Abstract
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The cases of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.
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Anick, D.J., Green, E.L.: On the homology of quotients of path algebras. Comm. Algebra 15(1-2), 309–341 (1987)
Artin, M., Schelter, W.F.: Graded algebras of global dimension 3. Adv. in Math. 66(2), 171–216 (1987)
Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. The Grothendieck Festschrift, vol. I, pp. 33–85, Progr Math., 86, Birkhäuser Boston, Boston, MA (1990)
Artin, M., Tate, J., Van den Bergh, M.: Modules over regular algebras of dimension 3. Invent. Math. 106(2), 335–388 (1991)
Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29(2), 178–218 (1978)
Buchweitz, R.-O., Leuschke, G.J., Van den Bergh, M.: Non-commutative desingularization of determinantal varieties I. Invent. Math. 182(1), 47–115 (2010)
Chouhy, S., Solotar, A.: Projective resolutions of associative algebras and ambiguities. J. Algebra 432, 22–61 (2015)
Eisenbud, D.: Commutative algebra With a view toward algebraic geometry Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York (1995)
Green, E.L.: The geometry of strong Koszul algebras. arXiv:1702.02918
Green, E., Huang, R.Q.: Projective resolutions of straightening closed algebras generated by minors. Adv. Math. 2, 314–333 (1995)
Green, E.L., Happel, D., Zacharia, D.: Projective resolutions over Artin algebras with zero relations. Illinois J. Math. 29(1), 180–90 (1985)
Green, E.L.: Multiplicative bases, Gröbner bases, and right Gröbner bases. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). J. Symbolic Comput. 29(4-5), 601–623 (2000)
Green, E.L.: Noncommutative Gröbner bases, and projective resolutions. Computational methods for representations of groups and algebras (Essen, 1997), 29–60, Progr. Math., 173, Birkhäuser, Basel (1999)
Green, E.L., Schroll, S.: On quasi-hereditary algebras. arXiv:1710.06674
Green, E. L., Solberg, Ø.: An algorithmic approach to resolutions. J. Symbolic Comput. 42(11-12), 1012–1033 (2007)
Iyama, O., Wemyss, M.: Auslander-Reiten duality for non-isolated singularities Maximal modifications. Invent. Math. 197(3), 521–586 (2014)
Kussin, D., Lenzing, H., Meltzer, H.: Nilpotent operators and weighted projective lines. J. Reine Angew. Math. 685, 33–71 (2013)
Kussin, D., Lenzing, H., Meltzer, H.: Triangle singularities ADE-chains, and weighted projective lines. Adv. Math. 237, 194–251 (2013)
Maclagan, D., Sturmfels, B.: Introduction to tropical geometry. Graduate Studies in Mathematics, 161. American Mathematical Society, Providence, RI (2015)
Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Second international colloquium on words, Languages and Combinatorics (Kyoto, 1992). Theoret. Comput. Sci. 134(1), 131–173 (1994)
Spenko, S., Van den Bergh, M.: Non-commutative resolutions of quotient singularities. Invent. Math. 210(1), 3–67 (2017)
van den Bergh, M.: Non-commutative crepant resolutions. The legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004)
Wilson, G.V.: The Cartan map on categories of graded modules. J. Algebra 85 (2), 390–398 (1983)
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Presented by: Henning Krause
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The first and third author were partially supported by an LMS scheme 4 grant. The third author is supported by the EPSRC through the Early Career Fellowship EP/P016294/1.
Appendix: Order resolutions
Appendix: Order resolutions
One of the main tools for proving the statements in Theorem 1.1 is based on the algorithmically constructed projective resolutions in [15], which we recall in this appendix under the name of order resolutions. In [1] and [15] two methods for creating projective resolutions of modules were presented. Although seemingly different, they turn out to be the same resolution. We adopt the approach found in [15] which employs Gröbner bases. We call these resolutions order resolutions since they are dependent on the chosen admissible order on the paths in Q. We note that if Λ is not monomial then the order resolution of a finitely generated Λ-module is not necessarily minimal. We will see however, that for a monomial algebra, the order resolution is the minimal projective resolution.
