Isogenies, power operations, and homotopy theory Charles Rezk - PowerPoint PPT Presentation

Properties of Λ-rings, 2 ψ n ( L ) = L ⊗ n in K ( X ) Line bundle L → X = ⇒ Multiplicative group scheme � � Z [ T , T − 1 ] G m = Spec � � ≈ Spec K (pt � U (1)) Adams operation = ⇒ isogeny of G m : � � ψ n [ n ] K (pt � U (1)) − → K (pt � U (1)) ⇐ ⇒ G m − → G m Isogeny: finite flat homomorphism of group schemes Remarks. � G m = Spf K ( BU (1)), multiplicative formal group These properties useful in classical applications (e.g., Adams work on vector fields on spheres, image of J , . . . ) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 6 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Other examples of power operations h ∗ ( − ) = generalized cohomology theory, commutative ring valued Would like to have h ∗ ( X ) P n refines of n th power x �→ x n → h ∗ Σ n ( X ) = h ∗ ( X × B Σ n ) − Do these exist? Yes if h ∗ ( − ) represented by a structured commutative ring spectrum (= commutative S -algebra = E ∞ -ring spectrum = . . . ) Examples. (Steenrod, 1953) reduced power operations in H ∗ ( − , F p ) ( Sq i for p = 2, P i for p odd) (Voevodsky, 2001) motivic reduced power operations (Quillen, 1971) power operations in bordism theories based on M �→ M × n � Σ n used to prove π ∗ MU classifies formal group laws Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 7 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

Elliptic cohomology What is elliptic cohomology? Theory K -theory elliptic cohomology Group scheme G m elliptic curve Cycles vector bundles ??? ??? = 2-dim conformal field theories? (Segal, . . . ) Examples: (Goerss-Hopkins-Miller) tmf = “topological modular forms” associated to universal elliptic curve over M Ell structured comm ring spectrum = ⇒ power operations! (Lurie) Equivariant elliptic cohomology theories Open question: Which equivariant elliptic cohomology theories admit power operations? Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 8 / 29

A nice example: Elliptic cohomology at the Tate curve ] = “ C × / q Z ”, defined over Spec Z [ Tate curve T [ [ q ] [ q ] ]. Equivariant elliptic cohomology at Tate curve � � L ghost ( X � G ) � U (1) Ell Tate ( X � G ) approx K := “ghost loops” = contstant loops; RHS is K of “twisted sectors” (see e.g., Ruan 2000, Lupercio-Uribe 2002) (Ganter, 2007, 2013) Power operations for Ell Tate Ell Tate ( X � G ) is an elliptic Λ -ring : two families of operations λ n : Ell Tate → Ell Tate , µ m : Ell Tate → Ell Tate ⊗ Z [ [ q 1 / m ] ] Z [ ] [ q ] { λ n } are Λ-ring structure, { µ m } are Λ-ring homomorphisms Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 9 / 29

A nice example: Elliptic cohomology at the Tate curve ] = “ C × / q Z ”, defined over Spec Z [ Tate curve T [ [ q ] [ q ] ]. Equivariant elliptic cohomology at Tate curve � � L ghost ( X � G ) � U (1) Ell Tate ( X � G ) approx K := “ghost loops” = contstant loops; RHS is K of “twisted sectors” (see e.g., Ruan 2000, Lupercio-Uribe 2002) (Ganter, 2007, 2013) Power operations for Ell Tate Ell Tate ( X � G ) is an elliptic Λ -ring : two families of operations λ n : Ell Tate → Ell Tate , µ m : Ell Tate → Ell Tate ⊗ Z [ [ q 1 / m ] ] Z [ ] [ q ] { λ n } are Λ-ring structure, { µ m } are Λ-ring homomorphisms Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 9 / 29

A nice example: Elliptic cohomology at the Tate curve ] = “ C × / q Z ”, defined over Spec Z [ Tate curve T [ [ q ] [ q ] ]. Equivariant elliptic cohomology at Tate curve � � L ghost ( X � G ) � U (1) Ell Tate ( X � G ) approx K := “ghost loops” = contstant loops; RHS is K of “twisted sectors” (see e.g., Ruan 2000, Lupercio-Uribe 2002) (Ganter, 2007, 2013) Power operations for Ell Tate Ell Tate ( X � G ) is an elliptic Λ -ring : two families of operations λ n : Ell Tate → Ell Tate , µ m : Ell Tate → Ell Tate ⊗ Z [ [ q 1 / m ] ] Z [ ] [ q ] { λ n } are Λ-ring structure, { µ m } are Λ-ring homomorphisms Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 9 / 29

A nice example: Elliptic cohomology at the Tate curve ] = “ C × / q Z ”, defined over Spec Z [ Tate curve T [ [ q ] [ q ] ]. Equivariant elliptic cohomology at Tate curve � � L ghost ( X � G ) � U (1) Ell Tate ( X � G ) approx K := “ghost loops” = contstant loops; RHS is K of “twisted sectors” (see e.g., Ruan 2000, Lupercio-Uribe 2002) (Ganter, 2007, 2013) Power operations for Ell Tate Ell Tate ( X � G ) is an elliptic Λ -ring : two families of operations λ n : Ell Tate → Ell Tate , µ m : Ell Tate → Ell Tate ⊗ Z [ [ q 1 / m ] ] Z [ ] [ q ] { λ n } are Λ-ring structure, { µ m } are Λ-ring homomorphisms Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 9 / 29

