arXiv subjects
Publications and source records attributed to Frauke M. Bleher.
We give an introduction to the deformation theory of linear representations of profinite groups which Mazur initiated in the 1980's. We then consider the case of representations of finite groups. We show how Brauer's generalized decomposition numbers can be used in some cases to explicitly determine universal deformation rings.
The versal deformation ring R(G,V) of a mod p representation V of a profinite group G encodes all isomorphism classes of lifts of V to representations of G over complete local commutative Noetherian rings. We introduce a new technique for determining R(G,V) when G is finite which involves Brauer's generalized decomposition numbers.
We provide a series of examples of finite groups G and mod p representations V of G whose stable endomorphisms are all given by scalars such that V has a universal deformation ring R(G,V) which is large in the sense that R(G,V)/pR(G,V) is isomorphic to a power series algebra in one variable.
Let k be an algebraically closed field, and let Λ be an algebra of dihedral type of polynomial growth as classified by Erdmann and Skowroński. We describe all finitely generated Λ-modules V whose stable endomorphism rings are isomorphic to k and determine their universal deformation rings R(Λ,V). We prove that only three isomorphism types occur for R(Λ,V): k, k[[t]]/(t^2) and k[[t]].
Let $\mathcal{W}$ be a complete local commutative Noetherian ring with residue field $k$ of positive characteristic $p$. We study the inverse problem for the versal deformation rings $R_{\mathcal{W}}(Γ,V)$ relative to $\mathcal{W}$ of finite dimensional representations $V$ of a profinite group $Γ$ over $k$. We show that for all $p$ and $n \ge 1$, the ring $\mathcal{W}[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse inverse problem' for the ring $\mathcal{W}[[t]]/(p^n t,t^2)$; this is to determine all pairs $(Γ, V)$ such that $R_{\mathcal{W}}(Γ,V)$ is isomorphic to this ring.
Let $k$ be a field, and let $Λ$ be a finite dimensional $k$-algebra. We prove that if $Λ$ is a self-injective algebra, then every finitely generated $Λ$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a universal deformation ring $R(Λ,V)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. If $Λ$ is also a Frobenius algebra, we show that $R(Λ,V)$ is stable under taking syzygies. We investigate a particular Frobenius algebra $Λ_0$ of dihedral type, as introduced by Erdmann, and we determine $R(Λ_0,V)$ for every finitely generated $Λ_0$-module $V$ whose stable endomorphism ring is isomorphic to $k$.
Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG with a dihedral defect group D such that there are precisely two isomorphism classes of simple B-modules. We determine the universal deformation ring R(G,V) for every finitely generated kG-module V which belongs to B and whose stable endomorphism ring is isomorphic to k. The description by Erdmann of the quiver and relations of the basic algebra of B is usually only determined up to a certain parameter c which is either 0 or 1. We show that R(G,V) is isomorphic to a subquotient ring of WD for all V as above if and only if c=0, giving an answer to a question raised by the first author and Chinburg in this case. Moreover, we prove that c=0 if and only if B is Morita equivalent to a principal block.
We consider deformations of bounded complexes of modules for a profinite group G over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex to be represented by a complex of G-modules that is strictly perfect over the associated versal deformation ring.
We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes $Z^\bullet$ of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic $\ell$.
Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ denote a non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalized quaternion. We determine the universal deformation rings $R(S_5,V)$ and $R(\hat{S}_5,V)$ for each mod 2 representation $V$ of $S_5$ that belongs to the principal 2-modular block of $S_5$ and whose stable endomorphism ring is given by scalars when it is inflated to $\hat{S}_5$. We show that for these $V$, a question raised by the first author and Chinburg concerning the relation of the universal deformation ring of $V$ to the Sylow 2-subgroups of $S_5$ and $\hat{S}_5$, respectively, has an affirmative answer.
We study the inverse problem for the versal deformation rings $R(Γ,V)$ of finite dimensional representations $V$ of a finite group $Γ$ over a field $k$ of positive characteristic $p$. This problem is to determine which complete local commutative Noetherian rings with residue field $k$ can arise up to isomorphism as such $R(Γ,V)$. We show that for all integers $n \ge 1$ and all complete local commutative Noetherian rings $\mathcal{W}$ with residue field $k$, the ring $\mathcal{W}[[t]]/(p^n t,t^2)$ arises in this way. This ring is not a local complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain an answer to a question of M. Flach in all characteristics.
We determine the universal deformation ring R(G,V) of certain mod 2 representations V of a finite group G which belong to a 2-modular block of G whose defect groups are isomorphic to a generalized quaternion group D. We show that for these V, a question raised by the author and Chinburg concerning the relation of R(G,V) to D has an affirmative answer. We also show that R(G,V) is a complete intersection even though R(G/N,V) need not be for certain normal subgroups N of G which act trivially on V.
Let $k$ be an algebraically closed field of characteristic 2, and let $G$ be a finite group. Suppose $B$ is a block of $kG$ with dihedral defect groups such that there are precisely two isomorphism classes of simple $B$-modules. The description by Erdmann of the quiver and relations of the basic algebra of $B$ is usually only given up to a certain parameter $c$ which is either 0 or 1. In this article, we show that $c=0$ if there exists a central extension $\hat{G}$ of $G$ by a group of order 2 together with a block $\hat{B}$ of $k\hat{G}$ with generalized quaternion defect groups such that $B$ is contained in the image of $\hat{B}$ under the natural surjection from $k\hat{G}$ onto $kG$. As a special case, we obtain that $c=0$ if $G=\mathrm{PGL}_2(\mathbb{F}_q)$ for some odd prime power $q$ and $B$ is the principal block of $k \mathrm{PGL}_2(\mathbb{F}_q)$.
Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.
Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2-modular block of a finite simple group. We determine the universal deformation ring R(G,V) for every kG-module V which belongs to B and has stable endomorphism ring k. It follows that R(G,V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.
Suppose $G$ is a finite group acting on a projective scheme $X$ over a commutative Noetherian ring $R$. We study the $RG$-modules $\HH^0(X,\mathcal{F} \otimes \mathcal{L}^n)$ when $n \ge 0$, and $\mathcal{F}$ and $\mathcal{L}$ are coherent $G$-sheaves on $X$ such that $\mathcal{L}$ is an ample line bundle. We show that the classes of these modules in the Grothendieck group $G_0(RG)$ of all finitely generated $RG$-modules lie in a finitely generated subgroup. Under various hypotheses, we show that there is a finite set of indecomposable $RG$-modules such that each $\HH^0(X,\mathcal{F} \otimes \mathcal{L}^n)$ is a direct sum of these indecomposables, with multiplicites given by generalized Hilbert polynomials for $n >> 0$.