We fix a quiver Q, an admissible order ≻ on \({\mathcal{B}}\), a tip-reduced subset \(\mathcal {T}\) of \({\mathcal{B}}\), and a vertex v ∈ Q0. Let Λ and \({\Lambda }^{\prime }\) be two algebras in \(\mathcal {V}_{\mathcal {T}}\). The goal of this section is to compare the order resolutions of the simple Λ-module Sv to the order resolution of the simple \({\Lambda }^{\prime }\)-module \(S^{\prime }_{v}\), where Sv and \(S^{\prime }_{v}\) are the one dimensional simple modules at v. For this we compare the order resolutions where \({\Lambda }^{\prime }={\Lambda }^{\prime }_{Mon}={\Lambda }_{Mon}\) with the last equality due to the assumption that both algebras are in \(\mathcal {V}_{\mathcal {T}}\).
We begin by recalling the general framework of the algorithmic construction of order resolutions of modules over Λ = KQ/I. Our specific goal is to show that the Betti numbers of the order resolutions of Sv and \(S_{v}^{\prime }\) are the same.
Let P be a finitely generated projective KQ-module. We fix a direct sum decomposition of \(P = \bigoplus _{i \in \mathcal {I}}v_{i} KQ\), where vi ∈ Q0 not necessarily distinct and \(\mathcal {I}\) is a finite indexing set [12]. We write elements of P as tuples. Let \({\mathcal{B}}^{*}\) be the set
where \(p[i]=(0,\dots ,0,p,0,\dots ,0)\) with p in the i-th component. Note that \({\mathcal{B}}^{*}\) is a K-basis of P. We extend the admissible order to \({\mathcal{B}}^{*}\). For this we order the set \(\mathcal {I}\) and set p[i] ≻ q[j] if p ≻ q or p = q and i > j.
Definition A.1
-
1)
Let \(x = {\sum }_{k} \alpha _{k} b_{k} \in P\) where αk ∈ K and \(b_{k} \in {\mathcal{B}}^{*}\). Set
$$ \text{tip}(x) = b_{m} \text{ where } \alpha_{m} \neq 0 \text{ and if } \alpha_{l} \neq 0\text{ and }l \neq m,\text{ then } b_{m} \succ b_{l}. $$ -
2)
If {0}≠X ⊂ P then
$$ \text{tip} (X) = \{ \text{tip} (x) \mid x \in X \}.$$ -
3)
We call an element x ∈ P right uniform, if there exists a vertex v ∈ Q0 such that xv = x.
-
4)
We say X ⊂ P is tip-reduced if for \(x, x^{\prime } \in X\), \(x \neq x^{\prime }\), and tip(x) = p[i] and \(\text {tip}(x^{\prime })=p^{\prime }[i]\), then p is not a prefix of \(p^{\prime }\).
If an element x ∈ P is right uniform then \(xKQ \simeq \mathfrak {t} (x) KQ\) where \( \mathfrak {t} (x)\) is the vertex v ∈ KQ such that xv = x.
Proposition A.2
[12] Let X be a tip-reduced right uniform generating set of a KQ-submodule L of a projective KQ-module P. Then
Next, we provide the general set-up for the construction of an order resolution as found in [15].
Let M be a KQ/I-module which is finitely presented as a KQ-module and let
be a finite KQ-presentation of M. Then we have the following commutative exact diagram.
The construction of the projective resolution of M is based on the construction of a sequence of sets, which we now introduce. Let \(\mathcal {F}^{0} = \{ v_{i}[i] \in P \mid i \in \mathcal {I} \}\). By [12], there is a right uniform tip-reduced generating set \(\widehat {\mathcal {F}}^{1}\) of L. Define \(\mathcal {F}^{1} = \{ f^{1} \in \widehat {\mathcal {F}}^{1} \mid f^{1} \notin PI \} \). Note that \(\mathcal {F}^{1} \subset \bigoplus _{f^{0} \in \mathcal {F}^{0}} f^{0} KQ\). Next, assume we have constructed \(\mathcal {F}^{n-1}\) and \(\mathcal {F}^{n-2}\). From this data, one constructs the tip-reduced set \(\mathcal {F}^{n}\) such that if \(f^{n}\in \mathcal {F}^{n}\) then fn is right uniform and \(f^{n}\in \bigoplus _{f^{n-1}\in \mathcal {F}^{n-1}} f^{n-1}KQ\).
