3 · Hochschild (co)homology¶

For a finite-dimensional algebra $A$, Hochschild cohomology $HH^\bullet(A)$ and homology $HH_\bullet(A)$ are computed here from the normalized bar complex. The bar complex is exponentially large — the degree-$n$ cochains of a $d$-dimensional algebra number $d(d-1)^n$ — so it is an honest oracle for small algebras only. But every entry it returns is an exact rank: no rounding, and the answer is characteristic-correct.

The entry points are A.hochschild_cohomology(top) and A.hochschild_homology(top), each returning an HHTable.

In [1]:

from quiverlab import CC, GF, truncated_polynomial, linear_path_algebra

from quiverlab import CC, GF, truncated_polynomial, linear_path_algebra

$k[x]/(x^n)$: the characteristic matters¶

Take $A = k[x]/(x^3)$. Over a field whose characteristic does not divide $n=3$, the Hochschild cohomology settles into a steady pattern.

In [2]:

cc = truncated_polynomial(3, field=CC).hochschild_cohomology(4)
print(cc)

cc = truncated_polynomial(3, field=CC).hochschild_cohomology(4) print(cc)

Worked steps: quiverlab_traces/HHc_49b354051845.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH^n dimensions for Algebra of dimension 3 over CC (computing exactly in QQ) (engine: normalized bar complex)
HH^0 = 3  HH^1 = 2  HH^2 = 2  HH^3 = 2  HH^4 = 2
refs: bar

Reading it: $HH^0(A)=Z(A)=A$ is the whole $3$-dimensional (commutative) algebra. For $i\ge 1$ the dimensions stabilize at $2$. This is the generic picture in characteristic $0$ (and in any characteristic prime to $n$).

Now change one word — the field — to $\mathbb{F}_3$, where $3\mid n$:

In [3]:

gf3 = truncated_polynomial(3, field=GF(3)).hochschild_cohomology(4)
print(gf3)

gf3 = truncated_polynomial(3, field=GF(3)).hochschild_cohomology(4) print(gf3)

Worked steps: quiverlab_traces/HHc_9b4510b323bf.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH^n dimensions for Algebra of dimension 3 over GF(3) (engine: hanlab engine (F_p fast rank))
HH^0 = 3  HH^1 = 3  HH^2 = 3  HH^3 = 3  HH^4 = 3
refs: bar

The $\operatorname{char}\mid n$ jump¶

Over $\mathbb{F}_3$ the dimensions are $[3,3,3,3,3]$ instead of $[3,2,2,2,\dots]$: every $HH^i$ with $i\ge1$ gains a dimension. The mechanism is classical — the connecting maps in the resolution of $k[x]/(x^n)$ carry factors of $n$ (think $x\mapsto n x^{n-1}$), and when $\operatorname{char}k\mid n$ those maps vanish, so nothing cancels. Same algebra, same presentation; only the field changed. This is why quiverlab treats the ground field as data, not decoration.

For contrast, $\mathbb{F}_2$ (where $2\nmid 3$) agrees with the characteristic-$0$ answer:

In [4]:

gf2 = truncated_polynomial(3, field=GF(2)).hochschild_cohomology(4)
print("HH^* over GF(2):", gf2.dims, " (2 does not divide 3 -> same as CC)")

gf2 = truncated_polynomial(3, field=GF(2)).hochschild_cohomology(4) print("HH^* over GF(2):", gf2.dims, " (2 does not divide 3 -> same as CC)")

Worked steps: quiverlab_traces/HHc_9ecf90dd9bff.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH^* over GF(2): [3, 2, 2, 2, 2]  (2 does not divide 3 -> same as CC)

Hereditary algebras: cohomology in low degree only¶

If $A$ is hereditary (global dimension $\le 1$) — for instance the path algebra of an acyclic quiver — then $HH^i(A)=0$ for all $i\ge 2$. Take the linear quiver $1\to2\to3$:

In [5]:

her = linear_path_algebra(3, field=CC).hochschild_cohomology(2)
print(her)

her = linear_path_algebra(3, field=CC).hochschild_cohomology(2) print(her)

Worked steps: quiverlab_traces/HHc_cbd9387a0be3.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH^n dimensions for Algebra of dimension 6 over CC (computing exactly in QQ) (engine: normalized bar complex)
HH^0 = 1  HH^1 = 0  HH^2 = 0
refs: bar

$HH^0=k$ because the algebra is connected (its center is just the scalars), $HH^1=0$ because the quiver is a tree (no oriented cycles, hence no outer derivations), and everything from degree $2$ up vanishes with the global dimension. Compare the truncated polynomial algebra above, whose cohomology never stops — it has infinite global dimension.