Morava E -theory: introduction Morava E-theories are “designer cohomology theories” — manufactured using homotopy theory, not coming from “nature” some arise as completions of “natural” theories, e.g. K ∧ Ell ∧ p , s.-s. point have rich theory of power operations (Ando, Hopkins, Strickland, R.) Goal: describe what we know about this theory (a lot) Recall: Power operations for K -theory are “controlled” by isogenies of G m Slogan Power operations for Morava E -theories are “controlled” by “deformations” of Frobenius isogenies of 1-dimensional formal groups Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 10 / 29

Morava E -theory: introduction Morava E-theories are “designer cohomology theories” — manufactured using homotopy theory, not coming from “nature” some arise as completions of “natural” theories, e.g. K ∧ Ell ∧ p , s.-s. point have rich theory of power operations (Ando, Hopkins, Strickland, R.) Goal: describe what we know about this theory (a lot) Recall: Power operations for K -theory are “controlled” by isogenies of G m Slogan Power operations for Morava E -theories are “controlled” by “deformations” of Frobenius isogenies of 1-dimensional formal groups Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 10 / 29

Morava E -theory: introduction Morava E-theories are “designer cohomology theories” — manufactured using homotopy theory, not coming from “nature” some arise as completions of “natural” theories, e.g. K ∧ Ell ∧ p , s.-s. point have rich theory of power operations (Ando, Hopkins, Strickland, R.) Goal: describe what we know about this theory (a lot) Recall: Power operations for K -theory are “controlled” by isogenies of G m Slogan Power operations for Morava E -theories are “controlled” by “deformations” of Frobenius isogenies of 1-dimensional formal groups Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 10 / 29

Morava E -theory: introduction Morava E-theories are “designer cohomology theories” — manufactured using homotopy theory, not coming from “nature” some arise as completions of “natural” theories, e.g. K ∧ Ell ∧ p , s.-s. point have rich theory of power operations (Ando, Hopkins, Strickland, R.) Goal: describe what we know about this theory (a lot) Recall: Power operations for K -theory are “controlled” by isogenies of G m Slogan Power operations for Morava E -theories are “controlled” by “deformations” of Frobenius isogenies of 1-dimensional formal groups Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 10 / 29

Morava E -theory: introduction Morava E-theories are “designer cohomology theories” — manufactured using homotopy theory, not coming from “nature” some arise as completions of “natural” theories, e.g. K ∧ Ell ∧ p , s.-s. point have rich theory of power operations (Ando, Hopkins, Strickland, R.) Goal: describe what we know about this theory (a lot) Recall: Power operations for K -theory are “controlled” by isogenies of G m Slogan Power operations for Morava E -theories are “controlled” by “deformations” of Frobenius isogenies of 1-dimensional formal groups Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 10 / 29

Morava E -theory: introduction Morava E-theories are “designer cohomology theories” — manufactured using homotopy theory, not coming from “nature” some arise as completions of “natural” theories, e.g. K ∧ Ell ∧ p , s.-s. point have rich theory of power operations (Ando, Hopkins, Strickland, R.) Goal: describe what we know about this theory (a lot) Recall: Power operations for K -theory are “controlled” by isogenies of G m Slogan Power operations for Morava E -theories are “controlled” by “deformations” of Frobenius isogenies of 1-dimensional formal groups Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 10 / 29

Morava E -theory: summary Let G 0 / F p = one dimensional commutative formal group of height n ∈ { 1 , 2 , . . . } . (Morava, 1978; Goerss-Hopkins-Miller 1993–2004) There exists a cohomology theory E G 0 ( Morava E-theory ) which is represented by a structured commutative ring spectrum is complex orientable; formal group Spf ( E 0 CP ∞ ) = universal deformation of G 0 (in sense of Lubin-Tate) E 0 G 0 (pt) = Z p [ [ a 1 , . . . , a n − 1 ] ] G 0 (pt) = E 0 G 0 (pt)[ u , u − 1 ], u ∈ E 2 E ∗ G 0 (pt) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 11 / 29

Morava E -theory: summary Let G 0 / F p = one dimensional commutative formal group of height n ∈ { 1 , 2 , . . . } . (Morava, 1978; Goerss-Hopkins-Miller 1993–2004) There exists a cohomology theory E G 0 ( Morava E-theory ) which is represented by a structured commutative ring spectrum is complex orientable; formal group Spf ( E 0 CP ∞ ) = universal deformation of G 0 (in sense of Lubin-Tate) E 0 G 0 (pt) = Z p [ [ a 1 , . . . , a n − 1 ] ] G 0 (pt) = E 0 G 0 (pt)[ u , u − 1 ], u ∈ E 2 E ∗ G 0 (pt) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 11 / 29

Morava E -theory: summary Let G 0 / F p = one dimensional commutative formal group of height n ∈ { 1 , 2 , . . . } . (Morava, 1978; Goerss-Hopkins-Miller 1993–2004) There exists a cohomology theory E G 0 ( Morava E-theory ) which is represented by a structured commutative ring spectrum is complex orientable; formal group Spf ( E 0 CP ∞ ) = universal deformation of G 0 (in sense of Lubin-Tate) E 0 G 0 (pt) = Z p [ [ a 1 , . . . , a n − 1 ] ] G 0 (pt) = E 0 G 0 (pt)[ u , u − 1 ], u ∈ E 2 E ∗ G 0 (pt) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 11 / 29

