Symmetric algebra of a cochain complex: cheat-sheet Gabriele Vezzosi July 2019 Abstract These are notes for students. In the cheat-sheet tradition, the focus will be on giving correct and general enough definitions and statements, rather than on proofs, which are most of the times straightforward or in any case elementary; they are thus mostly left to the student. This should hopefully insert a grain of pedagogical virtue in the weird act of a teacher handling cheat-sheets to his own students. Contents 1 Overview 1 2 Adjunctions 2 3 Bigradings and formulae 3 4 Graded derivations 5 5 Symmetric algebra of a dg-module 5 6 Examples 7 7 Comparison with Σ-coinvariants 10 8 Another approach 11 9 Graded modules over a graded-commutative algebra 12 10 dg-modules over a cdga 13 1 Overview We work over an arbitrary commutative (ungraded) ring R. In order to compute the symmetric algebra of an R-dg-module (E•; d), we proceed in two steps: first we forget about the differential d and compute the symmetric algebra of the graded R-module E•, then we use a further property of this symmetric algebra in • order to deduce from d a graded differential @d on the symmetric algebra of E . The pair consisting of the • symmetric algebra of the graded R-module E and @d will satisfy the universal property of the symmetric algebra of the dg-module (E•; d). We also state the straightforward generalizations to the case of dg-modules over a commutative differential graded R-algebra (= R-cdga). 1 2 Adjunctions Let (C; ⊗;') be a symmetric monoidal R-linear category (' denoting the symmetry morphism) having arbitrary coproducts, and with ⊗ distributing along coproducts. Definition 2.1 We define the symmetric monoidal category (grC; ⊗gr; gr') as follows: • Y grC := C = Fun(Zdiscr; C) Z • i gr j p q (X )i2Z ⊗ (Y )j2Z := (⊕p+q=n(X ⊗ Y ))n2Z • i gr j j gr i gr'X;Y :(X )i2Z ⊗ (Y )j2Z −! (Y )j2Z ⊗ (X )i2Z is the map in grC whose n-th component (gr'X;Y )n is determined by the maps (gr'X;Y )n;(p;q), for (p+q) = n, defined by the compositions ' p q p q X ;Y q p can q0 p0 (gr'X;Y )n;(p;q) : X ⊗ Y / Y ⊗ X / ⊕p0+q0=n(Y ⊗ X ) : • The coproduct functor ⊕i2Z : grC −! C will sometimes be referred to as the underlying C-object functor. i i+r • For any r 2 Z, there is a r-shift functor (−)(r): grC −! grC sending (X )i2Z to (X )i2Z and f : i i r (X )i2Z ! (Y )i2Z to (−1) f (in the unique obvious sense). • For r 2 Z, X;Y 2 grC, a map of weight r in grC from X to Y is a morphism X ! Y (r) in grC. Remark 2.2 Our notion of graded object in C, as an object in grC, compares as follows to the usual one, i i defined as an object X 2 C plus a decomposition X = ⊕i2ZX . To (X )i2Z 2 grC, we associate its underlying C-object together with its tautological decomposition. Proposition 2.3 Let (C; ⊗;') be as above. 1. The functor (−)1 : CAlg(grC; ⊗gr; gr') −! C defined as the composition 1 U (−) CAlg(grC; ⊗gr; gr') / grC / C has a left adjoint that will be denoted by Symgr;C. gr 2. The coproduct functor ⊕i2Z :(grC; ⊗ ; gr') −! (C; ⊗;') is symmetric monoidal. Remark 2.4 Note that Symgr;C is actually N-graded, i.e. it factors as grN gr grmodR −! CAlg(grNC; ⊗ ; grN') −! CAlg(grC; ⊗ ; gr'): gr gr One could have also chosen to work with (grNC; ⊗ N ; grN') rather than with CAlg(grC; ⊗ ; gr') from the beginning. 2 Note that, in particular, Symgr;C preserves coproducts, and the coproduct functor induces a functor (−)int : CAlg(grC; ⊗gr; gr') −! CAlg(C; ⊗;'): Here the super-script \int" stands for internal, in the sense that we are tracing out (i.e. summing up on) all the \external weights": e.g. C might be itself \graded", by some \internal weights", as it will be the case in our main C of interest (C = grmodR, see below). int Corollary 2.5 The functor (Symgr;C) : C −! CAlg(C; ⊗;') is left adjoint to the forgetful functor. Remark 2.6 Both Proposition 2.3 and Corollary 2.5 are quite classical when C = modR (endowed with its clas- sical monoidal structure and its classical obviously-no-signs-switch symmetry). In this case, CAlg(grC; ⊗gr; gr') is the category of strictly commutative (not graded-commutative !) graded R-algebras, since gr' is again the • no-signs-switch symmetry, and, for