Issue with Checking Golod Property Using Koszul Complex

[Shrikant Shekhar 1910404]

Dear all,
Following ring R is not a Golod ring as H(A) does not have trivial multiplication where A=koszulComplexDGA(R).

R = ZZ/101[x,y,z]/ideal (x^3,y^3,z^3,x^2*y^2*z^2);

"But using Infinite free resolutions by Avramov Proposition-5.2.4.(i) R is Golod if HH_{\geq 1}(A) is generated by a set of cycles Z, such that Z^2=0. Now I am applying this criterion using following codes for which I'm getting a zero list."

cycleList = getGenerators(A)
apply(#c-1,i->(c_i)^2)

I am not sure where I am making a mistake. Any guidance would be greatly appreciated.
Sincerely,
Shrikant.

[David Eisenbud]

I guess Z^2 includes c_ic_j for all i,j.

DE

--

David Eisenbud

Professor of Mathematics

University of California Berkeley

https://eisenbud.github.io/

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[Frank Moore]

You should use something like the last line below:

R = ZZ/101[x,y,z]/ideal (x^3,y^3,z^3,x^2*y^2*z^2);

A = koszulComplexDGA R
c = getGenerators(A)
apply(subsets(#c,2), p -> c_(p_0) * c_(p_1))

Also, note that getGenerators returns a list of cycles that generate HH(A) as an algebra, and it is still possible that c_i * c_j is nonzero, but a boundary in A (and hence zero in HH(A)).

Frank

To view this discussion visit https://groups.google.com/d/msgid/macaulay2/CAFbL_pYFYTjpgyacdUYZG1tWucpfbRjKFSVG2cAxWsmcr4m9HQ%40mail.gmail.com.

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Dr. W. Frank Moore
Associate Professor

Department of Mathematics and Statistics

Wake Forest University

email: moo...@wfu.edu

[Shrikant Shekhar 1910404]

Thanks a lot for your help.

Shrikant.