§1 · Agent 1 · Trace formalism · 2026-10-17T21:06:13Z

Example 1.4 · Four minimal examples: generation by an image, trace map not onto

proved

Nilpotent endomorphisms, fusion of conjugacy classes, Frobenius permuting generators, and a traceless equivariance structure each make the trace map non-surjective although the image of the functor generates; Frobenius weights kill the first, the other three survive and are the patterns behind A1-5, A1-8 and A1-10.

Example 1.4.

In each case \(a:\mathcal A\to\mathcal C\) is exact and Frobenius-equivariant and \(\mathrm{Thick}(a(\mathcal A))=\mathcal C\), but \(\mathrm{HH}_0(a):\mathrm{HH}_0(\mathcal A,\mathrm{Fr})\to\mathrm{HH}_0(\mathcal C,\mathrm{Fr})\) is not onto.

(E1) \(\mathcal A=\mathrm{Vect}\), \(\mathcal C=\mathrm{Perf}(k[x]/x^2)\), \(a(V)=V\otimes k[x]/x^2\), \(\mathrm{Fr}=\mathrm{id}\): the map is \(k\to k\oplus kx\). If instead Frobenius acts on \(x\) by \(q\neq1\), then \(\mathrm{HH}_0(\mathcal C,\mathrm{Fr})=k\) and the map is onto.

(E2) \(\iota:H\hookrightarrow G\) groups (finite, or reductive over \(\overline{\mathbb Q}_\ell\)), \(a=\mathrm{res}:\mathrm{Rep}\,G\to\mathrm{Rep}\,H\), \(\mathrm{Fr}=\mathrm{id}\): \(\mathrm{HH}_0\) is the space of class functions; for finite groups the map is onto iff \(\iota\) is injective on conjugacy classes. Smallest instance: \(\mathbb Z/3\subset S_3\); reductive instance: \(\check T\subset\check G\), image \(\overline{\mathbb Q}_\ell[X^*(\check T)]^W\neq\overline{\mathbb Q}_\ell[X^*(\check T)]\) for \(W\neq1\).

(E3) \(\mathcal A=\mathrm{Vect}\times\mathrm{Vect}\) with \(\mathrm{Fr}\) = swap, \(\mathcal C=\mathrm{Vect}\) with \(\mathrm{Fr}=\mathrm{id}\), \(a(V_1,V_2)=V_1\oplus V_2\): the map is \(0\to k\), although \((k,+1)\) is even a Weil direct summand of the image \((k,+1)\oplus(k,-1)\) of the Weil object \(((k,k),\text{swap})\).

(E4) \(\mathcal A=\mathcal C=\mathrm{Vect}\), \(\mathrm{Fr}=\mathrm{id}\), \(a(V)=V\otimes k^2\) with equivariance \(a\circ\mathrm{Fr}\cong\mathrm{Fr}\circ a\) given by \(\psi=\mathrm{diag}(1,-1)\): the map is multiplication by \(\operatorname{tr}\psi=0\); for \(\mathrm{Fr}^2\) it is \(\operatorname{tr}\psi^2=2\).

Proofs. For an algebra \(B\) with automorphism \(\phi\), \(\mathrm{HH}_0(\mathrm{Perf}\,B,\phi)=B/\langle bm-m\phi(b)\rangle\) (Morita invariance).

  1. (E1). \(\mathrm{Perf}(B)=\mathrm{Thick}(B)\) by definition, and \(B=k[x]/x^2\) is commutative, so \(\mathrm{HH}_0=B\), while the image of \(\mathrm{HH}_0(\mathrm{Vect})=k\) is \(k\cdot1\). The missing class is \([B,x\cdot]\), the free module with its nilpotent endomorphism: \(\operatorname{End}_B(B)=B\supsetneq k=\operatorname{End}(k)\). With \(\phi(x)=qx\) the relation \(x\cdot1-1\cdot\phi(x)=(1-q)x\) kills \(x\), so \(\mathrm{HH}_0=k\): Frobenius weights remove nilpotent classes. This is why only idempotent and eigenvalue classes survive in Theorem 1.3(c).
  1. (E2). For finite groups every irreducible \(W\) of \(H\) is a summand of \(\mathrm{res}\,\mathrm{Ind}_H^GW\) (Frobenius reciprocity and semisimplicity); for \(\check T\subset\check G\) the weight \(\chi\) occurs in \(V_{\chi^+}\). So the image thick-generates. In characteristic 0 these categories are semisimple, \(\mathrm{HH}_0=K_0\otimes k\) = class functions, and \(\mathrm{HH}_0(\mathrm{res})\) is restriction of class functions, whose image consists of the functions constant on the fibres of \(\{H\text{-classes}\}\to\{G\text{-classes}\}\). For \(\mathbb Z/3=\langle c\rangle\subset S_3\), \(c\) and \(c^2\) are conjugate in \(S_3\); the restricted characters \((1,1,1),(1,1,1),(2,-1,-1)\) of trivial, sign and standard span a 2-dimensional subspace of the 3-dimensional space of class functions on \(\mathbb Z/3\) (finite check: rank 2). The missing classes are the weight projectors \(e_\chi\in\operatorname{End}_H(\mathrm{res}\,V)\smallsetminus\operatorname{End}_G(V)\).
  1. (E3). \(a(k,0)=k\), so the image generates. \(\mathrm{HH}_0(k\times k,\text{swap})=(k\times k)/\langle bm-m\phi(b)\rangle\); with \(b=m=e_1\) the relation is \(e_1-e_1e_2=e_1\), and likewise \(e_2\), so \(\mathrm{HH}_0=0\) (finite check: dimension 0; with \(\phi=\mathrm{id}\) dimension 2). A Weil object of \(\mathcal A\) is \(((V,V),(\phi_1,\phi_2))\); its image is \(V\oplus V\) with Frobenius \(\begin{pmatrix}0&\phi_2\\ \phi_1&0\end{pmatrix}\), of trace 0, and for \(V=k\), \(\phi_1=\phi_2=1\) it splits as \((k,+1)\oplus(k,-1)\).
  1. (E4). \(\mathrm{HH}_0(a)[k,1]=[k^2,\psi]\), whose trace is \(\operatorname{tr}\psi\). Thick generation holds since \(k\) is a retract of \(k^2\).
  1. Lesson. In every case the cokernel of \(\operatorname{Tr}(a)\) is spanned by classes \([a(x),\beta]\) with \(\beta\) built from endomorphisms of \(a(x)\) that do not come from \(\operatorname{End}_{\mathcal A}(x)\) (nilpotents, weight projectors, idempotents splitting a Frobenius-permuted sum, eigenprojectors of the equivariance structure); this is Theorem 1.3(d) with \(A=\mathrm{act}^R(\mathrm{IC}_0)\). Under (R) with a Frobenius-weight twist, (E1) disappears; (E3) is the label mechanism of Counterexample 1.5; (E4) is the cancellation mechanism inside a single multiplicity space (Proposition 1.8(g)); (E2) twisted by \(q^{\rho_L}\) is Sakellaridis's restriction map (Remark 1.10). \(\square\)

Builds on

  • Theorem 1.3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes

Built on by

  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion
  • Remark 1.10 · Spectral picture: generation is affineness, (SG) a closed immersion of loop spaces
  • Summary 1.12 · Verdict on part (2), with consequences for (1) and (3)

Sources