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On the deformation theory of finite dimensional algebras. (English) Zbl 0851.16011

We are concerned with finite dimensional algebras \(A\) for which some of the low cohomology groups \(H^i(A)\) (\(i=1,2,3\)) vanish. In section 1, we relate the groups \(H^i(A)\) with local properties of the scheme \(\text{alg}_d\) of \(d\)-dimensional \(k\)-algebra structures (for \(d=\dim_k A\)). For example, we show that if \(H^3(A)=0\), then the point \(\alpha\) in \(\text{alg}_d\) corresponding to \(A\) is smooth. As a consequence, if \(H^1(A)=0\) and \(H^3(A)=0\), then there is an open neighbourhood \(\mathcal U\) of \(\alpha\) in \(\text{alg}_d\) such that for every \(\beta\) in \(\mathcal U\), the set of points corresponding to algebras isomorphic to \(B\) has dimension \(\dim_k H^2(A)\). Therefore, in this situation the different concepts of rigidity considered in the literature coincide.
In section 2 we concentrate on strongly simply connected tame algebras \(A\) of polynomial growth. For these algebras \(H^1(A)=0\) and \(H^3(A)=0\). We calculate \(\dim_k H^2(A)\) and therefore we find ridigity conditions. We construct explicitly a geometric \(\dim_k H^2(A)\)-parametric family of strongly simply connected algebras of polynomial growth.

MSC:

16G20 Representations of quivers and partially ordered sets
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16S80 Deformations of associative rings
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References:

[1] I. Assem and J. A. de la Peña. The fundamental groups of a triangular algebra. Preprint. Mexico (1994) · Zbl 0880.16007
[2] I. Assem and A. Skowroński. Indecomposable modules over multicoil algebras. Math. Scand.71, 31–61 (1992) · Zbl 0796.16014
[3] I. Assem and A. Skowroński. Multicoil algebras. Proc. ICRA VI Canadian Math. Soc. Conference Proc.14, 29–67 (1993) · Zbl 0827.16010
[4] K. Bongartz. Minimal singularities for representations of Dynkin quivers. Comment. Math. Helvetici69, 575–611 (1994) · Zbl 0832.16008 · doi:10.1007/BF02564505
[5] M. Demazure and P. Gabriel. Introduction to algebraic geometry and algebraic groups. Mathematics Studies39. North Holland (1980) · Zbl 0431.14015
[6] P. Gabriel. Finite representation type is open. In: Representations of Algebras, Springer LNM488, 132–155 (1975) · Zbl 0313.16034
[7] P. Gabriel. Auslander-Reiten sequences and representation-finite algebras. Proc. ICRA II (Ottawa 1979). In Representation of Algebras, Springer LNM831, 1–71 (1980)
[8] P. Gabriel and J. A. de la Peña. On algebras: wild and tame. In: Duration and change. Fifty years at Oberwolfach. Springer 177–210 (1994) · Zbl 0820.16016
[9] Ch. Geiss. On degenerations of tame and wild algebras. Arch. Math.64, 11–16 (1995) · Zbl 0828.16013 · doi:10.1007/BF01193544
[10] M. Gerstenhaber. On the deformation of rings and algebras. Ann. Math.79, 59–103 (1964) · Zbl 0123.03101 · doi:10.2307/1970484
[11] M. Gerstenhaber and M. Schack. Relative Hochschild cohomology, rigid algebras and the Bockstein. J. Pure Appl. Alg.43, 53–74 (1986) · Zbl 0603.16021 · doi:10.1016/0022-4049(86)90004-6
[12] D. Happel. Hochschild cohomology of finite dimensional algebras. Séminaire M.-P. Malliavin (Paris, 1987–88), LNM1404, Springer-Verlag 108–126 (1989)
[13] J. C. Jantzen. Representation of algebraic groups. Pure Appl. Mat. V.131 Ac. Press (1989)
[14] J. A. de la Peña. On the representation type of one-point extensions of tame concealed algebras. Manuscr. Math.61, 183–194 (1988) · Zbl 0647.16021 · doi:10.1007/BF01259327
[15] J. A. de la Peña. On the corank of the Tits form of a tame algebra. To appear in J. Pure Appl. Alg. · Zbl 0851.16014
[16] J.A. de la Peña and A. Skowroński. Forbidden subcategories of non-polynomial growth tame simply connected algebras. To appear Canadian J. Math. · Zbl 0870.16010
[17] J. A. de la Peña and A. Skowroński. Geometric and homological characterizations of polynomial growth simply connected algebras. Preprint. Mexico (1994) · Zbl 0883.16007
[18] C. M. Ringel. Tame algebras and integral quadratic forms. Springer, Berlin LNM1099 (1984) · Zbl 0546.16013
[19] A. Skowroński. Simply connected algebras and Hochschild cohomology. Proc. ICRA VI (Ottawa, 1992), Can. Math. Soc. Proc. Vol.14, 431–447 (1993) · Zbl 0806.16012
[20] A. Skowroński. Cycle-finite algebras. To appear J. Pure Appl. Alg. · Zbl 0841.16020
[21] A. Skowroński. Criteria for polynomial growth of algebras. To appear Bull. Polish Ac.
[22] D. Voigt. Induzierte Darstellungen in der Theorie der endlichen algebraischen Gruppen. Springer LNM592 (1977) · Zbl 0374.14010
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