Hochschild Complexes¶

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

$f_{n,i}(m \otimes a_1 \otimes \cdots \otimes a_n) = \begin{cases} m a_1 \otimes \cdots \otimes a_n & \text{if } i = 0, \\ a_n m \otimes a_1 \otimes \cdots \otimes a_{n-1} & \text{if } i = n, \\ m \otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n & \text{otherwise.} \end{cases}$

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

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

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

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

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

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

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

Hochschild Complexes¶

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