Slot sums on tensor products and contractions in characteristic two #

For a finite family of modules M i and endomorphisms d i, the slot sum is the endomorphism of ⨂[R] i, M i applying d i in one tensor factor at a time and adding the results. The naturality, reindexing, square-zero, and total-degree results below apply to dependent families. For a constant family they describe tensor powers. In that case:

D (⨂ₜ i, m i) = ∑ i, m₁ ⊗ ⋯ ⊗ d (m i) ⊗ ⋯ ⊗ mₗ.

Over a general ring the differential of a tensor product of complexes carries Koszul signs, which depend on an ordering of the factors and are permuted by reindexing. In characteristic two the signs disappear, so D does not depend on any ordering of ι: it commutes with reindexing the factors and with every family of factorwise maps commuting with d (TauCeti.PiTensorProduct.slotSum_comp_reindex, TauCeti.PiTensorProduct.slotSum_comp_map), and it squares to zero when d does (TauCeti.PiTensorProduct.slotSum_comp_slotSum). This is what makes the tensor power of a complex a complex on which a symmetric group, and hence a wreath product, acts without signs.

Characteristic two is needed only of the module, not of the scalars: the identities below assume that every element of M is its own additive inverse, which holds over a ring of characteristic two (TauCeti.add_self_eq_zero_of_charTwo) and also, for instance, for a module of 𝔽₂-valued cochains regarded over ℤ. A tensor power with at least one factor then inherits the property (PiTensorProduct.add_self_eq_zero), so terms appearing twice cancel.

Contractions also pass to a tensor product of a dependent family. If each h i is a homotopy with d i * h i + h i * d i = 1 - e i, where e i commutes with d i (typically the composite of an augmentation and a retraction), then for a linear order on ι the slot homotopy

H = ∑ i, e ⊗ ⋯ ⊗ e ⊗ h ⊗ 1 ⊗ ⋯ ⊗ 1        (`e` before the slot `i`, `h` at `i`, `1` after it)

satisfies D H + H D = 1 - e ⊗ ⋯ ⊗ e (TauCeti.PiTensorProduct.slotSum_comp_slotHomotopy_add): cross terms appear twice and cancel, and the diagonal terms telescope. This is the sign-free, characteristic-two form of the tensor trick, whose Koszul-signed version on words of every length is formalized in TauCeti/Algebra/Homology/Contraction/TensorTrick.lean. The homotopy depends on the ordering, but it need not be equivariant for anything. Like the slot sum, it is natural in factorwise maps intertwining the data e and h (TauCeti.PiTensorProduct.slotHomotopy_comp_map). Dependent factors arise when tensor induction is regrouped by double cosets: each orbit contributes its own tensor-induced resolution.

Finally, gradings ℳ i of the factors M i induce the total-degree grading of ⨂[R] i, M i (TauCeti.PiTensorProduct.totalDegreePiece), spanned by pure tensors of homogeneous factors whose degrees add up to the given degree. Factorwise maps of degrees a i shift total degree by ∑ i, a i, while the slot sum of a family f i of common degree a shifts total degree by a. When all the h i have the same degree, the slot homotopy shifts total degree by that degree. For a grading without negative pieces, the total-degree grading has no negative pieces either, and the tensor product of retractions vanishing in positive degrees vanishes in positive total degree. The slot homotopy of homotopies vanishing in degree zero vanishes in total degree zero.

These are the data with which a tensor power of the total coinduced resolution, with its contraction by evaluation at 1, is again a contracted resolution: the input of Evens' multiplicative transfer, where the norm of a class is the class of the tensor power of a representing cocycle in the tensor-induced resolution.

Main definitions #

TauCeti.PiTensorProduct.slotSum: the sign-free extension of a family of endomorphisms f i to ⨂[R] i, M i.
TauCeti.PiTensorProduct.slotHomotopy: the ordered homotopy built from e and h.
TauCeti.PiTensorProduct.totalDegreePiece: the total-degree grading of ⨂[R] i, M i induced by gradings ℳ i.

Main results #

TauCeti.PiTensorProduct.slotSum_comp_slotSum: if every element of each M i is its own additive inverse, the slot sum of a family of square-zero endomorphisms f i squares to zero.
TauCeti.PiTensorProduct.slotSum_comp_slotHomotopy_add: if every element of each factor is its own additive inverse, the slot homotopy contracts the tensor product, D H + H D = 1 - ⨂ i, e i.
TauCeti.PiTensorProduct.slotSum_comp_map and TauCeti.PiTensorProduct.slotSum_comp_reindex: the slot sum commutes with factorwise maps and with reindexing.
TauCeti.PiTensorProduct.subsingletonEquiv_slotSum and TauCeti.PiTensorProduct.isHomogeneous_subsingletonEquiv: on a tensor product with a single factor, the slot sum is the endomorphism of that factor, and identifying the tensor product with that factor preserves the degree.
TauCeti.PiTensorProduct.isHomogeneous_slotSum, TauCeti.PiTensorProduct.isHomogeneous_map, TauCeti.PiTensorProduct.isHomogeneous_map_update, TauCeti.PiTensorProduct.isHomogeneous_reindex and TauCeti.PiTensorProduct.isHomogeneous_slotHomotopy: the degrees of these maps for the total-degree grading.
TauCeti.PiTensorProduct.totalDegreePiece_eq_bot_of_neg: gradings without negative pieces induce a total-degree grading without negative pieces.
TauCeti.PiTensorProduct.map_mem_totalDegreePiece_zero, TauCeti.PiTensorProduct.map_eq_zero_of_mem_totalDegreePiece_of_pos and TauCeti.PiTensorProduct.slotHomotopy_eq_zero_of_mem_totalDegreePiece_zero: the tensor powers of an augmentation and a retraction, and the slot homotopy, respect the extreme degrees.

References #

L. Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), §§2–5, for tensor powers of resolutions.
V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Perturbation theory in differential homological algebra II, Illinois Journal of Mathematics 35 (1991), 357--373, for the tensor trick.
J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Mathematische Zeitschrift 207 (1991), 245--280, for the tensor trick.