We note that if \(f\in \mathcal {F}^{n}\) then \(\mathfrak {t}(f)=\mathfrak {t}(\text {tip}(f))\) by uniformity. We briefly recall some facts about the sets \(\mathcal {F}^{n}\). The tip set of \(\mathcal {F}^{n}\) is determined by the tip sets of \(\mathcal {F}^{n-1}\), \(\mathcal {F}^{n-2}\), and the reduced Gröbner basis of I. The point is that if the tip sets of \(\mathcal {F}^{n-1}\) and \(\mathcal {F}^{n-2}\) coincide for the order resolutions of a Λ-module M and \({\Lambda }^{\prime }\)-module M, then the tip set of the set \(\mathcal {F}^{n}\) for the two modules will coincide [15].
The n-th projective \(\mathcal {P}^{n}\) in the KQ/I-order resolution of M is given by
Proposition A.3
Suppose S, respectively \(S^{\prime }\), is a simple one dimensional KQ/I-module, respectively KQ/IMon-module, associated to a vertex v ∈ Q. Let \(\mathcal {P}\) be an order resolution of S with generating sets \(\mathcal {F}^{n}\) and \(\mathcal {P}^{\prime }\) be an order resolution of \(S^{\prime }\) with generating sets \((\mathcal {F}^{\prime })^{n}\). Then for all n, \(\text {tip}(\mathcal {F}^{n}) = \text {tip}((\mathcal {F}^{\prime })^{n})\).
Proof
S and \(S^{\prime }\) have the same KQ-presentation, namely
and
Furthermore, if \(\mathcal {G}\) and \(\mathcal {G}^{\prime }\) are the Gröbner bases for I and IMon respectively then by definition \( \text {tip}(\mathcal {G}) = \text {tip}(\mathcal {G}^{\prime })\). Thus, \(\text {tip}(\mathcal {F}^{0}) = \text {tip}((\mathcal {F}^{\prime })^{0})\), \(\text {tip}(\mathcal {F}^{1}) = \text {tip}((\mathcal {F}^{\prime })^{1})\). By our earlier discussion, we conclude \(\text {tip}(\mathcal {F}^{2}) = \text {tip}((\mathcal {F}^{\prime })^{2})\). Continuing, we see that \(\text {tip}(\mathcal {F}^{n}) = \text {tip}((\mathcal {F}^{\prime })^{n})\), for n ≥ 0. □
Recall that the nth-Betti number of a projective resolution \(\mathcal {P}\) of aΛ-module M is the sequence \((a_{1}, \ldots , a_{\vert Q_{0} \vert })\) where ai is the number of \(f \in \mathcal {F}^{n}\) such that \( \mathfrak {t} (f) = v_{i}\).
The proof of the following Theorem follows directly from Proposition A.3
Theorem A.4
Let S be a simple KQ/I-module and \(S^{\prime }\) a simple KQ/IMon-module, both corresponding to the same vertex v in Q0. Let \(\mathcal {P}\) and \(\mathcal {P}^{\prime }\) be the order resolutions of S and \(S^{\prime }\) respectively. Then the Betti numbers of \(\mathcal {P}\) and \(\mathcal {P}^{\prime }\) coincide.
Corollary A.5
Let KQ/I and \(KQ/I^{\prime }\) be such that \(I_{Mon} = I^{\prime }_{Mon}\). Let S be a simple KQ/I-module and \(S^{\prime }\) a simple \(KQ/I^{\prime }\)-module, both corresponding to the same vertex v in Q0. Let \(\mathcal {P}\) and \(\mathcal {P}^{\prime }\) be the order resolutions of S and \(S^{\prime }\) respectively. Then the Betti numbers of \(\mathcal {P}\) and \(\mathcal {P}^{\prime }\) coincide.
For a monomial algebra KQ/IMon it is easy to see, using [11], that the order resolution of the simple KQ/IMon-modules is minimal in the sense that the image of the differential from the n th-projective module to the n − 1st projective module Pn− 1 is contained in Pn− 1J, where J is the ideal in KQ generated by the arrows of KQ.
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Green, E.L., Hille, L. & Schroll, S. Algebras and Varieties. Algebr Represent Theor 24, 367–388 (2021). https://doi.org/10.1007/s10468-020-09951-3
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DOI: https://doi.org/10.1007/s10468-020-09951-3
Keywords
- Representation theory of associative algebras
- Non-commutative Gröbner bases
- Global dimension
- Cartan conjecture