Morava E -theory: summary Let G 0 / F p = one dimensional commutative formal group of height n ∈ { 1 , 2 , . . . } . (Morava, 1978; Goerss-Hopkins-Miller 1993–2004) There exists a cohomology theory E G 0 ( Morava E-theory ) which is represented by a structured commutative ring spectrum is complex orientable; formal group Spf ( E 0 CP ∞ ) = universal deformation of G 0 (in sense of Lubin-Tate) E 0 G 0 (pt) = Z p [ [ a 1 , . . . , a n − 1 ] ] G 0 (pt) = E 0 G 0 (pt)[ u , u − 1 ], u ∈ E 2 E ∗ G 0 (pt) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 11 / 29

Morava E -theory: summary Let G 0 / F p = one dimensional commutative formal group of height n ∈ { 1 , 2 , . . . } . (Morava, 1978; Goerss-Hopkins-Miller 1993–2004) There exists a cohomology theory E G 0 ( Morava E-theory ) which is represented by a structured commutative ring spectrum is complex orientable; formal group Spf ( E 0 CP ∞ ) = universal deformation of G 0 (in sense of Lubin-Tate) E 0 G 0 (pt) = Z p [ [ a 1 , . . . , a n − 1 ] ] G 0 (pt) = E 0 G 0 (pt)[ u , u − 1 ], u ∈ E 2 E ∗ G 0 (pt) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 11 / 29

Formal groups and complex oriented theories Formal group is object locally described by a formal group law. Formal group law (commutative, 1-dimensional) S ( x , y ) ∈ R [ [ x , y ] ] satisfying axioms for abelian group: S ( x , 0) = x = S (0 , x ) , S ( x , y ) = S ( y , x ) , S ( S ( x , y ) , z ) = S ( x , S ( y , z )) . For future reference, we note the p-series of G 0 : [ p ]( x ) = S ( x , S ( x , . . . S ( x , x ))) � �� � x appears p times Complex oriented cohomology theory Ring-valued cohomology theory E such that E ∗ ( CP ∞ ) = E ∗ [ [ x ] ], and x restricts to fundamental class of CP 1 = S 2 . Examples: H ∗ ( − , Z ), K -theory, Ell , Morava E -theories ,. . . Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 12 / 29

Formal groups and complex oriented theories Formal group is object locally described by a formal group law. Formal group law (commutative, 1-dimensional) S ( x , y ) ∈ R [ [ x , y ] ] satisfying axioms for abelian group: S ( x , 0) = x = S (0 , x ) , S ( x , y ) = S ( y , x ) , S ( S ( x , y ) , z ) = S ( x , S ( y , z )) . For future reference, we note the p-series of G 0 : [ p ]( x ) = S ( x , S ( x , . . . S ( x , x ))) � �� � x appears p times Complex oriented cohomology theory Ring-valued cohomology theory E such that E ∗ ( CP ∞ ) = E ∗ [ [ x ] ], and x restricts to fundamental class of CP 1 = S 2 . Examples: H ∗ ( − , Z ), K -theory, Ell , Morava E -theories ,. . . Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 12 / 29

Formal groups and complex oriented theories Formal group is object locally described by a formal group law. Formal group law (commutative, 1-dimensional) S ( x , y ) ∈ R [ [ x , y ] ] satisfying axioms for abelian group: S ( x , 0) = x = S (0 , x ) , S ( x , y ) = S ( y , x ) , S ( S ( x , y ) , z ) = S ( x , S ( y , z )) . For future reference, we note the p-series of G 0 : [ p ]( x ) = S ( x , S ( x , . . . S ( x , x ))) � �� � x appears p times Complex oriented cohomology theory Ring-valued cohomology theory E such that E ∗ ( CP ∞ ) = E ∗ [ [ x ] ], and x restricts to fundamental class of CP 1 = S 2 . Examples: H ∗ ( − , Z ), K -theory, Ell , Morava E -theories ,. . . Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 12 / 29

Deformations of formal groups G 0 / F p formal group of height n (i.e., [ p ] G 0 ( x ) = c x p n + O ( x p n +1 ), c � = 0) R = complete local ring, F p ⊂ R / m Groupoid Def 0 G 0 ( R ) of deformations of G 0 / F p to R Deformation ( G , α ): G is a formal group over R , ∼ iso α : G 0 − → G R / m of formal groups over F p Isomorphism ( G , α ) → ( G ′ , α ′ ) of deformations: iso f : G → G ′ compatible with id of G 0 Classified up to canonical iso by Lubin and Tate: (Lubin-Tate, 1966) ∃ universal deformation ( G univ , α univ ) over A ≈ Z p [ [ a 1 , . . . , a n − 1 ] ] G univ is the formal group of Morava E -theory E G 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 13 / 29

Deformations of formal groups G 0 / F p formal group of height n (i.e., [ p ] G 0 ( x ) = c x p n + O ( x p n +1 ), c � = 0) R = complete local ring, F p ⊂ R / m Groupoid Def 0 G 0 ( R ) of deformations of G 0 / F p to R Deformation ( G , α ): G is a formal group over R , ∼ iso α : G 0 − → G R / m of formal groups over F p Isomorphism ( G , α ) → ( G ′ , α ′ ) of deformations: iso f : G → G ′ compatible with id of G 0 Classified up to canonical iso by Lubin and Tate: (Lubin-Tate, 1966) ∃ universal deformation ( G univ , α univ ) over A ≈ Z p [ [ a 1 , . . . , a n − 1 ] ] G univ is the formal group of Morava E -theory E G 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 13 / 29

