--- date: 2023-01-13 01:48 title: Hochschild Cohomology for Algebras (Witherspoon) aliases: - Hochschild Cohomology for Algebras (Witherspoon) annotation-target: status: ✅ Started page_current: 0 page_total: 100000 flashcard: created: 2023-01-13T01:48 updated: 2024-01-29T17:51 --- Last modified: `=this.file.mday` # Hochschild Cohomology for Algebras (Witherspoon) ## Appendices - [x] What is a dga? ✅ 2023-01-24 - [ ] $C$ a dg $\kmod$ where for $x \in C_i, y\in C_j$, $$d(xy) = d(x)y + (-1)^i x d(y)$$ - [x] What is a dgla? ✅ 2023-01-24 - [ ] $C$ a dg $\liealg$ where for $x\in C_i, y\in C_j$, $$d([xy]) = [d(x) y] + (-1)^i[x d(y)]$$ - [x] What is a counital dg coalgebra? ✅ 2023-01-24 - [ ] $C$ a graded coalgebra, so $\exists \Delta: C\to C\tensor_k C$ coassociative $(\Delta\tensor 1)\Delta = (1\tensor \Delta)\Delta$ with $$(d\tensor 1 + 1\tensor d)\Delta = \Delta d$$ - [ ] Counital if $\exists \eps: C\to k$ with $$(\eps \tensor 1)\Delta = 1 = (1\tensor\eps)\Delta$$ - [x] What is a formula for the pushout of module morphisms $\alpha: Y\to A$ and $\beta: Y\to B$? ✅ 2023-01-24 - [ ] A quotient module: $$A\oplus B\surjects A\glue{Y} B \cong { X\oplus B \over \gens{ (\,-\alpha(y)\,, \beta(y)\, ) \st y\in Y} }$$ - [x] What is a formula for the pullback of module morphisms $\phi: A\to X$ and $\psi: B\to X$? ✅ 2023-01-24 - [ ] A submodule: $$A\fiberprod{X} B = \gens{(a,b) \in A\oplus B \st \phi(a) = \psi(b)} \subseteq A\oplus B$$ - [x] Describe limits and colimits in terms of quotients and submodules. ✅ 2023-01-24 - [ ] Colimits $\sim$ quotients, e.g. pushouts (glue by equivalence) - [ ] Limits $\sim$ submodules, e.g. pullbacks (impose equality) - [x] What is a (left) hereditary ring? Give an example. ✅ 2023-01-24 - [ ] Every (left) ideal is projective as a submodule, or $\mathrm{gldim}_{\mathrm{left}} R \leq 1$. - [ ] $k[x]$ is hereditary, since $\mathrm{gldim}_{\mathrm{left}} \kxn = n$ by the Hilbert syzygy theorem. - [x] What is the left global dimenson of a ring? ✅ 2023-01-24 - [ ] $$\mathrm{gldim}_{\mathrm{left}} R \da \sup_{M\in \rmod} \mathrm{projdim}_R M$$ where the projective dimension is the smallest length of a projective resolution. - [x] What is the variance/contravariance of homs? ✅ 2023-01-24 - [ ] $[P_1\to P_0, N] = [P_0, N] \to [P_1, N]$, so $\Hom(\wait, N)$ is contravariant (like a dual module $V\dual$) - [ ] $[M, I_1\to I_0] = [M, I_0]\to [M, I_1]$, so $\Hom(M, \wait)$ is covariant. - [x] How do you compute $\Ext_R^*(M, N)$? ✅ 2023-01-24 - [ ] $H_*(\Hom_R(P_M, N))$ where $P_M\covers M$ is a projective resolution. - [ ] $H_*(\Hom_R(M, I_N))$ where $N\injects I_N$ is an injective resolution. - [x] How do you compute $\Tor^R_*(M, N)$? ✅ 2023-01-24 - [ ] $H_*(M\tensor_R P_N)$ where $P_N\covers N$ is a projective resolution. - [x] What is the Kunneth theorem? ✅ 2023-01-24 - [ ] Theorem A.5.2 (Künneth Theorem). Let $C$. and $D$. be complexes of right and left $R$-modules, respectively, for which $C_n$ and $d\left(C_n\right)$ are flat $R$-modules for all $n \in \mathbb{Z}$. Then for all $n \in \mathbb{Z}$, there is a short exact sequence: $$\begin{aligned} 0 \longrightarrow \bigoplus_{i+j=n} \mathrm{H}_i(C) \otimes_R \mathrm{H}_j(D) \longrightarrow & \mathrm{H}_n\left(C \otimes_R D\right) \longrightarrow \\ & \bigoplus_{i+j=n-1} \operatorname{Tor}_1^R\left(\mathrm{H}_i(C), \mathrm{H}_j(D)\right) \longrightarrow 0 \end{aligned}$$ - [x] What is the UCT? ✅ 2023-01-24 - [ ] Theorem A.5.4 (Universal Coefficients Theorem). Let $C$ be a complex of right $R$-modules in which all $C_n, d\left(C_n\right)$ are flat, and let $M$ be a left $R$ module. There is a short exact sequence $$0 \longrightarrow \mathrm{H}_n(C) \otimes_R M \longrightarrow \mathrm{H}_n\left(C \otimes_R M\right) \longrightarrow \operatorname{Tor}_1^R\left(\mathrm{H}_{n-1}(C), M\right) \longrightarrow 0 .$$ ## Preface - [ ] What is a Gerstenhaber algebra? - [ ] What is a Koszul algebra? - [ ] What is a quiver? - [ ] What is a skew group algebra? - [ ] What is a twisted tensor product algebra? - [ ] What is the HKR theorem? - [ ] What is Hochschild dimension? - [ ] What is smoothness of an algebra? - [ ] What are noncommutative differential forms? - [ ] What is Van den Bergh duality? - [ ] What are CY algebras? - [ ] What is the Connes differential? - [ ] What does "rigidity" mean for algebras? - [ ] What are the Maurer-Cartan equations? - [ ] What is deformation quantization? - [ ] What is a support variety? - [ ] What is an infinity algebra? - [ ] What is a tensor category? - [ ] What is a quantum group? - [ ] How is topological Hochschild homology defined? - [ ] How is HH used in functional analysis? - [ ] How is HH connected to $K\dash$theory? - [ ] What is the derived Picard group of an algebra? - [ ] What is a dg category? ### Ch. 1 ### 1.1 - [x] What is the enveloping algebra $A^e$ of $A\in \kalg$? ✅ 2023-01-23 - [ ] $A\tensor_k A^{\op}$ where $(a_1\tensor b_1)\cdot (a_2\tensor b_2) = (a_1a_2)\tensor(b_2b_1)$. - [x] How are enveloping algebras related to bimodules? ✅ 2023-01-23 - [ ] $\bimods{A}{A} \iso \mods{A^e}$ where if $M\in \bimods{A}{A}$ is sent to $M$ with $(a\tensor b).m = amb$. - [x] What is the $\mods{A^e}$ structure on $A\tensorpowerk{n}$? ✅ 2023-01-23 - [ ] Act on the first and last components: $$(a \otimes b) \cdot\left(c_1 \otimes c_2 \otimes \cdots \otimes c_{n-1} \otimes c_n\right)=a c_1 \otimes c_2 \otimes \cdots \otimes c_{n-1} \otimes c_n b$$ - [x] How is the bar complex of $A\in \mods{A^e}$ defined? ✅ 2023-01-23 - [ ] $B(A): A^{\tensor \bullet} \surjectsvia{\pi} A$ where $\pi(a\tensor b) = ab$, $d_1(a\tensor b\tensor c) = ab\tensor c - a\tensor bc$ and the general differential is a sum over collapses: $$d_n\left(a_0 \otimes a_1 \otimes \cdots \otimes a_{n+1}\right)=\sum_{i=0}^n(-1)^i a_0 \otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_{n+1}$$ If $A\in \kmod^\free$ then $B(A)$ is an exact free resolution of $A$ with $H^*(B(A)) = A \cdot t^0$. - [x] How are Hochschild chains and homology defined? ✅ 2023-01-23 - [ ] For $M\in \mods{A^e}$, write $$C_*(A; M) = \bigoplus_{n\geq 0} M\tensor_{A^e} B_n(A) \cong \bigoplus_{n\geq 0} M\tensor_k A\tensorpowerk{n}$$ I.e. apply $\wait\tensor_{A^e} M$ to $B(A)$ levelwise, with differential $\id_M \tensor d_n$. - [ ] Then $\HoH(A; M) = H^*(C_*(A; M))$, i.e. $\HoH_n(A; M) = H_n(M\tensor A^{\tensor \bullet})$. - [x] What are cycles? Boundaries? ✅ 2023-01-23 - [ ] Cycles $=\ker d_n$, boundaries $=\im d_{n}$. - [x] How are Hochschild cochains and cohomology defined? ✅ 2023-01-23 - [ ] Apply $\mods{A^e}(\wait, M)$ to $B(A)$, has differentials $d_n^*(f) \da fd_n$ for $f\in \mods{A^e}(A\tensorpowerk{n+1}\to M )$, $\HoH^*(A; M)$ is its homology. - [x] How is $\HoH_*(A)$ defined? ✅ 2023-01-23 - [ ] Regard $A\in\mods{A^e}$ and take $\HoH_*(A; A)$. - [x] Discuss functoriality of $\HoH^*(\wait)$. ✅ 2023-01-23 - [ ] Not a functor! Instead $(A, M)\mapsto \HoH^*(A, M)$ defines a bifunctor equivalent to $\Ext_{A^e}^*(\wait, \wait)$ which is contravariant in $A$ and covariant in $M$. - [x] How are $\HoH_*$ and $\HoH^*$ related to ext and tor? ✅ 2023-01-23 - [ ] $\operatorname{HH}_n(A, M) \cong \operatorname{Tor}_n^{A^e}(M, A) \text { and } \operatorname{HH}^n(A, M) \cong \operatorname{Ext}_{A^e}^n(A, M)$ - [ ] First iso holds when $A\in\kmod^\flat$ and second when $A\in \kmod^{\proj}$. - [x] What is the reduced/normalized bar resolution? ✅ 2023-01-23 - [ ] $\bar B_n(A) = A\tensor (A/k)\tensorpowerk n \tensor A$. - [x] What is $\HoH^*(k[x])$? ✅ 2023-01-23 - [ ] $\HoH^*(k[x]) = k[x] t^0 \oplus k[x] t^1$, using the resolution $0\to R\tensorpowerk 2\mapsvia{\tensor 1 - 1\tensor x} R\tensorpowerk 2 \mapsvia{\pi} R\to 0$ where $R\da k[x]$ and applying $\Hom_{R^e}(\wait, R)$, using $\Hom_{R^e}(R\tensor R, R) \cong R$ in this case by $f\mapsto f(1\tensor 1)$. - [ ] Can write as a ring: $$\HoH^*(k[x]) \cong k[x, t]/\gens{t^2},\qquad \abs{x} = 0, \abs{t} = 1$$ - [x] What is $\HoH(k[x]/x^n)$? ✅ 2023-01-23 - [ ] See example 1.1.21: set $A\da k[x]/x^n$. One gets $\HoH^*(A) \cong \bigoplus_{i\geq 0} A t^i$ if $\characteristic k \mid n$, otherwise $$\HH^*(A) = At^0 + \bigoplus_{m\geq 0} \gens{x} t^{2m + 1} + \bigoplus_{m\geq 0} A/\gens{ x^{n-1} } t^{2m}$$ - [ ] What is Morita equivalence? - [ ] What are relative tor/ext? ### 1.2, 1.3 Interpretation in low degrees - [x] What are the interpretations of $\HoH^n(A)$ for small $n$? ✅ 2023-01-23 - [ ] $n=0$: $Z(A)$ - [ ] $n=1$: $\Out\Der_k(A)$, which equals $\Der_k(A)$ when $A$ is commutative since the only inner derivation is zero. - [ ] $n=2$: Infinitesimal deformations of $A$ - [ ] $n=3$: obstructions to lifting deformations to formal deformations. - [x] What are the interpretations of $\HoH^n(A; M)$ for small $n$? ✅ 2023-01-23 - [ ] $n=0$: $Z_M(A) \da \ts{m\in M\st am=ma \, \forall a\in A}$. - [ ] $n=1$: $\Out\Der_k(A\to M) \da \Der_k(A\to M)/\Inn\Der_k(A\to M)$. - [ ] $n=2$: Infinitesimal deformations of $A$ - [ ] $n=3$: obstructions to lifting deformations to formal deformations. - [x] What is a Hochschild 2-cocycle? 