×

Tameness and complexity of finite group schemes. (English) Zbl 1121.16015

Let \(\Lambda\) be a self-injective tame algebra over an algebraically closed field \(k\). A result of J.Rickard [Bull. Lond. Math. Soc. 22, No. 6, 540-546 (1990; Zbl 0742.16007)] states that if the complexity \(\text{cx}_\Lambda(M)\) of a finite dimensional \(\Lambda\)-module \(M\) is at least \(3\) then \(M\) is a wild algebra (or, equivalently, if \(M\) is tame then \(\text{cx}_\Lambda(M)\leq 2\)). However, the first lemma of that paper is a test for wildness which was shown to be invalid in 2004, and therefore the statement above remains unproven.
In this work, the author establishes this theorem in the case where \(\Lambda=\mathcal B\) is a block of a cocommutative Hopf algebra \(H\). This is accomplished using support varieties. Recall that, for \(\mathcal G\) a finite group scheme with algebra of measures \(H\), the cohomological support variety for a simple \(\mathcal B\)-module \(S\) is the radical of the kernel of the homomorphism \([f]\mapsto[f\otimes\text{id}_M]\colon H^\bullet(\mathcal G,k)\to\text{Ext}_{\mathcal G}^\bullet(S,S)\). Furthermore, the support variety \(\mathcal V_{\mathcal B}\) for the block \(\mathcal B\) is the union of the support varieties above for a collection \(\{S_i\}\) of simple \(\mathcal B\)-modules which form a complete set of such modules. The main theorem is that if \(\dim\mathcal V _{\mathcal B}\geq 3\) then \(\mathcal B\) is wild. The relationship of complexity to this dimension establishes the desired result.
There is an application of the above theorem to reduced enveloping algebras of finite-dimensional restricted Lie algebras. Given \((\mathfrak g,[p])\) a finite-dimensional restricted Lie algebra and \(\chi\in\mathfrak g^*\) we may form the \(\chi\)-reduced enveloping algebra \(U_\chi(\mathfrak g):=U(\mathfrak g)/I_\chi\), where \(U(\mathfrak g)\) is the universal enveloping algebra of \(\mathfrak g\) and \(I_\chi\) is an ideal generated by \(\{x^p-x^{[p]}-\chi(x)^p1\mid x\in\mathfrak g\}\). After defining the support variety for a block \(\mathcal B\subset U_\chi(\mathfrak g)\) the work above quickly establishes that if \(\mathcal B\) is tame then \(\text{cx}_{U_\chi(\mathfrak g)}(M)\leq 2\).

MSC:

16G60 Representation type (finite, tame, wild, etc.) of associative algebras
14L15 Group schemes
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B50 Modular Lie (super)algebras
20G05 Representation theory for linear algebraic groups

Citations:

Zbl 0742.16007
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Auslander, Representation theory of artin algebras (1995) · Zbl 0834.16001 · doi:10.1017/CBO9780511623608
[2] Benson, Representations and cohomology, I (1991)
[3] Benson, Representations and cohomology, II (1991) · Zbl 0731.20001
[4] Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 pp 104– (1983) · Zbl 0526.20040 · doi:10.1016/0021-8693(83)90121-7
[5] Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 pp 291– (1984) · Zbl 0543.20032 · doi:10.1007/BF01388448
[6] W. Crawley-Boevey On tame algebras and bocses 1988 56 Proc. London Math. Soc. 451 483
[7] Demazure, Groupes algébriques I (1970)
[8] Drozd, Tame and wild matrix problems, in: Representations and quadratic forms pp 39– (1979) · Zbl 0454.16014
[9] Eisenbud, Commutative algebra (1996)
[10] Erdmann, Blocks of tame representation type and related algebras (1990) · Zbl 0696.20001 · doi:10.1007/BFb0084003
[11] Farnsteiner, Periodicity and representation type of modular Lie algebras, J. Reine Angew. Math. 464 pp 47– (1995) · Zbl 0823.17025
[12] Farnsteiner, On the Auslander-Reiten quiver of an infinitesimal group, Nagoya Math. J. 160 pp 103– (2000) · Zbl 0976.16015 · doi:10.1017/S0027763000007704
[13] Farnsteiner, Block representation type of Frobenius kernels of smooth groups, J. Reine Angew. Math. 586 pp 45– (2005) · Zbl 1102.20029 · doi:10.1515/crll.2005.2005.586.45
[14] Farnsteiner, Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks, Invent. Math. 166 pp 27– (2006) · Zbl 1115.20035 · doi:10.1007/s00222-006-0506-z
[15] Farnsteiner, Classification of restricted Lie algebras with tame principal block, J. Reine Angew. Math. 546 pp 1– (2002) · Zbl 0989.17011 · doi:10.1515/crll.2002.043
[16] Farnsteiner, The tame infinitesimal groups of odd characteristic, Adv. in Math. 205 pp 229– (2006) · Zbl 1100.14037 · doi:10.1016/j.aim.2005.07.008
[17] Farnsteiner, On infinitesimal groups of tame representation type, Math. Z. 244 pp 479– (2003) · Zbl 1031.16022 · doi:10.1007/s00209-003-0491-5
[18] Friedlander, Modular representation theory of Lie algebras, Amer. J. Math. 110 pp 1055– (1988) · Zbl 0673.17010 · doi:10.2307/2374686
[19] Friedlander, Representation-theoretic support spaces for finite group schemes, Amer. J. Math. 127 pp 379– (2005) · Zbl 1072.20009 · doi:10.1353/ajm.2005.0010
[20] Friedlander, Cohomology of finite group schemes over a field, Invent. Math. 127 pp 209– (1997) · Zbl 0945.14028 · doi:10.1007/s002220050119
[21] Gordon, Block representation type of reduced enveloping algebras, Trans. Amer. Math. Soc. 354 pp 1549– (2001) · Zbl 1054.20025 · doi:10.1090/S0002-9947-01-02826-4
[22] Heller, Indecomposable representations and the loop space operation, Proc. Amer. Math. Soc. 12 pp 640– (1961) · Zbl 0100.26501 · doi:10.1090/S0002-9939-1961-0126480-2
[23] Jantzen, Representation of algebraic groups, Mathematical Surveys and Monographs 107 (2003) · Zbl 1034.20041
[24] Premet, Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms, Math. Z. 228 pp 255– (1998) · Zbl 0913.17011 · doi:10.1007/PL00004612
[25] Rickard, The representation type of self-injective algebras, Bull. London Math. Soc. 22 pp 540– (1990) · Zbl 0742.16007 · doi:10.1112/blms/22.6.540
[26] Suslin, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 pp 729– (1997) · Zbl 0960.14024 · doi:10.1090/S0894-0347-97-00239-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.