Deformations of formal groups G 0 / F p formal group of height n (i.e., [ p ] G 0 ( x ) = c x p n + O ( x p n +1 ), c � = 0) R = complete local ring, F p ⊂ R / m Groupoid Def 0 G 0 ( R ) of deformations of G 0 / F p to R Deformation ( G , α ): G is a formal group over R , ∼ iso α : G 0 − → G R / m of formal groups over F p Isomorphism ( G , α ) → ( G ′ , α ′ ) of deformations: iso f : G → G ′ compatible with id of G 0 Classified up to canonical iso by Lubin and Tate: (Lubin-Tate, 1966) ∃ universal deformation ( G univ , α univ ) over A ≈ Z p [ [ a 1 , . . . , a n − 1 ] ] G univ is the formal group of Morava E -theory E G 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 13 / 29

Deformations of formal groups G 0 / F p formal group of height n (i.e., [ p ] G 0 ( x ) = c x p n + O ( x p n +1 ), c � = 0) R = complete local ring, F p ⊂ R / m Groupoid Def 0 G 0 ( R ) of deformations of G 0 / F p to R Deformation ( G , α ): G is a formal group over R , ∼ iso α : G 0 − → G R / m of formal groups over F p Isomorphism ( G , α ) → ( G ′ , α ′ ) of deformations: iso f : G → G ′ compatible with id of G 0 Classified up to canonical iso by Lubin and Tate: (Lubin-Tate, 1966) ∃ universal deformation ( G univ , α univ ) over A ≈ Z p [ [ a 1 , . . . , a n − 1 ] ] G univ is the formal group of Morava E -theory E G 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 13 / 29

Deformations of formal groups G 0 / F p formal group of height n (i.e., [ p ] G 0 ( x ) = c x p n + O ( x p n +1 ), c � = 0) R = complete local ring, F p ⊂ R / m Groupoid Def 0 G 0 ( R ) of deformations of G 0 / F p to R Deformation ( G , α ): G is a formal group over R , ∼ iso α : G 0 − → G R / m of formal groups over F p Isomorphism ( G , α ) → ( G ′ , α ′ ) of deformations: iso f : G → G ′ compatible with id of G 0 Classified up to canonical iso by Lubin and Tate: (Lubin-Tate, 1966) ∃ universal deformation ( G univ , α univ ) over A ≈ Z p [ [ a 1 , . . . , a n − 1 ] ] G univ is the formal group of Morava E -theory E G 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 13 / 29

Isogenies Isogeny of formal groups over R Homomorphism f : G → G ′ given locally over R by f ( x ) = cx n + higher degree terms, c ∈ R × . ( n = deg f ) G 0 / F p has a distinguished family of Frobenius isogenies Frob r : G 0 → G 0 , r ≥ 0 , given locally by Frob r ( x ) = x p r . Category Def G 0 ( R ) of deformations of Frobenius Objects: (= objects of Def 0 deformations ( G , α ) to R G 0 ( R )) Morphisms ( G , α ) → ( G ′ , α ′ ): isogenies f : G → G ′ compatible with Frob r : G 0 → G 0 , some r ≥ 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 14 / 29

Isogenies Isogeny of formal groups over R Homomorphism f : G → G ′ given locally over R by f ( x ) = cx n + higher degree terms, c ∈ R × . ( n = deg f ) G 0 / F p has a distinguished family of Frobenius isogenies Frob r : G 0 → G 0 , r ≥ 0 , given locally by Frob r ( x ) = x p r . Category Def G 0 ( R ) of deformations of Frobenius Objects: (= objects of Def 0 deformations ( G , α ) to R G 0 ( R )) Morphisms ( G , α ) → ( G ′ , α ′ ): isogenies f : G → G ′ compatible with Frob r : G 0 → G 0 , some r ≥ 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 14 / 29

Isogenies Isogeny of formal groups over R Homomorphism f : G → G ′ given locally over R by f ( x ) = cx n + higher degree terms, c ∈ R × . ( n = deg f ) G 0 / F p has a distinguished family of Frobenius isogenies Frob r : G 0 → G 0 , r ≥ 0 , given locally by Frob r ( x ) = x p r . Category Def G 0 ( R ) of deformations of Frobenius Objects: (= objects of Def 0 deformations ( G , α ) to R G 0 ( R )) Morphisms ( G , α ) → ( G ′ , α ′ ): isogenies f : G → G ′ compatible with Frob r : G 0 → G 0 , some r ≥ 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 14 / 29

Isogenies Isogeny of formal groups over R Homomorphism f : G → G ′ given locally over R by f ( x ) = cx n + higher degree terms, c ∈ R × . ( n = deg f ) G 0 / F p has a distinguished family of Frobenius isogenies Frob r : G 0 → G 0 , r ≥ 0 , given locally by Frob r ( x ) = x p r . Category Def G 0 ( R ) of deformations of Frobenius Objects: (= objects of Def 0 deformations ( G , α ) to R G 0 ( R )) Morphisms ( G , α ) → ( G ′ , α ′ ): isogenies f : G → G ′ compatible with Frob r : G 0 → G 0 , some r ≥ 0 Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 14 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

The “pile” Def = Def G 0 We have assignments complete local ring R � = ⇒ category Def ( R ) local homomorphism R → R ′ functor Def ( R ) → Def ( R ′ ) � = ⇒ If Def ( R ) were a groupoid, we would call it a (pre-)stack Def is the “pile” of deformations of powers of Frob Sheaves on Def A sheaf of modules on Def is a collection of functors � � A R : Def ( R ) → R -modules with compatibility wrt base change along local homomorphisms R → R ′ Likewise, a sheaf of commutative rings on Def is . . . Notation: Mod ( Def ), Com ( Def ). Mod ( Def ) = Mod (Γ) for a certain ring Γ Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 15 / 29