M an R-module, Symgr(M) is the commutative algebra S (M), equipped int • with its canonical N-grading, while (Symgr(M)) is S(M), i.e. S (M) without its grading. One can check that M 7! S•(M) is left adjoint to the functor sending a strictly commutative graded R-algebra to its weight 1 part, while M 7! S(M) is left adjoint to the forgetful functor sending a(n ungraded) commutative R-algebra A to its underlying R-module. We will be mainly interested in the special case where (C; ⊗;') = (grmodR; ⊗gr;' = Koszul symmetry), i.e. • i j p q (E )i2Z ⊗gr (F )j2Z := (⊕p+q=n(E ⊗R F ))n2Z • '(xi ⊗ yj) := (−1)ijyj ⊗ xi, for wt(xi) = i, wt(yj) = j. In this situation, Proposition 2.3 gives us the left adjoint functor Sym := Sym : grmod −! CAlg(gr(grmod ); ⊗gr; gr') gr gr; grmodR R R to the A 7! (A)1 functor, and Corollary 2.5 provides us the left adjoint functor int n (Symgr) = ⊕n≥0Symgr : grmodR −! CAlg(grmodR; ⊗gr;' = Koszul symmetry) to the forgetful functor. Note that CAlg(grmodR; ⊗gr;' = Koszul symmetry) is the usual category of Z-graded-commutative R-algebras, • • int hence, if E is a Z-graded R-module, then (Symgr(E )) is a Z-graded-commutative R-algebra. Moreover, Symgr is fully faithful, and, equivalently, the unit map • 1 • • 1 E −! Symgr(E ) := (Symgr(E )) is an isomorphism in grmodR. 3 Bigradings and formulae In our case of interest, i.e. (C; ⊗;') = (grmodR; ⊗gr;' = Koszul symmetry), we have equivalences of categories Y Y Y grC' modR ' modR: Z Z Z×Z 3 Elements in the first Z factor in Z × Z are called external weights, while elements in the second Z are called internal weights: they form bi-weights (i; j) 2 Z × Z. Direct summation over external weights gives the underlying graded module functor. gr i;j In particular, any A 2 CAlg(grC; ⊗ ; gr') has bi-weighted components (A )(i;j)2Z×Z, where i is the external weight, and j the internal weight. The algebra product in A is bi-weighted, i.e. satisfies Ai;j · Ah;k ⊆ Ai+h;j+k and the graded-commutativity property reads as ai;j · a0h;k = (−1)jka0h;k · ai;j for ai;j 2 Ai;j ; a0h;k 2 Ah;k: Note that the external grading is immaterial in the graded commutativity constraint. int i;j Moreover, we have A = (⊕i2ZA )j2Z, and this formula together with the above commutativity constraint give another proof of the fact that Aint is indeed a graded-commutative R-algebra. For the same reason, A0;• is a graded-commutative R-algebra, as well. • If E 2 C = grmodR, we have Sym (E•) = Symn (E•) := (Sym (E•))n;• gr gr gr n2N n • where each Symgr(E ) 2 grmodR (its weight-grading is given by the internal weight) is given by Symn (E•) = Symn;m(E•) := (Sym (E•))n;m gr gr gr m2Z n;m • where Symgr (E ) 2 modR is defined as 0 1 M O p i O q j @ ^ i (E ) S j (E )A n P P P P o i2 odd j2 even (p;q)2 Zodd × Zeven j ipi+ jqj =m ; pi+ qj =n Z Z N N i2Zodd j2Zeven i2Zodd j2Zeven Remark 3.1 According to the convention that a graded object in C is a direct sum of its components (in our language this graded object is the underlying C object), a more common way of writing the previous formula is the following n • M M O pi i O qj j Symgr(E ) = ^ (E ) S (E ) 2 grmodR: P P p+q=n pi=p; qj =q i2 odd j2 even i2Zodd j2Zeven Z Z However, according to our definition of grC as Q C, the formula before this remark is definitely more correct Z (and it gives explicitly the internal grading). • The multiplication in Symgr(E ) is given by a family (µn;m;n0;m0 )(n;m;n0;m0)2N×Z×N×Z of maps of R-modules n;m • n0;m0 • n+n0;m+m0 • µn;m;n0;m0 : Symgr (E ) ⊗R Symgr (E ) −! Symgr (E ) 0 0 while its commutativity constraint is expressed by the following family, indexed by (n; m; n ; m ) 2 N×Z×N×Z, of commutative diagrams of R-modules n;m • n0;m0 • σ n0;m0 • n;m • Symgr (E ) ⊗R Symgr (E ) / Symgr (E ) ⊗R Symgr (E ) µn;m;n0;m0 µn0;m0;n;m n+n0;m+m0 • n+n0;m+m0 • Symgr (E ) 0 / Symgr (E ) (−1)mm where σ is the usual no-sign-switch symmetry in the symmetric monoidal category modR. 4 4 Graded derivations gr Definition 4.1 Let (C; ⊗;') = (grmodR; ⊗gr;' = Koszul symmetry), A 2 CAlg(grC; ⊗ ; gr'), and M 2 gr ModA(grC; ⊗ ; gr').A graded derivation of weight r 2 Z of A into M is a map @ : A ! M of R-modules of bi-weight (0; r), i.e.