rational spaces Topology 30(1991) no 2 205ndash214 DOI 1010160040-9383(91)90006-P MR1098914

[4] Michael Jollenbeck and Volkmar Welker Minimal resolutions via algebraic discreteMorse theory Mem Amer Math Soc 197 (2009) no 923 vi+74 MR2488864

[5] Martin Markl Transferring Ainfin (strongly homotopy associative) structures (2004)14 pp available at arXivmath0401007[mathAT]

[6] Emil Skoldberg Morse theory from an algebraic viewpoint Trans Amer MathSoc 358 (2006) no 1 115ndash129 DOI 101090S0002-9947-05-04079-1MR2171225

Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result


Resolutions of objects

What we do in the shadows

Generally we want to study an object Y which is linear in naturefor example a module over an algebra

A resolution of Y is loosely speaking a way to unpack Y into asequence of objects which are much more well behaved than Y

We relate these objects by a sequence of boundary maps Theseencode the ldquohomological propertiesrdquo of Y in a non-trivial way


Some examples
























































The plan





























i=1

















TorA BAi































































n leaves




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result


Some examples
























































The plan





























i=1

















TorA BAi































































n leaves




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result





















































The plan





























i=1

















TorA BAi































































n leaves




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result
























i=1

















TorA BAi































































n leaves




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result
















TorA BAi































































n leaves




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result






























































n leaves




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result




Proposition (T)











Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result








Theorem (T)









Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result








Theorem (T)









Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result








Theorem (T)





Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result




Theorem (T)


dX (γ) =sumngt2










sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result








sums+t=n

xsyxt minussum

s+t=ntgt1


dxn+1 =sum

s+t=n

(minus1)sxsxt







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result







Thank you


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result


References







Motivation


GHZ-A ADMT

dga algebras

HTT and trees

Main result

Pedro Tamaro Trinity College Dublin London, November 2018pedro/CITY2018.pdf · Trinity College...

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Transcript of Pedro Tamaro Trinity College Dublin London, November 2018pedro/CITY2018.pdf · Trinity College...