Morava E -theory takes values in sheaves on Def (Ando-Hopkins-Strickland 2004; see R. 2009) Power operations make Morava E -cohomology E G 0 a functor E ∗ ( − ): Spaces → Com ∗ ( Def ) Key step (Strickland 1997, 1998) : E 0 B Σ p r / I classifies subgroups of rank p r of deformations Broader context: We have E ∗ ( X ) = π ∗ ( E X + ) where A = E X + is (i) a structured commutative E -algebra spectrum, (ii) K ( n )-local ( ⇔ π ∗ A complete wrt ( a 1 , . . . , a n − 1 ) in a suitable sense) The real theorem is (ibid) π ∗ lifts to a functor π ∗ : h Com ( E ) K ( n ) → Com ∗ ( Def ) on homotopy category of K ( n )-local commutative E -algebra spectra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 16 / 29

Morava E -theory takes values in sheaves on Def (Ando-Hopkins-Strickland 2004; see R. 2009) Power operations make Morava E -cohomology E G 0 a functor E ∗ ( − ): Spaces → Com ∗ ( Def ) Key step (Strickland 1997, 1998) : E 0 B Σ p r / I classifies subgroups of rank p r of deformations Broader context: We have E ∗ ( X ) = π ∗ ( E X + ) where A = E X + is (i) a structured commutative E -algebra spectrum, (ii) K ( n )-local ( ⇔ π ∗ A complete wrt ( a 1 , . . . , a n − 1 ) in a suitable sense) The real theorem is (ibid) π ∗ lifts to a functor π ∗ : h Com ( E ) K ( n ) → Com ∗ ( Def ) on homotopy category of K ( n )-local commutative E -algebra spectra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 16 / 29

Morava E -theory takes values in sheaves on Def (Ando-Hopkins-Strickland 2004; see R. 2009) Power operations make Morava E -cohomology E G 0 a functor E ∗ ( − ): Spaces → Com ∗ ( Def ) Key step (Strickland 1997, 1998) : E 0 B Σ p r / I classifies subgroups of rank p r of deformations Broader context: We have E ∗ ( X ) = π ∗ ( E X + ) where A = E X + is (i) a structured commutative E -algebra spectrum, (ii) K ( n )-local ( ⇔ π ∗ A complete wrt ( a 1 , . . . , a n − 1 ) in a suitable sense) The real theorem is (ibid) π ∗ lifts to a functor π ∗ : h Com ( E ) K ( n ) → Com ∗ ( Def ) on homotopy category of K ( n )-local commutative E -algebra spectra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 16 / 29

Morava E -theory takes values in sheaves on Def (Ando-Hopkins-Strickland 2004; see R. 2009) Power operations make Morava E -cohomology E G 0 a functor E ∗ ( − ): Spaces → Com ∗ ( Def ) Key step (Strickland 1997, 1998) : E 0 B Σ p r / I classifies subgroups of rank p r of deformations Broader context: We have E ∗ ( X ) = π ∗ ( E X + ) where A = E X + is (i) a structured commutative E -algebra spectrum, (ii) K ( n )-local ( ⇔ π ∗ A complete wrt ( a 1 , . . . , a n − 1 ) in a suitable sense) The real theorem is (ibid) π ∗ lifts to a functor π ∗ : h Com ( E ) K ( n ) → Com ∗ ( Def ) on homotopy category of K ( n )-local commutative E -algebra spectra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 16 / 29

Examples Mod ( Def ) = modules for a certain ring Γ Height 1 G 0 = multiplicative formal group; E G 0 = K ∧ p Γ = Z p [ ψ p ] gen. by Adams operation ψ p Height 2 (R., arXiv:0812.1320) G 0 / F 2 = completion of s.-s. elliptic curve y 2 + y = x 3 over F 2   a 2 Q 0 − 2 a Q 1 + 6 Q 2 Q 0 a = � Q 1 a = 3 Q 0 + a Q 2   Q 2 a = − a Q 0 + 3 Q 1   Γ = Z 2 [ [ a ] ] � Q 0 , Q 1 , Q 2 �   Q 1 Q 0 = 2 Q 2 Q 1 − 2 Q 0 Q 2 = Q 0 Q 1 + a Q 0 Q 2 − 2 Q 1 Q 2 Q 2 Q 0 (Y. Zhu, 2014) gives similar description at height 2, p = 3 There is a uniform description of Γ / p at height 2, all primes p Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 17 / 29

Examples Mod ( Def ) = modules for a certain ring Γ Height 1 G 0 = multiplicative formal group; E G 0 = K ∧ p Γ = Z p [ ψ p ] gen. by Adams operation ψ p Height 2 (R., arXiv:0812.1320) G 0 / F 2 = completion of s.-s. elliptic curve y 2 + y = x 3 over F 2   a 2 Q 0 − 2 a Q 1 + 6 Q 2 Q 0 a = � Q 1 a = 3 Q 0 + a Q 2   Q 2 a = − a Q 0 + 3 Q 1   Γ = Z 2 [ [ a ] ] � Q 0 , Q 1 , Q 2 �   Q 1 Q 0 = 2 Q 2 Q 1 − 2 Q 0 Q 2 = Q 0 Q 1 + a Q 0 Q 2 − 2 Q 1 Q 2 Q 2 Q 0 (Y. Zhu, 2014) gives similar description at height 2, p = 3 There is a uniform description of Γ / p at height 2, all primes p Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 17 / 29

