2019
DOI: 10.1016/j.jalgebra.2019.08.029
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The first Hochschild (co)homology when adding arrows to a bound quiver algebra

Abstract: We provide a formula for the change of the dimension of the first Hochschild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first Hochschild homologies are isomorphic when adding new arrows.

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Cited by 2 publications
(5 citation statements)
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“…Recall that false(XSpfalse)/Lp,0=XBeSBp=sans-serifTor0Befalse(X,SBpfalse).The image of the last differential of Gp is then Lp,0, that is the last differential of Gp/Gp1 is surjective and H0false(Gp/Gp1false)=0. For p=1 it is proven in [10, Proposition 3.3] that there is a short exact sequence for Hochschild cohomology: 0H1false(Afalse|B,Xfalse)ιH1false(A,Xfalse)κH1false(B,Xfalse). Let V denote the dual of a vector space V. It is well known that for finite‐dimensional A and X, we have Hfalse(A,Xfalse)=(Hfalse(A,Xfalse)).…”
Section: Gap Of the Jacobi–zariski Long Nearly Exact Sequencementioning
confidence: 99%
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“…Recall that false(XSpfalse)/Lp,0=XBeSBp=sans-serifTor0Befalse(X,SBpfalse).The image of the last differential of Gp is then Lp,0, that is the last differential of Gp/Gp1 is surjective and H0false(Gp/Gp1false)=0. For p=1 it is proven in [10, Proposition 3.3] that there is a short exact sequence for Hochschild cohomology: 0H1false(Afalse|B,Xfalse)ιH1false(A,Xfalse)κH1false(B,Xfalse). Let V denote the dual of a vector space V. It is well known that for finite‐dimensional A and X, we have Hfalse(A,Xfalse)=(Hfalse(A,Xfalse)).…”
Section: Gap Of the Jacobi–zariski Long Nearly Exact Sequencementioning
confidence: 99%
“…However Kerκfalse|/Imιfalse| is intricate to describe. Nevertheless, we will obtain from [10] that if A and X are finite‐dimensional, the Jacobi–Zariski long nearly exact sequence is exact in degree 1.…”
Section: Gap Of the Jacobi–zariski Long Nearly Exact Sequencementioning
confidence: 99%
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“…However Kerκ | /Imι | is intricate to describe. Nevertheless, we will obtain from [8] that if A and X are finite dimensional, the Jacobi-Zariski long nearly exact sequence is exact in degree 1.…”
Section: 1)mentioning
confidence: 99%
“…In [7], the Jacobi-Zariski sequence in non commutative algebra has been useful in relation to Han's conjecture (see [12]). Moreover, it enables to give formulas for the change of dimension of Hochschild (co)homology when deleting or adding an inert arrow to the quiver of an algebra in [8,9].…”
Section: Introductionmentioning
confidence: 99%