All Questions
Tagged with deformation-theory homological-algebra
22 questions
11
votes
1
answer
1k
views
Motivation behind the definition of hochschild cohomology
For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
- 595
9
votes
1
answer
457
views
Deformations of Ext rings
Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
- 4,133
7
votes
1
answer
1k
views
Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)
Let $X_0$ be a locally noetherian scheme and $\mathcal{F}_0$ a coherent $\mathcal{O}_{X_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $...
- 7,789
7
votes
1
answer
614
views
Are exterior algebras intrinsically formal as associative dg algebras?
(Cross-posted from mathematics stackexchange.)
Fix a finite dimensional vector space $V$ over a field of characteristic zero, and let $R=Sym(V[1])$ be the free graded commutative algebra generated by ...
- 73
7
votes
1
answer
493
views
Pro-representability of deformation functor associated to a DG Lie algebra
Edit : There are several satisfying proofs in the case each $L^i$ is finite-dimensional. It is proven (for example, Hinich DG coalgebras as formal stacks) that for $A$ : local Artin ring then $\...
7
votes
1
answer
611
views
Extension of the formality theorem?
The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise ...
- 3,758
7
votes
0
answers
220
views
Cohomology of little disks and dg algebras over $\mathbb{F}_p$
This a alternative form of the question I posted some time ago.
We, the people who don't know topology, are told that in characteristic p the formalizm of DG algebras is not quite adequate for ...
6
votes
0
answers
138
views
Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?
Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{\...
- 9,019
5
votes
1
answer
192
views
Functoriality of the formality quasi-isomorphism of E-polydifferential operators
Given a smooth manifold $M$ and a Lie algebroid $E\rightarrow M$ we can consider the $E$-polydifferential operators $D_E$ and the $E$-polyvectorfields
$T_E$ as
$$D_E:=\bigoplus_{k=-1}^\infty\mathcal{...
- 397
4
votes
1
answer
423
views
Formality of the little $n$-disks operad and deformation theory
In [Another proof of M. Kontsevich formality theorem], Tamarkin provides a proof of the formality of the differential graded Lie algebra controlling the deformation of a polynomial associative algebra....
4
votes
1
answer
291
views
Intrinsic formality versus rigidity of a differential graded Lie algebra
Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential.
Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal ...
4
votes
1
answer
392
views
$Ext$-algebra of stable vector bundles
Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.
Question: What can we say about the algebra structure of ...
- 175
4
votes
1
answer
381
views
Classical deformation of algebras
Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$.
...
- 1,211
4
votes
0
answers
863
views
Hochschild cohomology and bar resolutions
I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here.
Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ ...
- 595
3
votes
1
answer
365
views
Brace algebra structure on the Hochschild complex of an associative algebra
As shown by Gerstenhaber and Voronov [Higher operations on the Hochschild complex], the Hochschild complex of an associative algebra is endowed with a natural structure of brace algebra. The first ...
2
votes
1
answer
203
views
Deformations of a complex trivial up to quasi-isomorphism
Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some ...
- 1,545
2
votes
0
answers
84
views
Infinity-morphisms for operadic algebras
Is there an already studied notion of $\infty$-morphism between algebras over a quasi-free operad $P = (T(E), \partial)$?
If the operad $P$ is Koszul, or of the form $\Omega C$ for $C$ a cooperad, ...
- 215
2
votes
0
answers
303
views
L-infinity algebra of deformations of an L-infinity algebra?
From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
- 784
2
votes
0
answers
223
views
Lie bialgebras cohomology
I am wondering if there exist a universal construction of (co)-chain complex associated to a given algebra to study deformations as Hochschild homology for associative algebras, or Chevalley-Eilenberg ...
- 442
1
vote
1
answer
293
views
Homological dimension of pure coherent sheaves and specialization
Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...
- 2,323
1
vote
0
answers
78
views
Compute action of the gauge group in deformation theory of an algebra
I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6.
Consider a vector space $A$ with a multiplication $m$ that makes it into ...
- 1,509
0
votes
0
answers
41
views
Is any deformation of an acyclic complex gauge equivalent to a trivial one?
This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...
- 9,019