Hochschild Complexes¶
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class
sage.homology.hochschild_complex.
HochschildComplex
(A, M)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.category_object.CategoryObject
The Hochschild complex.
Let \(A\) be an algebra over a commutative ring \(R\) such that \(A\) a projective \(R\)-module, and \(M\) an \(A\)-bimodule. The Hochschild complex is the chain complex given by
\[C_n(A, M) := M \otimes A^{\otimes n}\]with the boundary operators given as follows. For fixed \(n\), define the face maps
\[\begin{split}f_{n,i}(m \otimes a_1 \otimes \cdots \otimes a_n) = \begin{cases} m a_1 \otimes \cdots \otimes a_n & \text{if } i = 0, \\ a_n m \otimes a_1 \otimes \cdots \otimes a_{n-1} & \text{if } i = n, \\ m \otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n & \text{otherwise.} \end{cases}\end{split}\]We define the boundary operators as
\[d_n = \sum_{i=0}^n (-1)^i f_{n,i}.\]The Hochschild homology of \(A\) is the homology of this complex. Alternatively, the Hochschild homology can be described by \(HH_n(A, M) = \operatorname{Tor}_n^{A^e}(A, M)\), where \(A^e = A \otimes A^o\) (\(A^o\) is the opposite algebra of \(A\)) is the enveloping algebra of \(A\).
Hochschild cohomology is the homology of the dual complex and can be described by \(HH^n(A, M) = \operatorname{Ext}^n_{A^e}(A, M)\).
Another perspective on Hochschild homology is that \(f_{n,i}\) make the family \(C_n(A, M)\) a simplicial object in the category of \(R\)-modules, and the degeneracy maps are
\[s_i(a_0 \otimes \cdots \otimes a_n) = a_0 \otimes \cdots \otimes a_i \otimes 1 \otimes a_{i+1} \otimes \cdots \otimes a_n\]The Hochschild homology can also be constructed as the homology of this simplicial module.
REFERENCES:
[Redondo] Maria Julia Redondo. Hochschild cohomology: some methods for computations. http://inmabb.criba.edu.ar/gente/mredondo/crasp.pdf -
algebra
()¶ Return the defining algebra of
self
.EXAMPLES:
sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.algebra() Symmetric group algebra of order 3 over Rational Field
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boundary
(d)¶ Return the boundary operator in degree
d
.EXAMPLES:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: d1 = H.boundary(1) sage: z = d1.domain().an_element(); z 2*1 # 1 + 2*1 # x + 3*1 # y sage: d1(z) 0 sage: d1.matrix() [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 2 0 0 -2 0 0 0 0 0 0] sage: s = SymmetricFunctions(QQ).s() sage: H = s.hochschild_complex(s) sage: d1 = H.boundary(1) sage: x = d1.domain().an_element(); x 2*s[] # s[] + 2*s[] # s[1] + 3*s[] # s[2] sage: d1(x) 0 sage: y = tensor([s.an_element(), s.an_element()]) sage: d1(y) 0 sage: z = tensor([s[2,1] + s[3], s.an_element()]) sage: d1(z) 0
TESTS:
sage: def test_complex(H, n): ....: phi = H.boundary(n) ....: psi = H.boundary(n+1) ....: comp = phi * psi ....: zero = H.free_module(n-1).zero() ....: return all(comp(b) == zero for b in H.free_module(n+1).basis()) sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: H = SGA.hochschild_complex(SGA) sage: test_complex(H, 1) True sage: test_complex(H, 2) True sage: test_complex(H, 3) # long time True sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: test_complex(H, 1) True sage: test_complex(H, 2) True sage: test_complex(H, 3) True
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coboundary
(d)¶ Return the coboundary morphism of degree
d
.EXAMPLES:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: del1 = H.coboundary(1) sage: z = del1.domain().an_element(); z 2 + 2*x + 3*y sage: del1(z) 0 sage: del1.matrix() [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 2] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 -2] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0]
TESTS:
sage: def test_complex(H, n): ....: phi = H.coboundary(n) ....: psi = H.coboundary(n+1) ....: comp = psi * phi ....: zero = H.free_module(n+1).zero() ....: return all(comp(b) == zero for b in H.free_module(n-1).basis()) sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: H = SGA.hochschild_complex(SGA) sage: test_complex(H, 1) True sage: test_complex(H, 2) True sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: test_complex(H, 1) True sage: test_complex(H, 2) True sage: test_complex(H, 3) True
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coefficients
()¶ Return the coefficients of
self
.EXAMPLES:
sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.coefficients() Trivial representation of Standard permutations of 3 over Rational Field
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cohomology
(d)¶ Return the
d
-th cohomology group.EXAMPLES:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: H.cohomology(0) Vector space of dimension 3 over Rational Field sage: H.cohomology(1) Vector space of dimension 4 over Rational Field sage: H.cohomology(2) Vector space of dimension 6 over Rational Field sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.cohomology(0) Vector space of dimension 1 over Rational Field sage: H.cohomology(1) Vector space of dimension 0 over Rational Field sage: H.cohomology(2) Vector space of dimension 0 over Rational Field
When working over general rings (except \(\ZZ\)) and we can construct a unitriangular basis for the image quotient, we fallback to a slower implementation using (combinatorial) free modules:
sage: R.<x,y> = QQ[] sage: SGA = SymmetricGroupAlgebra(R, 2) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.cohomology(1) Free module generated by {} over Multivariate Polynomial Ring in x, y over Rational Field
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free_module
(d)¶ Return the free module in degree
d
.EXAMPLES:
sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.free_module(0) Trivial representation of Standard permutations of 3 over Rational Field sage: H.free_module(1) Trivial representation of Standard permutations of 3 over Rational Field # Symmetric group algebra of order 3 over Rational Field sage: H.free_module(2) Trivial representation of Standard permutations of 3 over Rational Field # Symmetric group algebra of order 3 over Rational Field # Symmetric group algebra of order 3 over Rational Field
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homology
(d)¶ Return the
d
-th homology group.EXAMPLES:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: H.homology(0) Vector space of dimension 3 over Rational Field sage: H.homology(1) Vector space of dimension 4 over Rational Field sage: H.homology(2) Vector space of dimension 6 over Rational Field sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.homology(0) Vector space of dimension 1 over Rational Field sage: H.homology(1) Vector space of dimension 0 over Rational Field sage: H.homology(2) Vector space of dimension 0 over Rational Field
When working over general rings (except \(\ZZ\)) and we can construct a unitriangular basis for the image quotient, we fallback to a slower implementation using (combinatorial) free modules:
sage: R.<x,y> = QQ[] sage: SGA = SymmetricGroupAlgebra(R, 2) sage: T = SGA.trivial_representation() sage: H = SGA.hochschild_complex(T) sage: H.homology(1) Free module generated by {} over Multivariate Polynomial Ring in x, y over Rational Field
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trivial_module
()¶ Return the trivial module of
self
.EXAMPLES:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: H = E.hochschild_complex(E) sage: H.trivial_module() Free module generated by {} over Rational Field
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