Examples Mod ( Def ) = modules for a certain ring Γ Height 1 G 0 = multiplicative formal group; E G 0 = K ∧ p Γ = Z p [ ψ p ] gen. by Adams operation ψ p Height 2 (R., arXiv:0812.1320) G 0 / F 2 = completion of s.-s. elliptic curve y 2 + y = x 3 over F 2   a 2 Q 0 − 2 a Q 1 + 6 Q 2 Q 0 a = � Q 1 a = 3 Q 0 + a Q 2   Q 2 a = − a Q 0 + 3 Q 1   Γ = Z 2 [ [ a ] ] � Q 0 , Q 1 , Q 2 �   Q 1 Q 0 = 2 Q 2 Q 1 − 2 Q 0 Q 2 = Q 0 Q 1 + a Q 0 Q 2 − 2 Q 1 Q 2 Q 2 Q 0 (Y. Zhu, 2014) gives similar description at height 2, p = 3 There is a uniform description of Γ / p at height 2, all primes p Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 17 / 29

Examples Mod ( Def ) = modules for a certain ring Γ Height 1 G 0 = multiplicative formal group; E G 0 = K ∧ p Γ = Z p [ ψ p ] gen. by Adams operation ψ p Height 2 (R., arXiv:0812.1320) G 0 / F 2 = completion of s.-s. elliptic curve y 2 + y = x 3 over F 2   a 2 Q 0 − 2 a Q 1 + 6 Q 2 Q 0 a = � Q 1 a = 3 Q 0 + a Q 2   Q 2 a = − a Q 0 + 3 Q 1   Γ = Z 2 [ [ a ] ] � Q 0 , Q 1 , Q 2 �   Q 1 Q 0 = 2 Q 2 Q 1 − 2 Q 0 Q 2 = Q 0 Q 1 + a Q 0 Q 2 − 2 Q 1 Q 2 Q 2 Q 0 (Y. Zhu, 2014) gives similar description at height 2, p = 3 There is a uniform description of Γ / p at height 2, all primes p Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 17 / 29

Examples Mod ( Def ) = modules for a certain ring Γ Height 1 G 0 = multiplicative formal group; E G 0 = K ∧ p Γ = Z p [ ψ p ] gen. by Adams operation ψ p Height 2 (R., arXiv:0812.1320) G 0 / F 2 = completion of s.-s. elliptic curve y 2 + y = x 3 over F 2   a 2 Q 0 − 2 a Q 1 + 6 Q 2 Q 0 a = � Q 1 a = 3 Q 0 + a Q 2   Q 2 a = − a Q 0 + 3 Q 1   Γ = Z 2 [ [ a ] ] � Q 0 , Q 1 , Q 2 �   Q 1 Q 0 = 2 Q 2 Q 1 − 2 Q 0 Q 2 = Q 0 Q 1 + a Q 0 Q 2 − 2 Q 1 Q 2 Q 2 Q 0 (Y. Zhu, 2014) gives similar description at height 2, p = 3 There is a uniform description of Γ / p at height 2, all primes p Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 17 / 29

Examples Mod ( Def ) = modules for a certain ring Γ Height 1 G 0 = multiplicative formal group; E G 0 = K ∧ p Γ = Z p [ ψ p ] gen. by Adams operation ψ p Height 2 (R., arXiv:0812.1320) G 0 / F 2 = completion of s.-s. elliptic curve y 2 + y = x 3 over F 2   a 2 Q 0 − 2 a Q 1 + 6 Q 2 Q 0 a = � Q 1 a = 3 Q 0 + a Q 2   Q 2 a = − a Q 0 + 3 Q 1   Γ = Z 2 [ [ a ] ] � Q 0 , Q 1 , Q 2 �   Q 1 Q 0 = 2 Q 2 Q 1 − 2 Q 0 Q 2 = Q 0 Q 1 + a Q 0 Q 2 − 2 Q 1 Q 2 Q 2 Q 0 (Y. Zhu, 2014) gives similar description at height 2, p = 3 There is a uniform description of Γ / p at height 2, all primes p Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 17 / 29

Properties of Γ n = height of G 0 / F p (Ando 1995) Center (Γ) = Z p [ ˜ T 1 , . . . , ˜ T n ], ( Hecke algebra ) (R. arXiv:1204.4831) Γ is quadratic , i.e., Γ ≈ Tensor alg.( C 1 ) / (ideal gen. by C 2 ) where C 1 and C 2 ⊆ C 1 ⊗ E 0 C 1 are E 0 = Z p [ [ a 1 , . . . , a n − 1 ] ] bimodules (ibid) Γ is Koszul : have Γ-bimodule resolution 0 ← Γ ← Γ ⊗ E 0 C 0 ⊗ E 0 Γ ← · · · ← Γ ⊗ E 0 C n ⊗ E 0 Γ ← 0 , each C k is E 0 -bimod, free and f.g. as right E 0 -mod; C 0 = E 0 = ⇒ gl . dim(Γ) = 2 n Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 18 / 29

Properties of Γ n = height of G 0 / F p (Ando 1995) Center (Γ) = Z p [ ˜ T 1 , . . . , ˜ T n ], ( Hecke algebra ) (R. arXiv:1204.4831) Γ is quadratic , i.e., Γ ≈ Tensor alg.( C 1 ) / (ideal gen. by C 2 ) where C 1 and C 2 ⊆ C 1 ⊗ E 0 C 1 are E 0 = Z p [ [ a 1 , . . . , a n − 1 ] ] bimodules (ibid) Γ is Koszul : have Γ-bimodule resolution 0 ← Γ ← Γ ⊗ E 0 C 0 ⊗ E 0 Γ ← · · · ← Γ ⊗ E 0 C n ⊗ E 0 Γ ← 0 , each C k is E 0 -bimod, free and f.g. as right E 0 -mod; C 0 = E 0 = ⇒ gl . dim(Γ) = 2 n Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 18 / 29