2-coboundary? ✅ 2023-01-23 - [ ] Functions $f\in \Hom_{A^e}(A\tensorpowerk 2 \to M)$ which satisfy $$a f(b \otimes c)+f(a \otimes b c)=f(a b \otimes c)+f(a \otimes b) c$$ - [ ] These define $\algs{A}$ structures on $A\oplus M$, i.e. square-zero extensions. - [ ] Coboundaries are deformations isomorphic to the original algebra. - [x] What are orthogonal central idempotents? ✅ 2023-01-23 - [ ] Each $e_j\in Z(A)$ and $e_j e_\ell = \delta_{j, \ell} e_j$. - [x] What is the cup product on $\HoH^*$? ✅ 2023-01-23 - [ ] $$\begin{align} \smile: \Hom_k(A\tensorpowerk m\to A) \tensor_k \Hom_k(A\tensorpowerk m\to A) &\to \Hom_k(A\tensorpowerk {m+n}\to A) \\ (f \smile g)\left(a_1 \otimes \cdots \otimes a_{m+n}\right)&=(-1)^{m n} f\left(a_1 \otimes \cdots \otimes a_m\right) g\left(a_{m+1} \otimes \cdots \otimes a_{m+n}\right)\end{align}$$ - [ ] This makes $C^*(A, A)$ into a dga. - [x] What is a consequence of $\HoH^1(A; M) = 0$? ✅ 2023-01-23 - [ ] Every derivation $A\to M$ is inner. - [x] How do deformations of $A$ arise? ✅ 2023-01-23 - [ ] From noncommutative Poisson structures: elements $x\in \HoH^2(A)$ with $\ts{a, a} = 0$. ### 1.4 Gerstenhaber Bracket - [x] What is the circle product? ✅ 2023-01-23 - [ ] $$\begin{aligned}& (f \circ g)\left(a_1 \otimes \cdots \otimes a_{m+n-1}\right) \\ & =\sum_{i=1}^m(-1)^u f\left(a_1 \otimes \cdots \otimes a_{i-1} \otimes g\left(a_i \otimes \cdots \otimes a_{i+n-1}\right) \otimes a_{i+n} \otimes \cdots \otimes a_{m+n-1}\right) \end{aligned}$$ - [x] How is the Gerstenhaber bracket defined? ✅ 2023-01-23 - [ ] $$\begin{align} [\wait, \wait]: \Hom(A\tensorpowerk m \to A) \tensor_k \Hom(A\tensorpowerk n \to A) &\to \Hom(A\tensorpowerk{m+n-1}\to A) \\ [f, g]& =f \circ g-(-1)^{(m-1)(n-1)} g \circ f\end{align}$$ - [x] What is a Gerstenhaber algebra? ✅ 2023-01-23 - [ ] Definition 1.4.8. A Gerstenhaber algebra $(H, \smile,[]$,$) is a free \mathbb{Z}$-graded $k$-module $H$ for which $(H, \smile)$ is a graded commutative associative algebra, $(H,[]$,$) is a graded Lie algebra with bracket [$,$] of degree -1$ and corresponding degree shift by $-1$ on elements, and $$[\gamma, \alpha \smile \beta]=[\gamma, \alpha] \smile \beta+(-1)^{|\alpha|(|\gamma|-1)} \alpha \smile[\gamma, \beta]$$ for all homogeneous $\alpha, \beta, \gamma$ in $H$. - [x] What is the cap product? ✅ 2023-01-23 - [ ] $$\begin{align} \frown: \mathrm{HH}_n(A) \otimes \mathrm{HH}^m(A) &\too \mathrm{HH}_{n-m}(A) \\ \left(a_0 \otimes a_1 \otimes \cdots \otimes a_n\right) \frown f&=(-1)^{m(n-m)} a_0 f\left(a_1 \otimes \cdots \otimes a_m\right) \otimes a_{m+1} \otimes \cdots \otimes a_n\end{align}$$ - [x] What is a $(p, q)$ shuffle? ✅ 2023-01-23 - [ ] For nonnegative integers $p$ and $q$, a $(p, q)$-shuffle is an element $\sigma$ of the symmetric group $S_{p+q}$ for which $\sigma(i)<\sigma(j)$ whenever $1 \leq i