Homology vs. cohomology; symmetric algebras¶

$k[x]/(x^n)$ is a commutative Frobenius, hence symmetric, algebra. For a symmetric algebra there is a duality making $HH_i(A)\cong HH^i(A)$. We can watch the two sequences coincide:

In [6]:

A = truncated_polynomial(3, field=CC)
co = A.hochschild_cohomology(4)
ho = A.hochschild_homology(4)
print("HH^* :", co.dims)
print("HH_* :", ho.dims)
print("same dimensions in every degree?", co.dims == ho.dims)

A = truncated_polynomial(3, field=CC) co = A.hochschild_cohomology(4) ho = A.hochschild_homology(4) print("HH^* :", co.dims) print("HH_* :", ho.dims) print("same dimensions in every degree?", co.dims == ho.dims)

Worked steps: quiverlab_traces/HHc_49b354051845.html (self-contained HTML, no JavaScript; pairs with the JSON record)
Worked steps: quiverlab_traces/HHh_559dfd31a332.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH^* : [3, 2, 2, 2, 2]
HH_* : [3, 2, 2, 2, 2]
same dimensions in every degree? True

The two sequences agree degree by degree — the symmetric-algebra duality, made visible. It is not an artifact of characteristic $0$; over $\mathbb{F}_3$ homology and cohomology both become $[3,3,3,3,3]$, still equal:

In [7]:

B = truncated_polynomial(3, field=GF(3))
print("HH^* over GF(3):", B.hochschild_cohomology(4).dims)
print("HH_* over GF(3):", B.hochschild_homology(4).dims)

B = truncated_polynomial(3, field=GF(3)) print("HH^* over GF(3):", B.hochschild_cohomology(4).dims) print("HH_* over GF(3):", B.hochschild_homology(4).dims)

Worked steps: quiverlab_traces/HHc_9b4510b323bf.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH^* over GF(3): [3, 3, 3, 3, 3]
Worked steps: quiverlab_traces/HHh_d3e22318e283.html (self-contained HTML, no JavaScript; pairs with the JSON record)
HH_* over GF(3): [3, 3, 3, 3, 3]

Reading an HHTable¶

The returned object is a thin, honest record: a list of dimensions plus enough metadata to know what you are looking at. It indexes and iterates by degree, and its repr names the engine that produced it.

In [8]:

t = truncated_polynomial(3, field=CC).hochschild_cohomology(4)
print("kind       :", t.kind)          # "HH^" for cohomology, "HH_" for homology
print("top degree :", t.top)
print("dims       :", t.dims)
print("t[2]       :", t[2])            # HH^2
print("list(t)    :", list(t))
print()
print(repr(t))

t = truncated_polynomial(3, field=CC).hochschild_cohomology(4) print("kind :", t.kind) # "HH^" for cohomology, "HH_" for homology print("top degree :", t.top) print("dims :", t.dims) print("t[2] :", t[2]) # HH^2 print("list(t) :", list(t)) print() print(repr(t))

Worked steps: quiverlab_traces/HHc_49b354051845.html (self-contained HTML, no JavaScript; pairs with the JSON record)
kind       : HH^
top degree : 4
dims       : [3, 2, 2, 2, 2]
t[2]       : 2
list(t)    : [3, 2, 2, 2, 2]

HH^n dimensions for Algebra of dimension 3 over CC (computing exactly in QQ) (engine: normalized bar complex)
HH^0 = 3  HH^1 = 2  HH^2 = 2  HH^3 = 2  HH^4 = 2
refs: bar

Honesty about scope¶

The bar oracle is exponential: a max_cells guard raises DepthLimitError rather than quietly grinding through an intractable matrix, so you always know when you have left the certified range. Deeper, polynomial resolutions (Bardzell, minimal, Chouhy–Solotar) are future work. Within range, though, every number on these tables is an exact rank over an exact field — reproducible, and honest about the characteristic it was computed in.