Hochschild Complexes

class sage.homology.hochschild_complex.HochschildComplex(A, M)

Bases: UniqueRepresentation, Parent

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:

• Wikipedia article Hochschild_homology

• https://ncatlab.org/nlab/show/Hochschild+cohomology

• [Red2001]

class Element(parent, vectors)

Bases: ModuleElement

A chain of the Hochschild complex.

INPUT:

Can be one of the following:

• A dictionary whose keys are the degree and whose \(d\)-th value is an element in the degree \(d\) module.

• An element in the coefficient module \(M\).

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
sage: H(T.an_element())
Chain(0: 2*B['v'])
sage: H({0: T.an_element()})
Chain(0: 2*B['v'])
sage: H({1: H.module(1).an_element()})
Chain(1: 2*B['v'] # [1, 2, 3] + 2*B['v'] # [1, 3, 2] + 3*B['v'] # [2, 1, 3])
sage: H({0: H.module(0).an_element(), 3: H.module(3).an_element()})
Chain with 2 nonzero terms over Rational Field

sage: F.<x,y> = FreeAlgebra(ZZ)
sage: H = F.hochschild_complex(F)
sage: H(x + 2*y^2)
Chain(0: F[x] + 2*F[y^2])
sage: H({0: x*y - x})
Chain(0: -F[x] + F[x*y])
sage: H(2)
Chain(0: 2*F[1])
sage: H({0: x-y, 2: H.module(2).basis().an_element()})
Chain with 2 nonzero terms over Integer Ring

vector(degree)

Return the free module element in degree.

EXAMPLES:

sage: F.<x,y> = FreeAlgebra(ZZ)
sage: H = F.hochschild_complex(F)
sage: a = H({0: x-y, 2: H.module(2).basis().an_element()})
sage: [a.vector(i) for i in range(3)]
[F[x] - F[y], 0, F[1] # F[1] # F[1]]

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

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

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]

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

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

differential(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

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

module(d)

Return the module in degree d.

EXAMPLES:

sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
sage: H.module(0)
Trivial representation of Standard permutations of 3 over Rational Field
sage: H.module(1)
Trivial representation of Standard permutations of 3 over Rational Field
 # Symmetric group algebra of order 3 over Rational Field
sage: H.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

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