Properties of Γ n = height of G 0 / F p (Ando 1995) Center (Γ) = Z p [ ˜ T 1 , . . . , ˜ T n ], ( Hecke algebra ) (R. arXiv:1204.4831) Γ is quadratic , i.e., Γ ≈ Tensor alg.( C 1 ) / (ideal gen. by C 2 ) where C 1 and C 2 ⊆ C 1 ⊗ E 0 C 1 are E 0 = Z p [ [ a 1 , . . . , a n − 1 ] ] bimodules (ibid) Γ is Koszul : have Γ-bimodule resolution 0 ← Γ ← Γ ⊗ E 0 C 0 ⊗ E 0 Γ ← · · · ← Γ ⊗ E 0 C n ⊗ E 0 Γ ← 0 , each C k is E 0 -bimod, free and f.g. as right E 0 -mod; C 0 = E 0 = ⇒ gl . dim(Γ) = 2 n Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 18 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Remark about “Koszul” property (R. ibid) Γ is Koszul This was conjectured by Ando-Hopkins-Strickland It is purely a theorem about formal algebraic geometry Only general proof is a purely “topological” proof, using ingredients: (1) Γ = “primitives” of the Hopf algebra � m ≥ 0 E 0 ( B Σ m ) (Strickland) (2) bar complex of Γ in degree k is “primitives” in � m 1 ,..., m k E 0 B (Σ m 1 ≀ · · · ≀ Σ m k ) (3) vanishing results for Bredon homology of partition complexes with coeff. in appropriate Mackey functors (Arone-Dwyer-Lesh 2013) Proof inspired by role of partition complexes as “derivatives of identity functor” in Goodwillie’s functor calculus (R. 2012) purely alg. geom. proof in height 2 case, using results on moduli of subgroups of elliptic curves Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 19 / 29

Congruence criterion Homotopy groups of K ( n )-local E -algebras have some more structure: � � π ∗ : h Com ( E ) K ( n ) → T -algebras “ T -algebras” = a complicated algebraic catgeory (like Λ-rings) (R. 2009) R p -torsion free commutative E ∗ -algebra: � A ∈ Com ∗ ( Def ) with A ( G univ ) = R � � � T -algebra ↔ structures on R satisfying “Frobenius congruence” Frobenius congruence: Qx ≡ x p mod pR for a certain Q ∈ Γ There is a (non-additive) witness to the Frobenius congruence: Qx = x p + p θ ( x ) θ : R → R satisfying where R is a T -algebra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 20 / 29

Congruence criterion Homotopy groups of K ( n )-local E -algebras have some more structure: � � π ∗ : h Com ( E ) K ( n ) → T -algebras “ T -algebras” = a complicated algebraic catgeory (like Λ-rings) (R. 2009) R p -torsion free commutative E ∗ -algebra: � A ∈ Com ∗ ( Def ) with A ( G univ ) = R � � � T -algebra ↔ structures on R satisfying “Frobenius congruence” Frobenius congruence: Qx ≡ x p mod pR for a certain Q ∈ Γ There is a (non-additive) witness to the Frobenius congruence: Qx = x p + p θ ( x ) θ : R → R satisfying where R is a T -algebra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 20 / 29

Congruence criterion Homotopy groups of K ( n )-local E -algebras have some more structure: � � π ∗ : h Com ( E ) K ( n ) → T -algebras “ T -algebras” = a complicated algebraic catgeory (like Λ-rings) (R. 2009) R p -torsion free commutative E ∗ -algebra: � A ∈ Com ∗ ( Def ) with A ( G univ ) = R � � � T -algebra ↔ structures on R satisfying “Frobenius congruence” Frobenius congruence: Qx ≡ x p mod pR for a certain Q ∈ Γ There is a (non-additive) witness to the Frobenius congruence: Qx = x p + p θ ( x ) θ : R → R satisfying where R is a T -algebra Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 20 / 29

Application 1: nilpotence Easy consequence of existence of “witness” θ such that Q ( x ) = x p + p θ ( x ), Q ( x + y ) = Q ( x ) + Q ( y ): If A ∈ Com ( E ) K ( n ) , then x ( p +1) r = 0 . x ∈ π ∗ A , p r x = 0 = ⇒ Idea: deduce relation θ ( px ) = p p − 1 x − Q ( x ) = ( p p − 1 − 1) x p − p θ ( x ). If px = 0, then 0 = x θ ( px ) = − x p +1 . Mathew-Noel-Naumann observe this, and use it (with Nilpotence Theorem of Devinatz-Hopkins-Smith) to give an easy proof of a conjecture of May: (Mathhew-Noel-Naumann 2014) If R = structured commutative ring spectrum, then the kernel of the Hurewicz map π ∗ R → H ∗ ( R , Z ) consists of nilpotent elements Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 21 / 29

