The first Hochschild (co)homology when adding arrows to a bound quiver algebra. (English) Zbl 1423.18051
The first Hochschild cohomology vector space \(HH^1(B)\) of an algebra \(B\) over a field \(k\) is isomorphic to the quotient of the \(k\)-derivations of the algebra by the inner ones. For a bound quiver algebra \(B = kQ/I\), given some hypotheses on \(I\), J. A. de la Peña and M. Saorín [Manuscr. Math. 104, No. 4, 431–442 (2001; Zbl 0985.16008)] obtained formulas computing the dimension of \(HH^1(B)\).
The authors study the change in both the Hochschild cohomology and Hochschild homology of an algebra given by quiver and relations, when arrows to the quiver are added. A main tool for the work is relative cohomology as defined by G. Hochschild [Trans. Am. Math. Soc. 82, 246–269 (1956; Zbl 0070.26903)] and used for instance by M. Auslander and Ø. Solberg [Commun. Algebra 21, No. 9, 2995–3031 (1993; Zbl 0792.16017)] in the context of representation theory. For this purpose they show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology.
The first Hochschild homologies are isomorphic when adding new arrows. As a by-product of the presented formula, a new computation of the dimension of \(HH^1(kQ)\) for a quiver \(Q\) without cycles is obtained.
The authors study the change in both the Hochschild cohomology and Hochschild homology of an algebra given by quiver and relations, when arrows to the quiver are added. A main tool for the work is relative cohomology as defined by G. Hochschild [Trans. Am. Math. Soc. 82, 246–269 (1956; Zbl 0070.26903)] and used for instance by M. Auslander and Ø. Solberg [Commun. Algebra 21, No. 9, 2995–3031 (1993; Zbl 0792.16017)] in the context of representation theory. For this purpose they show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology.
The first Hochschild homologies are isomorphic when adding new arrows. As a by-product of the presented formula, a new computation of the dimension of \(HH^1(kQ)\) for a quiver \(Q\) without cycles is obtained.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |
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