Application 1: nilpotence Easy consequence of existence of “witness” θ such that Q ( x ) = x p + p θ ( x ), Q ( x + y ) = Q ( x ) + Q ( y ): If A ∈ Com ( E ) K ( n ) , then x ( p +1) r = 0 . x ∈ π ∗ A , p r x = 0 = ⇒ Idea: deduce relation θ ( px ) = p p − 1 x − Q ( x ) = ( p p − 1 − 1) x p − p θ ( x ). If px = 0, then 0 = x θ ( px ) = − x p +1 . Mathew-Noel-Naumann observe this, and use it (with Nilpotence Theorem of Devinatz-Hopkins-Smith) to give an easy proof of a conjecture of May: (Mathhew-Noel-Naumann 2014) If R = structured commutative ring spectrum, then the kernel of the Hurewicz map π ∗ R → H ∗ ( R , Z ) consists of nilpotent elements Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 21 / 29

Application 1: nilpotence Easy consequence of existence of “witness” θ such that Q ( x ) = x p + p θ ( x ), Q ( x + y ) = Q ( x ) + Q ( y ): If A ∈ Com ( E ) K ( n ) , then x ( p +1) r = 0 . x ∈ π ∗ A , p r x = 0 = ⇒ Idea: deduce relation θ ( px ) = p p − 1 x − Q ( x ) = ( p p − 1 − 1) x p − p θ ( x ). If px = 0, then 0 = x θ ( px ) = − x p +1 . Mathew-Noel-Naumann observe this, and use it (with Nilpotence Theorem of Devinatz-Hopkins-Smith) to give an easy proof of a conjecture of May: (Mathhew-Noel-Naumann 2014) If R = structured commutative ring spectrum, then the kernel of the Hurewicz map π ∗ R → H ∗ ( R , Z ) consists of nilpotent elements Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 21 / 29

Application 2: units and orientations A = structured commutative ring = ⇒ units spectrum gl 1 A ( gl 1 A ) 0 ( X ) = ( A 0 ( X )) × Question. Does there exist structured commutative ring map MG → A , where MG = spectrum representing bordism ( G ∈ { U , SU , O , SO , Spin , . . . } )? Answer (May-Quinn-Ray-Tornehave 1977). Yes iff the composite g → o J − → gl 1 S → gl 1 A is null-homotopic as map of spectra, where g = infinite delooping of G (Ando-Hopkins-R.; see Hopkins 2002) There is a map of structured commutative ring spectra M String → tmf which realizes the “Witten genus”; String = six-connected cover of Spin Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 22 / 29

Application 2: units and orientations A = structured commutative ring = ⇒ units spectrum gl 1 A ( gl 1 A ) 0 ( X ) = ( A 0 ( X )) × Question. Does there exist structured commutative ring map MG → A , where MG = spectrum representing bordism ( G ∈ { U , SU , O , SO , Spin , . . . } )? Answer (May-Quinn-Ray-Tornehave 1977). Yes iff the composite g → o J − → gl 1 S → gl 1 A is null-homotopic as map of spectra, where g = infinite delooping of G (Ando-Hopkins-R.; see Hopkins 2002) There is a map of structured commutative ring spectra M String → tmf which realizes the “Witten genus”; String = six-connected cover of Spin Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 22 / 29

Application 2: units and orientations A = structured commutative ring = ⇒ units spectrum gl 1 A ( gl 1 A ) 0 ( X ) = ( A 0 ( X )) × Question. Does there exist structured commutative ring map MG → A , where MG = spectrum representing bordism ( G ∈ { U , SU , O , SO , Spin , . . . } )? Answer (May-Quinn-Ray-Tornehave 1977). Yes iff the composite g → o J − → gl 1 S → gl 1 A is null-homotopic as map of spectra, where g = infinite delooping of G (Ando-Hopkins-R.; see Hopkins 2002) There is a map of structured commutative ring spectra M String → tmf which realizes the “Witten genus”; String = six-connected cover of Spin Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 22 / 29

Application 2: units and orientations A = structured commutative ring = ⇒ units spectrum gl 1 A ( gl 1 A ) 0 ( X ) = ( A 0 ( X )) × Question. Does there exist structured commutative ring map MG → A , where MG = spectrum representing bordism ( G ∈ { U , SU , O , SO , Spin , . . . } )? Answer (May-Quinn-Ray-Tornehave 1977). Yes iff the composite g → o J − → gl 1 S → gl 1 A is null-homotopic as map of spectra, where g = infinite delooping of G (Ando-Hopkins-R.; see Hopkins 2002) There is a map of structured commutative ring spectra M String → tmf which realizes the “Witten genus”; String = six-connected cover of Spin Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 22 / 29

Logarithmic operations Logarithmic operation: spectrum map ℓ : gl 1 A → A (tom Dieck 1989) p ( X ) × → K ∧ A = K ∧ p : exists ℓ : gl 1 K ∧ p → K ∧ p , giving ℓ : K ∧ p ( X ) by ℓ ( x ) = log( x ) − 1 p log( ψ p ( x )) log = Taylor exp. at 1 � � ψ p ( x ) ≡ x p mod p = 1 x p /ψ p ( x ) p log = � m ≥ 1 ( − 1) m p m − 1 ψ p ( x ) = x p + p θ p ( x ) ( θ p ( x ) / x ) m m (R. 2006) E = E G 0 , height G 0 = n; exists ℓ : gl 1 E → E giving E 0 ( X ) × → E 0 ( X ) by n � ( − 1) k p ( k 2 ) − k log ˜ ℓ ( x ) = T k ( x ) k =0 where ˜ T k ∈ Z p [ ˜ T 1 , . . . , ˜ T n ] = Center (Γ) Charles Rezk (UIUC) Power operations Seoul, August 18, 2014 23 / 29